Theoretical Calculations for Predicted States of Heavy Quarkonium Via Non-Relativistic Frame Work

Volume 12 Number 32

EJTP Electronic Journal of Theoretical Physics

ISSN 1729-5254

Editors

Ignazio Licata Ammar Sakaji http://www.ejtp.com January, 2015 E-mail:[email protected]

Volume 12 Number 32

EJTP Electronic Journal of Theoretical Physics

ISSN 1729-5254

Editors

Ignazio Licata Ammar Sakaji http://www.ejtp.com January, 2015 E-mail:[email protected]

Editor in Chief

Ammar Sakaji

Theoretical Condensed Matter, Mathematical Physics International Institute for Theoretical Physics and Mathematics (IITPM), Prato, Italy Naval College, UAE And Tel:+971557967946 P. O. Box 48210 Abu Dhabi, UAE Email: info[AT]ejtp.com info[AT]ejtp.info

Co-Editor

Ignazio Licata

Foundations of Quantum Mechanics, Complex System & Computation in Physics and Biology, IxtuCyber for Complex Systems , and ISEM, Institute for Scientific Methodology, Palermo, Sicily – Italy

editor[AT]ejtp.info Email: ignazio.licata[AT]ejtp.info

ignazio.licata[AT]ixtucyber.org

Editorial Board

Gerardo F. Torres del Castillo

Mathematical Physics, Classical Mechanics, General Relativity, Universidad Autónoma de Puebla, México, Email:gtorres[AT]fcfm.buap.mx Torresdelcastillo[AT]gmail.com

Leonardo Chiatti

Medical Physics Laboratory AUSL VT Via Enrico Fermi 15, 01100 Viterbo (Italy) Tel : (0039) 0761 1711055 Fax (0039) 0761 1711055 Email: fisica1.san[AT]asl.vt.it chiatti[AT]ejtp.info

Francisco Javier Chinea

Differential Geometry & General Relativity, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Spain, E-mail: chinea[AT]fis.ucm.es

Maurizio Consoli

Non Perturbative Description of Spontaneous Symmetry Breaking as a Condensation Phenomenon, Emerging Gravity and Higgs Mechanism, Dip. Phys., Univ. CT, INFN,Italy

Email: Maurizio.Consoli[AT]ct.infn.it

Avshalom Elitzur

Foundations of Quantum Physics ISEM, Institute for Scientific Methodology, Palermo, Italy Email: Avshalom.Elitzur[AT]ejtp.info

Elvira Fortunato

Quantum Devices and Nanotechnology:

Departamento de Ciência dos Materiais CENIMAT, Centro de Investigação de Materiais I3N, Instituto de Nanoestruturas, Nanomodelação e Nanofabricação FCT-UNL Campus de Caparica 2829-516 Caparica Portugal

Tel: +351 212948562; Directo:+351 212949630 Fax: +351 212948558 Email:emf[AT]fct.unl.pt elvira.fortunato[AT]fct.unl.pt

Tepper L. Gill

Mathematical Physics, Quantum Field Theory Department of Electrical and Computer Engineering Howard University, Washington, DC, USA

Email: tgill[AT]Howard.edu tgill[AT]ejtp.info

Alessandro Giuliani

Mathematical Models for Molecular Biology Senior Scientist at Istituto Superiore di Sanità Roma-Italy

Email: alessandro.giuliani[AT]iss.it

Vitiello Giuseppe

Quantum Field Theories, Neutrino Oscillations, Biological Systems Dipartimento di Fisica Università di Salerno Baronissi (SA) - 84081 Italy Phone: +39 (0)89 965311 Fax : +39 (0)89 953804 Email: [email protected]

Richard Hammond

General Relativity High energy laser interactions with charged particles Classical equation of motion with radiation reaction Electromagnetic radiation reaction forces Department of Physics University of North Carolina at Chapel Hill, USA Email: rhammond[AT]email.unc.edu

Arbab Ibrahim

Theoretical Astrophysics and Cosmology Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321, Khartoum 11115, Sudan

Email: aiarbab[AT]uofk.edu arbab_ibrahim[AT]ejtp.info

Kirsty Kitto

Quantum Theory and Complexity Information Systems | Faculty of Science and Technology Queensland University of Technology Brisbane 4001 Australia

Email: kirsty.kitto[AT]qut.edu.au

Hagen Kleinert

Quantum Field Theory Institut für Theoretische Physik, Freie Universit¨at Berlin, 14195 Berlin, Germany

Email: h.k[AT]fu-berlin.de

Wai-ning Mei

Condensed matter Theory Physics Department University of Nebraska at Omaha,

Omaha, Nebraska, USA Email: wmei[AT]mail.unomaha.edu physmei[AT]unomaha.edu

Beny Neta

Applied Mathematics Department of Mathematics Naval Postgraduate School 1141 Cunningham Road Monterey, CA 93943, USA Email: byneta[AT]gmail.com

Peter O'Donnell

General Relativity & Mathematical Physics, Homerton College, University of Cambridge, Hills Road, Cambridge CB2 8PH, UK E-mail: po242[AT]cam.ac.uk

Rajeev Kumar Puri

Theoretical Nuclear Physics, Physics Department, Panjab University Chandigarh -160014, India Email: drrkpuri[AT]gmail.com rkpuri[AT]pu.ac.in

Haret C. Rosu

Advanced Materials Division Institute for Scientific and Technological Research (IPICyT) Camino a la Presa San José 2055 Col. Lomas 4a. sección, C.P. 78216 San Luis Potosí, San Luis Potosí, México Email: hcr[AT]titan.ipicyt.edu.mx

Zdenek Stuchlik

Relativistic Astrophysics Department of Physics, Faculty of Philosophy and Science, Silesian University, Bezru covo n´am. 13, 746 01 Opava, Czech Republic Email: Zdenek.Stuchlik[AT]fpf.slu.cz

S.I. Themelis

Atomic, Molecular & Optical Physics Foundation for Research and Technology - Hellas P.O. Box 1527, GR-711 10 Heraklion, Greece Email: stheme[AT]iesl.forth.gr

Yurij Yaremko

Special and General Relativity, Electrodynamics of classical charged particles, Mathematical Physics, Institute for Condensed Matter Physics of Ukrainian National Academy of Sciences 79011 Lviv, Svientsytskii Str. 1 Ukraine Email: yu.yaremko[AT]gmail.com yar[AT]icmp.lviv.ua

yar[AT]ph.icmp.lviv.ua

Nicola Yordanov

Physical Chemistry Bulgarian Academy of Sciences, BG-1113 Sofia, Bulgaria Telephone: (+359 2) 724917 , (+359 2) 9792546

Email: ndyepr[AT]ic.bas.bg ndyepr[AT]bas.bg

Former Editors:

Ammar Sakaji, Founder and Editor in Chief (2003- October 2009)

Ignazio Licata, Editor in Chief (October 2009- August 2012)

Losé Luis López-Bonilla, Co-Editor (2008-2012)

Ammar Sakaji, Editor in Chief (August 2012- )

Table of Contents

No Articles Page 1 Preface

i

2 Notes on High-Frequency Quantum Effect Andrea Claudio Levi CNISM and Dipartimento di Fisica dell'Università, Genova, Italy

1

3 Generalized Uncertainty Principle and Recent Cosmic Inflation Observations Abdel Nasser Tawfik and Abdel Magied Diab Egyptian Center for Theoretical Physics (ECTP), Modern University for Technology and Information (MTI), 11571 Cairo, Egypt. World Laboratory for Cosmology And Particle Physics (WLCAPP), Cairo, Egypt

9

4 The Shadow of a Black Hole from Heterotic String Theory at the Center of the Milky Way Alexis Larranaga National Astronomical Observatory. National University of Colombia, Colombia

31

5 Scattering of Waves in the Phase Space, Quantum Mechanics, and Irreversibility E. M. Beniaminov Russian State University for the Humanities, Miusskaya sq. 6, Moscow, GSP-3, 125993, Russia

43

6 Frequency Distribution of Spontaneous Emission Saul. M. Bergmann Potomac, Maryland, USA

61

7 Spatially Homogeneous Cosmological Models in $f(R,T)$ Theory of Gravity S. Chandel and Shri Ram Department of Applied Mathematics, Indian Institute of Technology, (Banaras Hindu University), Varanasi 221 005, India

69

8 Relativistic Pentametric Coordinates from Relativistic Localizing Systems and the Projective Geometry of the Spacetime Manifold Jacques L. Rubin Université de Nice--Sophia Antipolis, UFR Sciences Institut du Non-Linéaire de Nice, UMR7335 1361 route des Lucioles, F-06560 Valbonne, France

83

9 Theoretical Calculations for Predicted States of Heavy Quarkonium Via Non-Relativistic Frame Work T. A. Nahool, A. M. Yasser and G. S. Hassan Physics Department, Faculty of Science at Qena, South Valley University, Egypt Physics Department, Faculty of Science, Assuit University, Egypt

113

10 The One-dimensional Thermal Properties for the Relativistic Harmonic Oscillators Abdelmalek Boumali Laboratoire de Physique Appliquée et Théorique, Université de Tébessa, 12000, W. Tébessa, Algeria

121

11 The Effectiveness of Hénon Map for Chaotic Optimization Algorithms Using a Global Locally Averaged Strategy Tayeb Hamaizia Department of Mathematics, Faculty of Sciences, University Constantine -1-, Algeria

131

12 Weakton Model of Elementary Particles and Decay Mechanisms Tian Ma and Shouhong Wang Department of Mathematics, Sichuan University, Chengdu, P. R. China Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

139

13 Physics of Currents and Potentials II. Classical Singlet-Triplet Electroweak Theory with Non-point Particles Valerii Temnenko Tavrian National University, 95004, Simferopol, Crimea, Ukraine

179

14 From Quantum Mechanics to Intelligence Michail Zak Jet Propulsion Laboratory California Institute of Technology, Pasadena, CA 91109, USA

295

Electronic Journal of Theoretical Physics 12, No. 32 (2015) i

Welcome to the Year of Light 2015!

Dear EJTP Friends,

We start the new year with very rich issue with high quality papers, as a New Year

Greeting and celebration, we are working with our special project on the international

year of light 2015 entitled ” Bohr-Einstein Debates in the Year of Light 2015 ”. In fact,

the UNESCO General assembly has proclaimed 2015 as the International Year of Light

and Light-based Technology (IYL 2015). The opening ceremony will be held in Paris on

19-20 January 2015.

EJTP will celebrate the event with a Special Issue (http://www.ejtp.com/iyl2015.html),

in parallel with other contingent matters. The year of the photon is also the year of the

electrodynamics of the moving bodies, the magic triad of 1905 Einstein’s papers, respec-

tively: Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen

Gesichtspunk, Annalen der Physik 17, 132 (1905) [17 pp.] & Zur Elektrodynamik be-

wegter Korper, Annalen der Physik 17, 891 [31 pp]. From the contemporary viewpoint,

it is not just a coincidence at all.

The modern view of light nowadays is to look at highly complex matters regarding

the Gauge Theories’ underlying symmetries. R. Laughlin has effectively summed up: “If

Einstein were alive today, he would be horrified at this state of affairs (...) It would be

perfectly a challenge for him to re-examine the facts, toss them over in his mind, and

conclude that his beloved principle of relativity was not fundamental at all but emergent-a

collective property of the matter constituting space-time that becomes increasingly exact

at long length scales but fails at short ones”. [I am working at Planck scale just now!

(IL)].

It is an emergence that lead us to another important theme set by the Year of Light:

when Nature finds a good trick, it plays on different scales! The correspondences between

the BCS Theory of superconductivity and the Physics of Higgs boson, the mysterious cor-

nerstone supporting the gauge theories, are things too beautiful to be left unexplained.

At present, the physics of the analogy of GR & QM systems in Condensed Matter has

undergone a great development phase and the future of the field will give unexpected

theoretical and experimental possibilities to test theories born in microscopic and cos-

mological context in the mesoscopic one, promising technological scenarios that were

unthinkable few years ago.

Without forgetting the modern core of physics, last but not least, the problems of QM

foundation. A generalized crisis of Copenhagen (in a recent mail to a colleague I called

it poetically “the burning walls of Copenhagen”), should remind us that the so-called

QM interpretation is not a quirk for out of service physicists or recycling philosophers, a

game of intellectual exercises is to try imposing a meta-physics guarding the formalism.

It should be just a formalism to suggest the beables and the possibilities of our language,

as N. Bohr – who got a not operatively shortsighted vision of QM – warned. Actually,

the interpretation work of a physical theory cannot be isolated from the theoretical and

ii Electronic Journal of Theoretical Physics 12, No. 32 (2015)

experimental problems and its developments.

Naturally, taking in mind that the Optics by Newton (1704) is a masterpiece where

theory and experiment are intimately fused, we wish the pact between theoreticians and

experimentalists, it is an important to minimize the barriers and deviations, because

Physics is One science.

Now, I “pass the ball” to Ammar Sakaji who will present you the issue.

I would like to thank all editors and reviewers for helping me to prepare this issue

(Vo.12, No.32, 2015) of the Electronic Journal of Theoretical Physics EJTP, especially

my brother and friend Ignazio Licata, I greatly appreciate his notes and comments. I’ve

spent a lot of time sifting and editing the manuscripts of this issue, which covers impor-

tant topics: from Chaos Theory, Quantum Mechanics (relativistic and non-relativistic),

Elementary Particles Physics, Quantum Gravity at Planck’s Scale, String Theory, Gravi-

tational Cosmological Models, Black Holes, Weak Interaction up to Quantum Intelligence.

Most of the editorial work done in Abu Dhabi and the other part at Iowa.

This Issue includes 13 reviewed research papers on mathematical and theoretical

physics with the efforts of 18 authors in addition to the reviewers, referees and edi-

tors: Levi in his paper on the high–frequency quantum effect discuses the quantum phase

mechanism, energy exchange, and decoherence. Tawfik and Diab figure the GUP at the

Planck scale with view of latest experimental observations. Larranaga in his paper on

black holes addresses the Shadow of a Black Hole from Heterotic String theory to the

Center of the Milky Way. Beniaminov reviews some useful formalism of quantum me-

chanics, reversibility and Phase Space. Bergmann figures the spontaneous emission via

Fock space techniques. Chandel and Ram discus some gravitational cosmological mod-

els. Rubin describes in very beautiful way the relativistic pentametric coordinates with

generalized Cartan space. Nahool et al. show in their paper the non-relativistic Heavy

Quarkonium computations. Boumali derives some thermal properties of relativistic har-

monic oscillator. Hamaizia shows the map of optimization Algorithms especially Henon

map. Ma and Wang show the decay mechanisms of elementary particles via Weakton

Model. Temnenko addresses a comprehensive study of the singlet-triplet electroweak

theory. Finally Zak with his attractive paper on Quantum Mechanics and Intelligence

demonstration.

Enjoy reading

Ignazio Iicata & Ammar Sakaji.

EJTP 12, No. 32 (2015) 1–8 Electronic Journal of Theoretical Physics

Notes on High–Frequency Quantum Effect

Andrea Claudio Levi∗

CNISM and Dipartimento di Fisica dell’Università, Genova, Italy

Received 07 ○ctober 2014, Accepted 19 December 2014, Published 10 January 2015

Abstract: Although any quantum system subject to disturbances tends to approach aclassical behaviour because of decoherence, such decoherence may be largely quenched whenthe perturbations are characterized by high-frequency oscillations. This is related to a simpleproperty of Fourier transforms: if, as usually happens, the time dependence of the process issmooth, this implies a rapid decrease of the omega-dependent quantities at high frequencies. Asa consequence, in the case of scattering of a particle with a solid (surface or bulk) the contributionof high-frequency vibrations to the Debye-Waller factor is small. But the effect is much moregeneral: it is shown that, whenever a particle interacts with a large system, high-frequencyoscillations of the latter contribute little to decoherence, both in the absence (elastic case) andin the presence of energy exchange (thus, if such high-frequency oscillations are the only relevantperturbation, the process maintains quantum- mechanical features).c○ Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: High Frequency Quantum Effect; Quantum Phase; Energy Exchange; decoherencePACS (2010): 03.65.-w; 03.65.Vf; 03.65.Yz

1. Introduction

Any quantum system subject to disturbances tends to approach a classical behaviourbecause of decoherence, but it is interesting to see that, when the perturbations affectingit are characterized by high–frequency oscillations, decoherence may be largely quenchedand the system may behave almost coherently, i. e. quantum–mechanically. This can beconsidered as an instance of the so–called Zeno effect [1, 2]2.

∗ Email:[email protected] The Zeno effect is often described as the confirmation of quantum behaviour via repeated acts ofmeasurement, but in fact measurement (the intervention of a physicist!) has probably little to do with it.To confirm quantum behaviour it is important rather for disturbances to appear as repeated oscillations,whose effect is ultimately averaged out.

2 Electronic Journal of Theoretical Physics 12, No. 32 (2015) 1–8

In a different context, the fact that decoherence can be inhibited by subjecting thesystem to a sequence of very frequent pulses has been discussed by Viola and Lloyd [3]and by Vitali and Tombesi [4] (similar phenomena and techniques have been well–knownin nuclear magnetic resonance for many years [5, 6]).

Indeed, the quantum phase can be written as an action divided by �, and its randompart (representing the perturbation due to the relevant oscillating modes) tends to becomesmall at high frequencies, at least if the process under consideration can be written as theintegral of a convenient Lagrangian. In [7] some properties of the action for the case of amaterial particle (e. g. an atom) scattered off a large system are discussed (in that casethe main quantity on which the attention is to be focused is the Debye–Waller factor),but in the present study I will keep a more general point of view, trying to discuss theconditions under which the effect considered takes place.

An essential point is that, unless some discontinuous or catastrophic event occurs,the time dependence of the process is smooth, and that, due to well–known mathema-tical properties of Fourier transforms, this implies a rapid decrease of the ω–dependentquantities at high frequencies. Such mathematical properties are the following:

If a real function f(t) can be extended to an analytic function f(z) whose closestsingularity has a distance b from the real axis and which behaves at infinity as O(e−|(β−ε)x|)(for any ε), then its Fourier transform F(ω) behaves at infinity as O(e−|(b−δ)Rεω|) (for anyδ) (and its closest singularity has a distance β from the real axis).

If, on the other hand, f(z) is an entire function, then its Fourier transform tends tozero faster than any exponential for large |ω|.

This corresponds to, and follows from, Theorem 26 of Titchmarsh, page 44 [8]. Now itis clear that a number of physical processes (including most scattering processes) involvephysical quantities which vary with time in a continuous fashion, and take place within awell–defined time interval, so that the interactions are vanishingly small before and afterthat interval. Thus it is reasonable to assume that, for an extensive class of processes,the conditions of the above theorem are fulfilled. As a consequence, relevant physicalquantities describing such processes have Fourier transforms that tend to zero at leastexponentially at high |ω|.

2. The Phase

In the present Section, I will reconsider in a simple fashion the quantum–mechanical phaseη (or equivalently the action S, since there is only a factor � between the two quantities),as well as the physical role of these quantities. It is well known that under extremelygeneral conditions the quantum–mechanical amplitude can be formally written as a path

integral of a phase factor exp(iη) [9], so that the complications lie in the somewhatmysterious procedure of path integration, while the phase factor itself, in many cases, isrelatively simple. I mention this only to indicate that a semiclassical treatment of η = S/�

is often meaningful and that the mechanical properties of S can be considered and shownoften to possess the properties of continuity in time and limitation to a well–defined time

Electronic Journal of Theoretical Physics 12, No. 32 (2015) 1–8 3

interval mentioned in the Introduction.Two obvious complications, however, are to be discussed. First of all, to obtain the

amplitude an integral (in principle, a path integral, but in practical approximations anintegral over some set of relevant variables) is to be performed. Secondly, probabilitiesare of course not amplitudes, but are obtained from amplitudes taking the absolute valuessquared. These questions will be dealt with, in simple ways, in the following.

If the perturbations are sufficiently weak, the phase η may be written as a sum

η = η0 + δη (1)

where η0 is the unperturbed phase and δη is in some sense linear in the perturbations.The effect of η0 may be assumed to be coherent, while decoherence will arise from thestatistical properties of δη, i. e. of the perturbations.

The amplitude of a physical process will be the integral of a phase factor exp(iη)

along a path, integrated subsequently over paths; and the corresponding probability willbe the statistical average

P =< |∫eiηdV |2 >, (2)

where∫

dV indicates a (complex, but here unspecified) integration procedure.Splitting η as in (1), (2) may be rewritten as

P =∫ ∫

ei[η0(x)−η0(x′)] < ei[δη(x)−δη(x

′)] > dV dV ′ (3)

where all decoherence properties arise from the statistical average involving the δη’s.Let me focus on such average.

In many interesting cases, δη may be assumed to be linear in the disturbances (e.g.phonons or other relevant quanta), and the latter to have a harmonic nature and thereforea Gaussian distribution of amplitudes. In this case the well–known theorem for theaverages of Gaussian variables of zero mean

< eia >= e−12<a2> = e−

12<a∗a>, (4)

(valid for both real c-numbers and self–adjoint quantum variables) may be applied,with the result that (3) takes the simpler form

P =∫ ∫

ei[η0(x)−η0(x′)]e−

12<|δη(x)−(δη(x′)|2>dV dV ′, (5)

where all statistical properties appear only in the exponent.


3. Time Dependence

In (5) x, x′, dV, dV ′ are unspecified and may imply space and time variables or moregenerally path variables. A very simple, but important case is that where x′ differs fromx in time only along a path, so that the average occurring in (5) may be written

2W (t, t′) =< |δη(t)− δη(t′)|2 > (6)

(the factor of 2 is written here to respect the Debye–Waller tradition. . . ). Theproperties of W (t, t′) are considered below.

It is possible to write δη as an integral over frequencies

δη(t) =1

2πQ(t) =

1

2π

∫ ∞

−∞eiωta(ω)F (ω)dω (7)

where F (ω) is the Fourier transform of a function f(t) of time describing, say, ascattering event, while a(ω) is the amplitude of the (harmonic) disturbances.

The latter is a random quantity. If α∗ and α are creation and annihilation operatorsfor disturbances (e. g. phonons) of frequency ω, then |a(ω)|2 may typically be assumedto be proportional to �|ω|α∗α = �|ω|n(|ω|)g(|ω|), and

< a∗(ω)a(ω′) >= �ω < n(|ω)| > g(|ω|)δ(ω′ − ω), (8)

where g(ω) is the spectrum of disturbances. It is reasonable to assume that onlya(ω) has nontrivial statistical properties, related e. g. to the expectation values of thephonon variables at the temperature of interest, while (for a small amplitude of distur-bances) f(t) and F (ω) are mechanical quantities of a deterministic nature, f(t) beingthe time dependence of the mechanical process taking place and F (ω) being its Fouriertransform. The mechanical process considered (scattering or otherwise) is expected to behighly regular and strictly limited in time, so that the conditions for the validity of themathematical properties considered in the Introduction may be assumed to be fulfilled.As a consequence, F (ω) tends to zero at least exponentially at high frequencies.

Due to the form of (6) and (7), it is clear that W (t, t′) actually depends only on thetime difference t′ − t = τ , and will henceforth be indicated as such. Indeed

W (t, t′) =1

8π2

∫ ∫ ∞

−∞(e−iωt − e−iωt

′)(eiω

′t − eiω′t′) < a∗(ω)a(ω′) > F ∗(ω)F (ω′)dωdω′ (9)

which, because of (8), gives

W (t, t′) = W (τ) =�

2π2

∫ ∞

0(1− cosωτ) < n(ω) > g(ω)|F (ω)|2ωdω. (10)

(10) is the main result of the present Section.


4. Single Frequency

It is instructive (although obviously too simple) to consider the elementary case wherethe disturbance occurs at a single frequency ω0, g(ω) = δ(ω − ω0): a physical exampleof this might be the “Einstein model” of crystal vibrations. In this single–frequency caseW (τ) reduces to

W (τ) =�ω0

2π2(1− cosω0τ) < n(ω0) > |F (ω0)|2, (11)

exceedingly small if the frequency ω0 is high. At high frequencies, indeed, W becomessmall because F decreases exponentially. Notice that �ω0 < n(ω0) > is the mean energyof the disturbance (with the zero–point energy subtracted).

For example, if

f(t) =B

cosh(2t/τc)(12)

(a reasonable behaviour for a process having a duration of the order of τc), then

W (τ) =1

32τ 2cB

2 �ω0 < n(ω0) >

cosh2(πω0τc/4)(1− cosω0τ). (13)

5. Asymptotic Behaviour

Let us come back to the general formula (10) and consider the asymptotic behaviourat long τ ’s. This corresponds to vanishing energy exchange: for example, in scatteringphenomena, to elastic scattering.

Indeed, as I shall show in the next Section, the probability of a process with energyexchange Δ is proportional to the Fourier transform, with the exp(−iτΔ/�) factor, ofexp[−2W (τ)], with W (τ) given by (10). If τ is very long, only the Δ = 0 contributionsurvives.

But if τ is very long, in the integrals Q and Q∗ vastly different frequencies are relevant(except in the case, already considered, where a single frequency occurs), so that Q andQ∗ become uncorrelated and

< |Q|2 >=< Q∗Q >→ | < Q > |2. (14)

The elastic probability depends thus only on

< Q(τ) >=<∫

eiωτa(ω)F (ω)dω >=∫eiωτ < a(ω) > F (ω)dω, (15)

where, as above, I have supposed (as is reasonable) that only a(ω) has nontrivialstatistical properties.

In the case of elastic scattering, exp(−2| < Q > |2) has the meaning of a Debye–Wallerfactor, but the present formulation is in principle much more general.


6. General Case and Energy Exchange

In the previous section I have considered the case where the energy exchange vanishes;in general, however, there is of course a nonvanishing energy exchange Δ. I have givensuch general theory in several articles, through the years [10, 11, 12]. To fix ideas, letme assume that a particle (elementary or otherwise) interacts with a large system: if theparticle loses the energy Δ, then the latter goes to excite the large system from an initialstate α to a final state β such that Eβ = Eα +Δ (the large system will be assumed to belarge enough, and its levels dense enough, that this is possible for any Δ).

The probability for this to occur will be proportional to

∑α

Pα

∑β

|Tβ←α|2δ(Eβ − Eα −Δ), (16)

where T is the quantum–mechanical operator for the transition of the large systemfrom α to β, (while at the same time the particle loses the energy Δ), and Pα is thestatistical (e. g. canonical, Gibbs–Boltzmann) probability for the large system to be instate α.

A van Hove transformation [13] may now be applied3.Expanding the absolute value squared, and expressing the δ–function as an integral,

(16) becomes

1

2π�

∫ ∑a

Pα

∑β

(α|T ∗|β)eiEβt/�(β|T |α)e−iEαt/�e−itΔ/�dt. (17)

But eiEαt/� multiplying state |α) and eiEβt/� multiplying state |β) are equivalent tothe operator eiHt/� operating respectively on such states (where H is the Hamiltonian ofthe large system), so that the quantity considered amounts to

1

2π�

∫ ∑α

Pα

∑β

(α|T ∗eiHt/�|β)(β|Te−iHt/�|α)e−itΔ/�dt. (18)

The sum of |β) over all states is 1 and the sum of Pα|α) over all states correspondsto a statistical average, so that the result

1

2π�

∫< T ∗eiHt/�Te−iHt/� > e−itΔ/�dt (19)

is obtained; this may best be written in terms of time correlations:

1

2π�

∫e−itΔ/� < T ∗(0)T (τ) > dτ. (20)

3 In his original article [13], van Hove dealt with a perturbation approximation (valid for neutron singlescattering from condensed matter), but in fact his method is exact, provided the exact T operator isconsidered . Notice that the present discussion is nonperturbative in the variables describing the process(e.g. it may be applied to multiple scattering), although it is perturbative (linear) in the disturbances.


Time correlations depend on the time difference (which, in agreement with eq. (10),may be called τ). If the essential part of T (t) is a time–dependent4 phase factor exp[iη(t)],then the time correlation occurring in (20) appears to be simply proportional to< exp[i(η(0)− η(τ))] >. Using then eq. (1) and the methods leading to eq.’s (6) and (7),this may be written as

ei[η0(0)−η0(τ)]e−1/2<[δη(0)−δη(τ)]2>, (21)

in agreement with Section 1.Asymptotically, for long τ ’s, < δη(0)δη(τ) > vanishes and (21) becomes

ei[η0(0)−η0(τ)]e−<δη2>. (22)

δη may be replaced by an integral of the form 12πQ(τ) (see (7)) (strictly the latter is

complex, however) and the expression (10) for W (τ) = 12< δη2 > holds; from this the

discussion given in Section 4 follows. Alternatively, one may say that, in the elastic limitwhere Δ vanishes, the correlation between T–matrices effectively disappears:

< T ∗(0)T (τ) >→ | < T > |2, (23)

in agreement with (14), but this time without any restriction on frequencies.On the other hand, for finite τ , the average < [δη(0)− δη(τ)]2 > is typically less than

the asymptotic value 2 < δη2 >, so that the conclusion, according to which the exponentappearing as an average in (21) rapidly decreases for high frequencies ω, is a fortiori valid.

7. Conclusions

I have shown that, under quite general circumstances, perturbations corresponding tohigh–frequency oscillations scarcely cause decoherence. To be more precise, processeslasting for a long time τc, such that many oscillations take place during the process andthat ωτc is a large number, suffer little decoherence and behave quantum–mechanically.The oscillations are effectively averaged out. This has been used by me in [14, 15] toexplain the quantum behaviour of neon scattering from surfaces (due to the relativelyslow motion of neon atoms), and in particular the visibility of diffraction peaks.

References

[1] P. Facchi and S. Pascazio, J. Phys. A: Math. Gen. 41 (2008) 493001

[2] A. C. Levi, J. Phys.: Condens. Matter 22 (2010) 304003

[3] L. Viola and S. Lloyd, Phys. Rev. A58 (1998) 2733

[4] D. Vitali and P. Tombesi, Phys. Rev. A59 (1999) 4178

4 For each path time integration is to be performed, while further properties of T (t) correspond tointegration over a set of different paths.


[5] S. Pascazio, private communication

[6] P. Slichter, Principles of magnetic resonance, 3rd ed., Springer–Verlag, Berlin 1990

[7] A. C. Levi, J. Phys.: Condens. Matter 21 (2009) 405004

[8] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press,Oxford 1948

[9] R. P. Feynman and A: R: Hibbs, Quantum mechanics and path integrals, McGrawHill, New York 1965

[10] A. C. Levi, C. R. Acad. Sc. Paris 259 (1964) 3975

[11] A. C. Levi, Nuovo Cim. B54 (1979) 357

[12] V. Bortolani and A. C. Levi, Riv. Nuovo Cim. 9, n. 11(1986)

[13] L. van Hove, Phys. Rev. 95 (1954) 249

[14] A. C. Levi and H. Suhl, Surf. Sci. 88 (1979) 221

[15] A. C. Levi, Huang Congcong, W. Allison and D. A. MacLaren, J. Phys.:Condens. Matter 21 (2009) 225009


Generalized Uncertainty Principle and RecentCosmic Inflation Observations

Abdel Nasser Tawfik∗† and Abdel Magied Diab

Egyptian Center for Theoretical Physics (ECTP), Modern University for Technology andInformation (MTI), 11571 Cairo, Egypt

World Laboratory for Cosmology And Particle Physics (WLCAPP), Cairo, Egypt

Received 2 January 2015, Accepted 9 January 2015, Published 10 January 2015

Abstract: The recent background imaging of cosmic extragalactic polarization (BICEP2) observations

are believed as an evidence for the cosmic inflation. BICEP2 provided a first direct evidence for the

inflation, determined its energy scale and debriefed witnesses for the quantum gravitational processes.

The ratio of scalar-to-tensor fluctuations r which is the canonical measurement of the gravitational

waves, was estimated as r = 0.2+0.07−0.05. Apparently, this value agrees well with the upper bound value

corresponding to PLANCK r ≤ 0.012 and to WMAP9 experiment r = 0.2. It is believed that the

existence of a minimal length is one of the greatest predictions leading to modifications in the Heisenberg

uncertainty principle or a generalization of the uncertainty principle (GUP) at the Planck scale. In

the present work, we investigate the possibility of interpreting recent BICEP2 observations through

quantum gravity or GUP. We estimate the slow-roll parameters, the tensorial and the scalar density

fluctuations which are characterized by the scalar field φ. Taking into account the background (matter

and radiation) energy density, φ is assumed to interact with the gravity and with itself. We first

review the Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe and then suggest modification in

the Friedmann equation due to GUP. By using a single potential for a chaotic inflation model, various

inflationary parameters are estimated and compared with the PLANCK and BICEP2 observations. While

GUP is conjectured to break down the expansion of the early Universe (Hubble parameter and scale

factor), two inflation potentials based on certain minimal supersymmetric extension of the standard

model result in r and spectral index matching well with the observations. Corresponding to BICEP2

observations, our estimation for r depends on the inflation potential and the scalar field. A power-law

inflation potential does not.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Inflationary Universe; Quantum Gravity; Early Universe

PACS (2010): 98.80.Cq; 04.60.-m; 98.80.Cq

∗ Email:[email protected], [email protected], [email protected]† http://atawfik.net/


1. Introduction

Constrains to the inflationary cosmological models can be set by cosmological observations [1, 2].

The inflationary expansion not only solves various problems, especially in the early Universe such

as the Big Bang cosmology [3, 4, 5, 6, 7], but also provides an explanation for the large-scale

structure from the quantum fluctuation of an inflationary field, φ [8, 9, 10]. Furthermore, the

gravitational waves and the polarization due to the existence of the inflation was discovered in the

cosmic microwave background (CMB) [11].

Not only the physicists around the world are very aware of the existence of the background

imaging of cosmic extragalactic polarization (BICEP2) telescope at the south pole, but the world

public as well. It is believed that the BICEP2 observations offer an evidence for the cosmic inflation

[11]. Other confirmations from Planck [12, 13] and WMAP9 [14] measurements, for instance, are

likely in near future. BICEP2 did not only provide the first direct evidence for the inflation, but

also determined its energy scale and furthermore debriefed witnesses for the quantum gravitational

processes in the inflationary era, in which a primordial density and gravitational wave fluctuations

are created from the quantum fluctuations [15, 16]. The ratio of scalar-to-tensor fluctuation, r,

which is a canonical measurement of the gravitational waves [1, 2], was estimated by BICEP2,

r = 0.2+0.07−0.05 [11]. This value is apparently comparable with the upper bound value corresponding

to PLANCK r ≤ 0.012 and to WMAP9 experiment r = 0.2. On the other hand, the PLANCK

satellite [12, 13] has reported the scalar spectral index ns ≈ 0.96.

If these observations are true, then the hypothesis that our Universe should go through a

period of cosmic inflation will be confirmed and the energy scale of inflation should be very near

to the Planck scale [17]. The large value of tensor-to-scalar ratio, r, requires inflation fields as

large as the Planck scale. This idea is known as the Lyth bound [18, 19, 20], which estimates the

change of the inflationary field Δφ,

Δφ

Mp

=

√r

8ΔN, (1)

where Mp is the Planck mass and ΔN denotes the number of e-folds corresponding to the observed

scales in the CMB left the inflationary horizon. Since the Planckian effects become important and

need to be taken into account during the inflation era, as indicated by the Lyth bound, then Δφ

should be smaller than or comparable with the Planck scale | Δφ |�Mp. This constrain suggests

focusing on concrete inflation field models. In this case, the many corrections suppressed by the

Planck scale appear less problematic but come in tension with BICEP2 discovery. Thus, more

observations are required to confirm this conclusion.

Various approaches to the quantum gravity (QG) offer quantized description for some prob-

lems of gravity, for details readers can consult Ref. [21]. The effects of minimal length and

maximal momentum which are likely applicable at the Planck scale (inflation era) which lead to

modifications in the Heisenberg uncertainty principle appear in quadratic and/or linear terms of

momentum. These can be implemented at this energy scale. The quadratic GUP was predicted

in different theories such as string theory, black hole physics and loop QG [21, 22, 23, 24, 25, 26,

27, 28, 29, 30, 31, 32, 33, 34, 35]. The latter, the linear GUP, was introduced by doubly Special

Relativity (DSR), which suggests a minimal uncertainty in position and a maximum measurable

momentum [36, 37, 38, 21]. Accordingly, a minimum measurable length and a maximum measur-

able momentum [39, 40, 41] are simultaneously likely. This offers a major revision of the quantum


phenomena [42, 21, 43]. This approach has the genetic name, Generalized (gravitational) Uncer-

tainty Principle (GUP). Recently, various implications of GUP approaches on different physical

systems have been carried out [44, 45, 46, 47, 48, 49, 50], App. A.

In the present work, we estimate various inflationary parameters which are characterized by

the scalar field φ and apparently contribute to the total energy density. Taking into account the

background (matter and radiation) energy density, the scalar field is assumed to interact with

the gravity and with itself. The coupling of φ to gravity is assumed to result in total inflation

energy. We first review the Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe and then

suggest modifications in the Friedmann equation due to GUP. Using modified Friedmann equation

and a single potential for a chaotic inflation model, the inflationary parameters are estimated and

compared with PLANCK and BICEP2 observations.

The applicability of the GUP approaches in estimating inflationary parameters comparable

with the recent BICEP2 observations will be discussed. In section 2., we present Friedmann-

Lemaitre-Robertson-Walker (FLRW) Universe and introduce the modification of Friedmann equa-

tion due to GUP at planckian scale in matter and radiation background. In section 3., the modified

Friedmann equation in cosmic inflation will be introduced. Some inflation potentials for chaotic

inflation models will be surveyed. We suggest to implement the single inflationary field φ. In

the cosmic inflation models and quantum fluctuations, the inflationary parameters are given in

section 4.. The discussion and final conclusions will be outlined in section 5.. Appendix A1 gives

details about the higher order GUP with minimum length uncertainty and maximum measurable

momentum in Hilbert space. The applicability of GUP to the cosmic inflation will be elaborated

in Appendix A2. The modified dispersion relation (MDR) as an alternative to the GUP will be

introduced in App. B.

2. Generalized Uncertainty Principle in FLRW Background

In (n+ 1)-dimensional FLRW Universe, the metric can be described by the line element as [51]

ds2 = c2dt2 + a(t)2(

dr2

1− κ r2+ r2 dθ2 + r2sin2 dφ2

), (2)

where a(t) is the scale factor and κ is the curvature constant that measures the spatial flatness

±1 and 0. In Einstein-Hilbert space, the action reads

S =

∫ (1

8 π GLG + Lφ

)dΩ, (3)

where dΩ = dθ + sinθ dφ , G is the gravitational constant, LG is the geometrical Lagrangian

related to the line element of the FLRW Universe and Lφ [52] is Lagrangian coupled to the scalar

field φ

Lφ = − [gμ ν ∂μφ ∂νφ+ V (φ)] , (4)

where V (φ) is the potential and gμ ν is diagonal matrix diag {1,−1,−1,−1}. Under the assump-

tion of homogeneity and isotropy, a standard simplification of the variables leads to the FLRW

metric, where the gradient of the scalar field vanishes. The integration of the action, Eq. (3),


over a unit volume results in Lφ = −12a3φ2 − a3V (φ). Since the FLRW Lagrangian of scalar field

evaluated at vanishing mass, results in,

L =1

2a3 φ2 − 3

8 π G

(a a2 − a κ

), (5)

and the energy-momentum tensor reads

T μν = ∂ν ∂L

∂(∂μ φ)− gμ νL, (6)

while the four momentum tensor is given by P μ = T μ 0 and the Hamiltonian constraint H = P 0 =

T 0 0

H = π∂L

∂(∂0φ)− L, (7)

where the π = ∂L/∂φ is known as the canonical momentum conjugate for the scalar field φ. Thus,

the total Hamiltonian is given as

h =

∫d3 xH. (8)

The scalar field becomes equivalent to a perfect fluid with respectively energy density and pressure

ρ =φ

2+ V (φ), (9)

p =φ

2− V (φ). (10)

When taking into account the cosmological constant Λ, then the energy density ρ → ρ + ρv,

with ρv = Λ/8 π G. Using Eq. (5) and taking into account Eq. (9), the dynamics of such models

are summarized in the Hamiltonian constraint

H = −2π G

3

p2aa− 3

8 π Gκa + a3ρ ≡ 0. (11)

This equation is equivalent to the estimation for FLRW Universe [53, 54, 55], where the momenta

pa associated with the scalar factor are defined as

pa :=∂L∂a

=−34 π G

a a. (12)

The standard Friedmann equations can be extracted from the equations of motion which can be

derived from the extended Hamiltonian by exchanging the negative sign in Eq. (11) in order to

estimate the exact form of Friedmann equations

HE =2π G

3

p2aa

+3

8 π Gκa− a3ρ. (13)

Based on the relationship between the commutation relation and the Poisson bracket which was

first proposed by Dirac [56], we get for two quantum counterparts A and B and two observables

A and B that

[A, B] = i � {A,B}. (14)


In the standard case, the canonical uncertainty relation for variables of scale factor a and momenta

pa satisfies the Poisson bracket {a, pa } = 1. Then, the equations of motion read

a = {a, HE} = {a, pa}∂HE

∂pa=

(4 π G

3

)paa, (15)

pa = {pa, HE} = −{a, pa}∂HE

∂a=

(2 π G

3

)p2aa2− 3

8 π Gκ + 3 a2ρ + a3

∂ρ

∂a. (16)

From Eqs. (15) and (16) and the Hamiltonian constrain, Eq. (11), then the Friedmann equation

is given as

H2 =

(8 π G

3

)ρ − κ

a2, (17)

where H = a/a is the Hubble parameter. For a cosmic fluid, the energy density is combined from

a contribution due to the inflation ρ(φ), Eq. (9) and another part related to the inclusion of the

cosmological constant, ρv.

Now, we consider the higher-order GUP in deformed Poisson algebra in order to study classical

approaches, such as Friedmann equations, Appendix A. We introduce GUP in terms of first order

α [21]. Accordingly, the Poisson bracket between the scale factor a and momenta pa reads

{a , pa} = 1− 2α pa. (18)

We follow the same procedure as in Eq. (17), but for a modified term of QG, we will use the

extended Hamiltonian with the Poisson brackets to get the modified equations of motion

a = {a, pa}∂HE

∂pa= (1− 2αpa)

4πG

3

paa, (19)

pa = {a, pa}∂HE

∂a= (1− 2αpa)

(2πG

3

p2aa2− 3

8πGκ+ 3a2ρ+ a3

dρ

da

). (20)

By using Eqs. (19) and (20) with the Hamiltonian constraint, Eq. (11), we obtain the modified

Friedmann equation

H2 =

(8πG

3ρ− κ

a2

) [1 − 3α a2

πG

(8πG

3ρ− κ

a2

)1/2]. (21)

By considering the standard case, Eq. (17), in which α vanishes and for κ = 0, we find that the

modified Friedmann equation reads

H2 =8 π G

3ρ

[1− 3α a2

√8

3 π Gρ1/2

]. (22)

2.1 Bounds on GUP Parameter

The GUP parameter is given as α = α0/(Mpc) = α0�p/�, where c, � and Mp are speed of light

and Planck constant and mass, respectively. The Planck length �p ≈ 10−35 m and the Planck

energy Mpc2 ≈ 1019 GeV. α0, the proportionality constant, is conjectured to be dimensionless

[39]. In natural units c = � = 1, α will be in GeV−1, while in the physical units, α should be


in GeV−1 times c. The bounds on α0, which was summarized in Ref. [41, 57, 58], should be a

subject of precise astronomical observations, for instance gamma ray bursts [48].

• Other alternatives were provided by the tunnelling current in scanning tunnelling microscope

and the potential barrier problem [59], where the energy of the electron beam is close to the

Fermi level. We found that the varying tunnelling current relative to its initial value is shifted

due to the GUP effect [57, 59], δI/I0 ≈ 2.7×10−35 times α20 . In case of electric current density

J relative to the wave function Ψ, the current accuracy of precision measurements reaches the

level of 10−5. Thus, the upper bound α0 < 1017. Apparently, α tends to order 10−2 GeV−1 innatural units or 10−2 GeV−1 times c in physical units. This quantum-mechanically-derived

bound is consistent with the one at the electroweak scale [57, 59, 58]. Therefore, this could

signal an intermediate length scale between the electroweak and the Planck scales [57, 59, 58].

• On the other hand, for a particle with mass m mass, electric charge e affected by a constant

magnetic field �B = Bz ≈ 10 Tesla, vector potential �A = B x y and cyclotron frequency

ωc = eB/m, the Landau energy is shifted due to the GUP effect [57, 59] by

ΔEn(GUP )

En

= −√8m α (�ωc)

12

(n+

1

2

) 12

≈ −10−27 α0. (23)

Thus, we conclude that if α0 ∼ 1, then ΔEn(GUP )/En is too tiny to be measured. But with the

current measurement accuracy of 1 in 103, the upper bound on α0 < 1024 leads to α = 10−5

in natural units or α = 10−5 times c in the physical units.

• Similarly, for the Hydrogen atom with Hamiltonian H = H0 +H1, where standard Hamilto-

nian H0 = p20/(2m) − k/r and the first perturbation Hamiltonian H1 = −α p30/m, it can be

shown that the GUP effect on the Lamb Shift [57, 59] reads

ΔEn(GUP )

ΔEn

≈ 10−24 α0. (24)

Again, if α0 ∼ 1, then ΔEn(GUP )/En is too small to be measured, while the current measure-

ment accuracy gives 1012. Thus, we assume that α0 > 10−10.In light of this discussion, should we assume that the dimensionless α0 has the order of unity in

natural units, then α equals to the Planck length ≈ 10−35 m. The current experiments seem not

be able to register discreteness smaller than about 10−3-th fm, ≈ 10−18 m [57, 59]. We conclude

that the assumption that α0 ∼ 1 seems to contradict various observations [48] and experiments

[57, 59]. Therefore, such an assumption should be relaxed to meet the accuracy of the given

experiments. Accordingly, the lower bounds on α ranges from 10−10 to 10−2 GeV−1. This means

that α0 ranges between 109 c to 1017 c.

2.2 Standard Model Solution of Universe Expansion

In a toy model [60, 61], the prefect cosmic fluid contributing to the stress tensor Tμν can be

characterized by symmetries of the metric, homogeneity and isotropy of the cosmic Universe.

Thus, the total stress-energy tensor Tμν must be diagonal and the spatial components will be

given as

Tμν = diag(ρ,−p,−p,−p). (25)


Assuming that all types of energies in the early Universe are heat Q captured in a closed sphere

with radius equal to scale factor a of volume V = 4π a3/3, the energy density during the expansion

ρ = U/V , where U is internal energy [60, 61]. The first law of thermodynamic satisfies of the total

energy conservation

dQ = dU + p dV = 0. (26)

By substituting the totally differential of the energy density, d ρ = dU/V −U dV/V 2 into Eq. (26),

we get

d ρ = −3daa(ρ+ p). (27)

Dividing both sides over dt results in

ρ = −3H(ρ+ p). (28)

For a very simple equation of state, ω = p/ρ, where ω is independent of time, the energy density

reads

ρ ∼ a−3(1+ω). (29)

The radiation-dominated phase is characteristic by ω = 1/3 or p = ρ/3. Therefore, ρ ∼ a−4, thescaling factor, a ∼ const. t1/2 and the Hubble parameter, H = 1/(2t). In the matter-dominated

phase, ω = 0, i.e. p� ρ. Therefore, ρ ∼ a−3, a ∼ const. t2/3 and H = 2/(3t).

The left-hand panel (a) of Fig. 1 shows the Hubble parameter, H in dependence on the scale

factor, a. The standard (without GUP), Eq. (17) are compared with the modified (with GUP)

characterizations of the cosmic fluid, Eq. (22) in the flat universe. It is obvious that H in both

cases (with/without GUP) diverges at vanishing a. This would mean that a singularity exists at

the beginning. The GUP has the effect to slightly slow down the expansion rate of the Universe.

This is valid for both cases of cosmic background, radiation and matter.

In the right-panel (b) of Fig. 1, the dependence of the scale factor, a, on the cosmic time, t, is

given for both cases of cosmic matters, radiation and matter with and without GUP. Apparently,

the GUP is not sensitive to the matter-dominated phase but has a clear effect on the radiation-

dominated phase. The GUP breaks down the expansion.

3. Cosmic Inflation

Here, we estimate various inflation parameters, which are characterized by the scalar field φ and

apparently contribute to the total energy density [1, 2]. Also, taken into account the background

(matter and radiation) energy density, the scalar field is assumed to interact with the gravity and

with itself [6, 62, 63, 1, 2]. In order to reproduce the basics of the field theory, the coupling of φ

to gravitation results in total inflation energy

1

2

(φ2 + (∇φ)2

)+ V (φ). (30)

The dynamics of the inflation can be described by two types of equations:

• the Friedmann equation, which describes the contraction and expansion of the Universe and


0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

H

a

matter with GUPradiation with GUP

matter without GUPradiation without GUP

0

2.5

5

7.5

10

12.5

0 1.5 3 4.5 6 7.5 9

a

t

(b)

matter with GUP radiation with GUP

matter without GUP radiation without GUP

Fig. 1 (Color online) Left-hand panel (a) presents the variation of the Hubble parameter H with respectto the scale factor a. Matter- and radiation-dominated phases with and without GUP are compared witheach other. Right-hand panel (b) presents the scale factor a as function of the cosmic time t. Variousparameters are fixed, G = 1 and α = 10−2 GeV−1.

• the Klein-Gordon equation, which is the simplest equation of motion for a spatially homoge-

neous scalar field

φ+ 3H φ+ ∂φV (φ) = 0, (31)

where ∂φ ≡ ∂/∂φ.

In a flat Universe, κ = 0, the total inflation energy, Eq. (30) and the energy density due to the

cosmological constant ρv = Λ/8 π G, can be substituted in the modified Friedmann equation, Eq.

(21),

H2 =8πG

3

[φ2 + (∇φ)2

2+ V (φ) + ρv

] ⎡⎣1− 3αa2√

8

3πG

(φ2 + (∇φ)2

2+ V (φ) + ρv

)1/2⎤⎦ . (32)

In rapidly expanding Universe and if the inflation field starts out sufficiently homogeneous, the

inflation field becomes minimum, very slow [1, 2]. This would be modelled by a sphere in a viscous

medium, where both the energy densities due to matter ρm and radiation ρr are neglected

(∇φ)2 � V (φ), (33)

φ � 3H φ, (34)

φ2 � V (φ). (35)

The first inequality, Eq. (33), is obtained under the assumption of homogeneity and isotropy

of the FLRW Universe [1, 2], while the second inequality, Eq. (34), states that the scalar field

changes very slowly so that the acceleration would be neglected [1, 2]. The third inequality, Eq.

(35), gives a principle condition for the expansion. Accordingly, the kinetic energy is much less

than the potential energy [1, 2]. Apparently, the Universe expansion accelerates [6]. Therefore,

the modified Friedmann equation, Eq. (32) and the Klein-Gordon equation, Eq. (31), respectively

read

H2 =8π G

3(V (φ) + ρv)

[1− 3α a2

√8

3π G(V (φ) + ρv)

1/2

], (36)

φ = − 1

3H∂φV (φ). (37)


The cosmological constant characterizes the minimum mass that is related to the Planck mass

MP =√�c/G. The Planck length �p =

√�G/c3 [64] is also related to the mass quanta, where

quantized mass [65] is proportional to the GUP parameter α. The cosmological constant Λ is

one of the foundation of gravity [65]. It related the Planck (quantum scale) and the Einstein (in

cosmological scale) masses, MP and ME, respectively, with each other [65]

Mp =

(h

c

)(Λ

3

)1/2

, (38)

ME =

(c2

G

)(3

Λ

)1/2

. (39)

By using natural units � = c = 1, the modified Friedmann equation, Eq. (36), becomes

H2 =4π

3 M2p

{[V (φ) +

3M4p

4 π

]− 3α a2

√16M2

p

3 π

[V (φ) +

3M4p

4 π

]3/2}. (40)

There are various inflation models such as chaotic inflation models, which suggest different inflation

potentials [6, 62, 63]. Now, it is believed that they are better motivated than other models

[6, 62, 63]. In this context, there are two main types of models; one with a single inflation field

and the other one combines two inflation fields. Here, we summarize some models requiring a

single inflation-field φ which in some regions satisfies the slow-roll conditions,

Polynomial chaotic inflation V (φ) =1

2m2φ2, V (φ) = λφ4, (41)

Power-law inflation V (φ) = V0 exp

[√16π G

pφ

], (42)

Natural inflation V (φ) = V0

(1 + cos

φ

f

), V (φ) ∝ φ−β. (43)

Based on this concept, we select three different inflation potential models, Eqs. (45), (46) and

(47). The first one is based on certain minimal supersymmetric extensions of the standard model

for elementary particles [66] and the related effects have been studied, recently [66, 68]. It has

two free parameters, m and λ,

V (φ) =

(m2

2

)φ2 −

(√2λ (n− 1)m

n

)φn +

(λ

4

)φ2(n−1), (44)

where n > 2 is an integer. At n = 3,

V1(φ) =

(m2

2

)φ2 −

(2√λm

3

)φ3 +

(λ

4

)φ4, (45)

which is an S-dual inflationary potential [69] with a free parameter f . The S duality has its origin

in the Dirac quantization condition of the electric and magnetic charges [70]. This would suggest

an equivalence in the description of the quantum electrodynamics [70],

V2(φ) = V0 sech

(φ

f

). (46)


For a power-law inflation with the free parameter d [71, 62],

V3(φ) =3M2

pd2

32π

[1− exp

(− 16π

3M2p

1/2

φ

)]2

. (47)

For these inflation potentials, Eqs. (45), (46) and (47), the inflation parameters such as

potential slow-roll parameters ε, η, tensorial pt and scalar ps density fluctuations, the ratio of

tensor-to-scalar fluctuations r, scalar spectral index ns and the number of e-folds with the inflation

era Ne can be estimated.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

V/V

0

φ/Mp

V1(φ)V2(φ)v3(φ)

Fig. 2 (Color online) The variation of inflation potentials V/V0 is given in dependence on scalar fieldφ/Mp at limited free constants. The solid, long-dashed and dotted line stands for V1(φ), V2(φ) and V3(φ),respectively.

Fig. 2 shows the variation of the different inflation potentials, Eqs. (45), (46) and (47),

normalized with respect to initial potential V0 with the single inflation field φ according to Lyth

bound during the inflation era [18, 19, 20] and normalized with respect to Mp. The inflation

field, φ ≡ Δφ = (φ0 − φend) should be smaller than or comparable with the Planck scale Mp.

This was confirmed by the BICEP2 observation conditionally with this bound of small scalar field

[18, 19, 20]. The potentials, Eqs. (45) and (47) increase with φ/Mp, while the third potential,

Eq. (46) , decreases. This means that the latter is finite at vanishing inflation field, φ, while the

earlier vanishes.

4. Fluctuations and Slow-roll Parameters in the Inflation Era

In very early Universe, the scaler field φ is assumed to derive the inflation [6, 62, 63]. The main

potential slow-roll parameters are given as

ε ≡M2

p

16 π

(∂φV (φ)

V (φ)

)2

, (48)

η ≡M2

p

8π

(∂2φV (φ)

V (φ)

). (49)

Fig. 3 shows the variation of the potential slow-roll parameters as functions of the scalar field.

Various inflation potentials, Eqs. (45), (46) and (47) are used to deduce the slow-roll parameters,


Eqs. (48) and (49). The scalar fields in left- (a) and right-hand panel (c) result in slow-roll

parameters, which start from large values at small field. Then, they rapidly decline (vanish) as

the scalar field increases. The field presented in the middle panel gives slow-roll parameters with

relatively very small values, but seem to remain stable with the field.

0

200

400

600

800

0 0.02 0.04 0.06 0.08 0.1

ε,η

φ/Mp

(a)slow roll parameters for V1(φ)

εη

-4.5

-3

-1.5

0

1.5

3

4.5

0 0.08 0.16 0.24 0.32 0.4

ε,η

φ/Mp

(b)slow roll parameters for V2(φ)

εη

0

200

400

600

800

0 0.02 0.04 0.06 0.08 0.1

ε,η

φ/Mp

(c)slow roll parameters for V3(φ)

εη

Fig. 3 (Color online) From the left- (a) middle (b) and right-hand (c) panels present the slow-roll param-eters associated with V1(φ) from Eq. (45), V2(φ) from Eq. (46) and V3(φ) from (47), respectively. Thesolid and dot-dashed curves represent ε and η parameters, respectively.

The tensorial and scalar density fluctuations are given as [6, 62, 63]

pt =

(H

2π

)2 [1− H

Λsin

(2Λ

H

)]=

(H

2 π

)2 [1− H

3M2p

sin

(6M2

p

H

)], (50)

ps =

(H

φ

)2 (H

2π

)2 [1− H

Λsin

(2Λ

H

)]=

(H

φ

)2 (H

2 π

)2 [1− H

3M2p

sin

(6M2

p

H

)]. (51)

Fig. 4 shows the dependence of tonsorial (top panel) and scalar (bottom) density fluctuations,

Eqs. (50) and (51), on the scalar field of inflation φ. We show the fluctuations corresponding

to the inflation potential and find that the tonsorial and scalar density fluctuations decrease as

scalar field of inflation φ increases. The tonsorial density fluctuations (top panel) corresponding

the inflation potentials, V1(φ) from Eq. (45), V2(φ) from Eq. (46) and V3(φ) from (47), look

similar. There is a rapid rise at small and a decrease at large φ/Mp. For the inflation potential

given in Eq. (47), φ/Mp at which the peak takes place is smaller than that for Eq. (45).

No systematic comparison can be done for the scalar (bottom panel) density fluctuations of

the different inflation potentials. The left-hand and the middle panels shows that the potential,

Eqs. (45) and (46), very rapidly decreases with φ/Mp. Then, increasing φ/Mp does not change the

fluctuations. The right-hand panel, Eq. (47), presents another type of scalar density fluctuations,

which remain almost unchanged for a wide range of φ/Mp. Then, the fluctuations are almost

damped, at large φ/Mp.

The results corresponding to α = 10−2 GeV−1 are depicted. Exactly the same curves are also

obtained at α = 10−19 GeV−1. The earlier value is related to α0 = 1017 c while the latter to

α0 = 1 c. In light of this, the bounds on α0 seem not affecting the evolution of both tonsorial and

scalar density fluctuations with the scalar field of inflation φ.

Therefore, we can now study the ratio of tensor-to-scalar fluctuations, r, which obviously reads

[6, 62, 63]

r =ptps

=

(φ

H

)2

, (52)

relating potential evolution with the Hubble parameter H. Corresponding to the tensor-to-scalar


0

0.02

0.04

0.06

0.08

0.1

0 0.08 0.16 0.24 0.32 0.4

P t

φ/Mp

(a): pt for V1(φ)

0

0.02

0.04

0.06

0.08

0.1

0 0.08 0.16 0.24 0.32 0.4

n s

φ/Mp

(b): Pt for V2(φ)

0

0.02

0.04

0.06

0.08

0.1

0 0.08 0.16 0.24 0.32 0.4

P t

φ/Mp

(c): Pt for V3(φ)

0

1

2

3

4

5

0 0.08 0.16 0.24 0.32 0.4

Ps

φ/Mp

(a): ps for V1(φ)

0

4

8

12

16

20

0 0.08 0.16 0.24 0.32 0.4

Ps

φ/Mp

(b): Ps for V2(φ)

-2.4

-1.2

0

1.2

2.4

0 0.08 0.16 0.24 0.32 0.4

Ps

φ/Mp

(c): Ps for V3(φ)

Fig. 4 (Color online) The top panels show the tonsorial density fluctuations, pt, in dependence on thescalar field φ/Mp. The bottom panels give the scalar density fluctuations ps in dependence on the scalarfield φ/Mp. Each column is associated with an inflation potential model. The results corresponding toα = 10−2 GeV−1 are depicted, only. Exactly same curves are also obtained at α = 10−19 GeV−1 (notshown here).

fluctuations, a spectral index ns can be defined

ns = 1−√

r

3. (53)

The number of e-folds is given by numbers of the Hubble Ne ≈ 60 [62] or the integral of the

expansion rate,

Ne =

∫ tf

ti

H(t) dt = −3∫ φf

φ

H2

∂φ V (φ)dφ, (54)

where

H(t) dt =H

φdφ = −3 H2

∂φ V (φ)dφ. (55)

In Fig. 5, the left-hand panel shows the ratio of tonsorial to scalar density fluctuations r in

dependence on φ/MP . The dashed curves are evaluated at α = 10−2 GeV−1, while the solid thick

curves at α = 10−19 GeV−1. The earlier value is corresponding to α0 = 1017 while the latter to

α0 = 1. It is obvious that the bounds on α0 do no affect the ratio of tonsorial to scalar density

fluctuations r in dependence on φ/MP . The behavior of the tonsorial to scalar ratio is limited by

the modified Friedmann equation (in the presence of GUP), where the GUP physics is related to

the gravitational effect on such model at the Planck scale. The GUP parameter α - appearing in

the modified Friedmann equation - should play an important role in bringing the value of r very

near to both PLANCK and BICEP2, r = 0.2+0.07−0.05. According to Eq. (40), α breaks (slows) down

the expansion rate, H, compared with Fig. 1. It is obvious that the parameters related to the

Gaussian sections of the three curves match nearly perfectly with the results estimated by the

PLANCK and BICEP2 collaborations (compare with Fig. 6).

The right-hand panel of Fig. 5 shows the variation of the spectral index, ns, with scalar field

for the three inflation potentials, Eqs. (45), (46) and (47). Again, the dashed curves are evaluated


at α = 10−2 GeV−1, while the solid thick curves at α = 10−19 GeV−1. It is obvious that the

bounds on α0 do no affect the dependence of spectral index, ns on φ/MP .

Figure 6 summarizes the observations of PLANCK [12, 13] and BICEP2 [11] collaborations

together with the parametric dependence of spectral index ns and the ratio r. Both parametric

quantities are functions of φ, Eqs. (52) and (53). We find that the region of PLANCK

at 1 σ [12, 13] and BICEP2 at 1 σ [11] observations for r and ns is crossed by our

parametric calculations for r vs. ns for two different potentials Eqs. (45) and (46).

For the inflation potential, Eq. (47), the parametric calculations for r vs. ns are very small. This

can be interpreted due to the large minimum in the right-hand panel of Fig. 5, which means that

main part of ns calculated for this potential is entirely excluded (out of the range). The related

part is obviously very small. The authors of Ref. [67] predict variations of the fluctuations tensor

with the spectral index at 55 e-folding corresponding to a for the chaotic inflation potential,

V (φ) =m2φ2

2(1− a φ+ a2bφ2)2. (56)

Our results fit well with the curves of Ref. [67] (open symbols in Fig. 6) which have an excellent

agreement with PLANCK and BICEP2 observations. It is worthwhile to highlight that they are

deduced using other methods than ours. The main difference is the varying chaotic potential

parameters at a constant inflation field. Furthermore, Ref. [67] gives ns(a) and r(a), while we

are varying various potential with the scalar field at constant potential parameters and estimate

ns(φ) and r(φ). The parametric dependence of ns(a) and r(a) is given in Fig. 6.

It is apparent that the graphical comparison in Fig. 6 presents an excellent agree-

ment between the observations of PLANCK [12, 13] and BICEP2 [11] and the para-

metric calculations, especially for the inflation potentials, Eqs. (45) and (46). The

agreement is apparently limited to the values given by the parametric calculations,

while the observations are much wider.

0

0.08

0.16

0.24

0.32

0 0.08 0.16 0.24 0.32 0.4

r

φ/Mp

V1(φ)

V2(φ)

V3(φ)

α = 10-2 GeV-1

α= 10-19 GeV-1

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.08 0.16 0.24 0.32 0.4

n s

φ/Mp

V2(φ)

V1(φ)

V3(φ)

α = 10-2 GeV-1

α= 10-19 GeV-1

Fig. 5 (Color online) Left-hand panel shows the ratio of tonsorial-to-scalar density fluctuations, r, independence on φ/Mp calculated for the inflation potentials V1(φ), V2(φ) and V3(φ). The right-hand panelgives the spectral index ns vs. φ/Mp. The dashed curves are evaluated at α = 10−2 GeV−1, while thesolid curves at α = 10−19 GeV−1.

Tab. 1 summarizes the results of r and ns at various scalar fields φ/Mp for the three inflation

potentials, V1(φ) from Eq. (45), V2(φ) from Eq. (46) and V3(φ) from (47). The BICEP2-relevant

results are r ranging from 0.15 from 0.27 and simultaneously ns between 0.94 and 0.98. It is

apparent that the results from V3(φ) do not appear in this r−ns window. The results from V1(φ)


0

0.08

0.16

0.24

0.32

0.4

0.94 0.96 0.98 1

r

ns

ns(a) vs. r(a) in chaotic V(φ) at b=0.34ns(a) vs. r(a) in chaotic V(φ) at b=5

ns(φ) vs. r(φ) in V1(φ)ns(φ) vs. r(φ) in V2(φ)ns(φ) vs. r(φ) in V3(φ)

PLANCK at 1 σPLANCK at 2 σBICEP2 at 1 σ

BICEP2 at 2 σ

Fig. 6 (Color online) Contours showing PLANCK and BICEP2 results at 1σ and 2σ confidence comparedwith the parametric calculations for r as function of scalar spectral index ns. The parametric calculationsfor chaotic inflation potential given in Ref. [67], the square (b = 0.34) and circle (b = 5) balls correspondsto (0.001 < a < 0.13) and inflation field φ ∼ 8.2 are also compared with.

ΔφMp

r for V1(φ) ns for V1(φ)ΔφMp

r for V2(φ) ns for V2(φ)ΔφMp

r for V3(φ) ns for V3(φ )

0.21 0.263 0.934 0.07 0.157 0.961 0.02 0.118 0.801

0.22 0.254 0.936 0.08 0.182 0.954 0.03 0.192 0.746

0.23 0.245 0.939 0.09 0.204 0.950 0.04 0.240 0.717

0.24 0.236 0.941 0.10 0.219 0.945 0.05 0.258 0.707

0.25 0.227 0.943 0.11 0.230 0.942 0.06 0.255 0.708

0.26 0.218 0.945 0.15 0.227 0.943 0.07 0.239 0.717

0.27 0.210 0.947 0.16 0.218 0.945 0.08 0.218 0.730

0.29 0.193 0.951 0.17 0.207 0.948 0.09 0.194 0.745

0.3 0.185 0.953 – – – 0.1 0.172 0.761

0.32 0.171 0.957 – – – 0.11 0.151 0.775

0.33 0.164 0.959 – – – – – –

0.35 0.151 0.962 – – – – – –

0.36 0.145 0.963 – – – – – –

Tab. 1 The ratio of tonsorial to scalar density, the fluctuations r and the spectral index ns associated withthe scalar field for different inflation potentials, V1(φ) from Eq. (45), V2(φ) from Eq. (46) and V3(φ) from(47).

and V2(φ) obviously do. While V1(φ) allows a wide range of φ, V2(φ) is only relevant for a narrower

one. Fig. 6 represents this comparison, graphically.


5. Discussion and Conclusions

The BICEP2 results announced on March 17, 2014 made the physicists around the globe having

another view about the evidence of Universe and its expansion, especially at about the inflation

era. The cosmic inflation is based on the assumption that an extreme inflationary phase should

take place after the Big Bang (at about the Planck time). Thus, the Universe should expand at a

superluminal speed. On the other hand, the inflation would result from a hypothetical field acting

as a cosmological constant to produce an acceleration expansion of the Universe.

Argumentation about the applicability of GUP on the inflation era will be elaborated in Ap-

pend. A2. Due to the very high energy (quantum or Planck scale), the Heisenberg uncertainty

principle should be modified in terms of momentum uncertainty. The QG approach in form of

GUP appears in the modified Friedmann equation - in terms of α. This term reduces the Hubble

parameter, which appears in the denominator of the ratio of tonsorial-to-scalar density fluctua-

tions. Thus, the fluctuations ratio increases due to decreasing H. The fluctuations ratio r has

been evaluated as function of the spectral index ns. We found that the calculations match well

with the PLANCK and BICEP2 observations. This is the main conclusion of the present work.

We believe that the results point to the importance of quantum correlation during the inflation

era.

The estimation of the ns(a) and r(a) at 55 e-folds for a chaotic potential for different values of

b and varying inflation as function of a. The parameters b = 0.34 (open squares) and b = 5 (open

circles) are corresponding to (0.001 < a < 0.13) and inflation field φ ∼ 8.2ss [67]. The authors

predict the variation of the fluctuation tensor with the spectral index. The best curves in Ref.

[67] agree well with PLANCK and BICEP2. These are deduced using another method, varying

a and selecting out the suitable scalar field. The main difference with our method is the is the

varying chaotic potential parameters at constant inflation field. We vary the inflation potential

with the scalar field at constant potential parameters.

We have reviewed different inflationary potentials and estimated the modifications of the Fried-

mann equation due to the GUP approach. We found that

• the first potential, Eq. (45), gives a power law of the scalar inflation-field. This is based on

certain minimal supersymmetric extensions of the standard model [66].

• The second potential, Eq. (46), hypothesizes that the potential should be invariant un-

der the S-duality constraint g → 1/g, or φ → −φ, where φ is the dilation/inflation and

g ≈ exp (φ/M) [69]. The S-duality had its roots in the Dirac quantization condition for the

electromagnetic field. Thus, it should be equivalence to the description of quantum electro-

dynamics as either a weakly coupled theory of electric charges or a strongly coupled theory

of magnetic monopoles [72]. The latter, Eq. (47) appeared in an exponential form with a

power-law inflation field. These inflationary potentials seem to agree well with of the ob-

servations of PLANCK and BICEP2 collaborations at different 1σ and 2 σ. In the range of

spectral index and fluctuation ratio.

• The potential, Eq. (47) disagrees. Few remarks are now in order. The agreement should

be limited to the values given by the parametric calculations. The PLANCK and BICEP2

observations are much wider but have uncertainties in r of order 25%. We have presented

through a conceivable way the effects of reasonably-sized GUP parameter of our estimation

for r.


We conclude that depending on the inflation potential V (φ) and the scalar field, φ, the GUP

approach seems to reproduce the BICEP2 observations r = 0.2+0.07−0.05, which also have been fitted

by using 55 e-folds for a chaotic potential for varying inflation and seem to agree well with the

upper bound value corresponding to PLANCK and to WMAP9 experiment.

A Generalized uncertainty principle (GUP)

A1 Minimal length uncertainty and maximum measurable momentum

The commutator relation [39, 40, 41], which are consistent with the string theory, the black holes

physics and DSR leads to

[xi, pj] = i�

[δij − α

(pδij +

pipjp

)+ α2

(p2δij + 3pipj

)], (A.1)

implying a minimal length uncertainty and a maximum measurable momentum when implement-

ing convenient representation of the commutation relations of the momentum space wave-functions

[42, 21]. The constant coefficient α = α0/(Mp c) = α0 lp/� is referring to the quantum-gravitational

effects on the Heisenberg uncertainty principle. The momentum pj and the position xi operators

are given as

xi Ψ(p) = x0i(1− α p0 + 2α2 p20)Ψ(p),

pj Ψ(p) = p0j Ψ(p). (A.2)

We notice that p20 =∑3

j p0j p0j satisfies the canonical commutation relations [x0i, p0j] = i � δij.

Then, the minimal length uncertainty [39, 40, 41] and maximum measurable momentum [42, 21],

respectively, read

Δx ≥ (Δx)min ≈ �α,

pmax ≈1

4α, (A.3)

where the maximum measurable momentum agrees with the value which was obtained in the

doubly special relativity (DSR) theory [36, 21]. By using natural units, the one-dimensional

uncertainty reads [39, 40, 41]

ΔxΔp ≥ �

2

(1− 2αΔp+ 4α2Δp2

). (A.4)

This representation of the operators product satisfies the non-commutative geometry of the space-

time [42]

[pi, pj] = 0,

[xi, xj] = −i �α(4α− 1

p

) (1− α p0 + 2α2 �p0

2)Lij. (A.5)

The rotational symmetry does not break by the commutation relations [42, 32]. In fact, the

rotation generators can still be expressed in terms of position and momentum operators as [42, 21]

Lij =Xi Pj − Xj Pi

1− α p0 + 2α2 �p02 . (A.6)


A2 Applicability of GUP to the cosmic inflation

The quantum aspects of the gravitational fields can emerge in the limit, where strong, weak

and electromagnetic interactions can be distinguished from each other. In the view of gedanken

experiments that have been designed to measure the apparent horizon area of a black hole in

QG [26], the uncertainty relation is found preformed [25]. The deformed or modified Heisenberg

algebra, which was suggested to investigate GUP, introduces a relation between QG and Poincare

algebra [26]. Nevertheless, GUP given in quadratic forms [25, 27], which fits well with the string

theory and the black hole physics and introduces a minimal length uncertainty and additional

linear terms of momenta [36] agrees also well with DSR and assumes that the momenta approach

maximum value at very high energy (Planck scale) [36].

There are several observations supporting the concept of GUP approaches and offering a possi-

bility of studying the influence of the minimal length on the properties of a wide range of physical

systems, especially at quantum scale [25, 26, 34]. The effects of linear GUP approach have been

studied on compact stars [46], Newtonian law of gravity [45], inflationary parameters and thermo-

dynamics of the early Universe [47], Lorentz invariance violation [48] and measurable maximum

energy and minimum time interval [73]. Furthermore, the effects of QG on the quark-gluon plasma

(QGP) are studied [49]. It was found that the GUP can potentially explain the small observed

violations of the weak equivalence principle in neutron interferometry experiments [74]. Also, it

was suggested [75] that GUP can be measured directly in Quantum Optics Lab [40, 41]. The cur-

rent researches of the quantum problems in the presence of gravitational field at very high energy

near to the Planck scale implies new physical laws and even corrections of the spacetime of our

Universe [27]. The quantum field theory in curved background can be normalized by introducing

a minimal observable length as an effective cut-off in ultraviolet domain [27]. It is conjectured

that the string can’t probe distances smaller than its own length. For cosmic inflation, is the

expansion of space in the early Universe. The inflationary epoch lasted from 10−36 seconds after

the Big Bang to sometime between 10−33 and 10−32 seconds near to Planck scale. Following the

inflationary period, the universe continues to expand, but at a less accelerated rate. Actually,

the GUP at very high energy Planck scale would likely be applicable to contract these approach

interpret of the quantum study of the inflationary of the universe.

B Modified Dispersion Relation (MDR)

Various observations support the conjectured that the Lorentz invariance might be violated. The

velocity of light should differ from c. Any tiny adjustment leads to modification of the energy-

momentum relation and modifies the dispersion relation in vacuum state by δv [76, 77, 78, 79].

In particular, at the Planck scale, the modifications of energy-momentum dispersion relation have

been considered in Refs. [80, 81]. Two functions p(E) as expansions with leading Planck-scale

correction of order Lp E3 and L2

pE4 respectively, reads [82],

�p2 � E2 −m2 + α1 Lp E3, (B.1)

�p2 � E2 −m2 + α2 L2p E

4. (B.2)

These are valid for a particle of mass M at rest, whose position is being measured by a procedure

involving a collision with a photon of energy E and momentum p. Since the relations are originated


from Heisenberg uncertainty principle for position with precision δx, one should use a photon with

momentum uncertainty δp ≥ 1/δx. Based on the argument of Ref. [83] in loop QG, we convert

δp ≥ 1/δx into δE ≥ 1/δx. By using the special-relativistic dispersion relation and δE ≥ 1/δx,

then M ≥ δE. If indeed loop QG hosts a Planck-scale-modified dispersion relation, Eq. (B.2),

thenδpγ ≥ 1/δx and this required that [82],

M ≥ 1

δx

(1− α2

3L2p

2(δx)2

). (B.3)

These results apply only to the measurement of the position of a particle at rest [83]. We can

generalize these results to measurement of the position of a particle of energy E.

• In case of standard dispersion relation, one obtains that E ≥ 1/δx as required for a linear

dependence of entropy on area, Eq. (B.2)

• For the dispersion relation, Eq. (B.2)

E ≥ 1

δx

(1− α2

3L2p

2(δx)2

). (B.4)

The requirements of these derivation lead in order of correction of log-area form.

• Furthermore,

E ≥ 1

δx

(1 + α1

Lp

δx

). (B.5)

In case of string theory, the ”reversed Bekenstein argument” leads to quadratic GUP, that fits well

with the string theory [84] and black holes physics,

δx ≥ 1

δp+ λ2

sδp. (B.6)

The scale λs in Eq. (B.6) is an effective string length giving the characteristic length scale which

be identical with Planck length. Many researches of loop QG [80, 81] support the possibility of the

existence of a minimal length uncertainty and a modification in the energy-momentum dispersion

relation at Planck scale.

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The Shadow of a Black Hole from Heterotic StringTheory at the Center of the Milky Way

Alexis Larranaga∗

National Astronomical Observatory. National University of Colombia

Received 24 October 2014, Accepted 20 December 2014, Published 10 January 2015

Abstract: We study the shadow produced by a charged rotating black hole arising from the

heterotic string theory. The size and the shape of the shadow, depending on the mass, electric

charge and angular momentum, are calculated. We find that very-long baseline interferometry

(VLBI) in the near future will be enough to extract the information about the electric charge

of this kind of black hole.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Black Holes; Strings; Branes

PACS (2010): 04.70.-s; 11.25.-w; 98.35.Jk

1. Introduction

Black holes are one of the most interesting predictions of general relativity, as well as

of other gravitational theories, and it is widely believed that there exist supermassive

black holes in the center of many galaxies. Due to the rotation of the host galaxies, these

black holes are generally thought to possess a spin parameter. In fact, the center of our

galaxy host the best constrained supermassive black hole known up to date, Sagittarius

A*. Its mass and the distance from us have been accurately determined but, unfortu-

nately, other characteristics such as its spin or electric charge are not known. Recently,

a possible method to constrains such parameters has emerged: the observation of black

hole shadows. The shadow of a black hole corresponds to the lensed image of the event

horizon and it is a two-dimensional dark zone seen from the observer, so it can be, in

principle, easily measured. For a non-rotating black hole, the shadow is a perfect circle

while for the rotating case, it has an elongated shape in the direction of the rotation axis

due to the dragging effect [1, 2].

∗ Email:[email protected]


This subject has gained attention recently [3, 4, 5, 6, 7, 8] because it is expected

that very long baseline radio interferometry can resolve it and observation results may

be obtained in the near future [9]. In the literature, the optical properties and apparent

shape of different rotating black holes have been investigated. For example, the shadow

of a Kerr black hole and a Kerr naked singularity was examined by Hioki and Maeda

[10] where it was suggested that the spin parameter of the black hole can be determined

from the observation. Schee and Stuchlik[11] studied the phenomena in a braneworld Kerr

black hole, and applied their results to Sagittarius A∗ (see also [12]). The same treatment

was extended to the rotating black hole in extended Chern-Simons modified gravity [13],

the Kaluza-Klein rotating dilaton black hole [14], rotating traversable wormholes [15], the

Einstein-Maxwell-Dilaton-Axion black hole [16], some cosmic strings solutions [17] and

regular black holes [18]. These results show that, beside the spin, other parameters (as

for example coupling constants or tidal charges) also affect the shadow of a black hole,

and therefore, observation of its characteristics provides a possible way to determine such

parameters.

On the other hand, it has been realized that the low-energy effective field theory

describing heterotic string theory contains black hole solutions which can have properties

which are qualitatively different from those that appear in Einstein gravity. In this work,

we propose that the observation of the shadow will be very useful to determine the true

nature of a particular black hole and to decide whether it comes from general relativity

or from string theory. Using the observables proposed in [10], we will study the apparent

shape of the shadow of the Sen black hole, which comes from heterotic string theory.

Assuming that the supermassive black hole Sagittarius A∗ can be described by the Sen

black hole metric, the astronomical observables and angular radius of the shadow are

obtained. The result shows that the resolution of 1 μas expected with the very-long

baseline interferometry (VLBI) in the near future [19, 20] will be enough to extract the

information about the electric charge of the black hole and thus discriminating the string

theory solution from the corresponding general relativity counterpart.

2. The Sen Black Hole

Sen [21, 22] found a charged, stationary, axially symmetric solution of the field equations

of heterotic string theory by using target space duality, applied to the classical Kerr

solution. The line element of this solution can be written, in generalized Boyer-Lindquist

coordinates, as

ds2 = −(1− 2Mr

ρ2

)dt2 + ρ2

(dr2

Δ+ dθ2

)− 4Mra sin2 θ

ρ2dtdϕ

+

(r (r + rα) + a2 +

2Mra2 sin2 θ

ρ2

)sin2 θdϕ2, (1)

where


Δ = r (r + rα)− 2Mr + a2 (2)

ρ2 = r (r + rα) + a2 cos2 θ. (3)

Here M is the mass of the black hole, a = JM

is the specific angular momentum of the

black hole and the electric charge is represented by the quantity

rα =Q2

M. (4)

Note that taking a = 0, the metric (1) coincides with the (Gibbons-Maeda-Garfinkle-

Horowitz-Strominger) GMGHS solution [23, 24, 25, 26] while setting rα = 0 in (1) gives

the Kerr solution. The Sen spacetime has a spherical event horizon given as the biggest

root of the equation Δ = 0,

rH =2M − rα +

√(2M − rα)

2 − 4a2

2

or in terms of the black hole parameters M,Q and J ,

rH = M − Q2

2M+

√(M − Q2

2M

)2

− J2

M2. (5)

Here one distinguishes three interesting cases. First, for a < 12|2M − rα| the event

horizon exists and the solution represents a black hole. Second, when a > 12|2M − rα|

the metric represents a naked singularity. Finally, the critical condition a = 12|2M − rα|

results in an extremal black hole. In this work we will restrict our considerations to the

existence of the event horizon. For simplicity, in what follows we will adopt M = 1 in all

equations, which is equivalent to adimensionalize all physical quantities with the mass of

the black hole.

3. Photon Orbits

When a photon coming from infinity passes near a black hole, there are two cases that

can be considered. The first one corresponds to a photon with large orbital angular

momentum. At some turning points, it will turn back to be received by an observer

located at infinity. The second case corresponds to a photon with small orbital angular

momentum, which falls into the black hole, without reaching the observer. This simple

analysis shows that there is a dark zone in the sky called the shadow of the black hole

and the boundary of such region is determined by the two cases described above. In order

to determine the motion of the photons in the black hole background, we will use the

Hamilton-Jacobi equation,

∂S

∂λ= −1

2gμν

∂S

∂xμ

∂S

∂xν, (6)


to determine the geodesic equation. In this equation λ is an affine parameter along

the geodesics, and S is the Jacobi action. In general, for a particle with rest mass m0

in the background of the Sen black hole described by the metric in equation (1), the

Hamilton-Jacobi equation is separable and it possesses a solution of the form

S =1

2m2

0λ− Et+ Lϕ+ Sr (r) + Sθ (θ) , (7)

where Sr and Sθ are the radial and angular contributions, respectively. The constants

of motion E and L are the energy at infinity and the angular momentum with respect

to the axis of symmetry of the spacetime, respectively. Obviously, the rest mass for the

photon is zero, m0 = 0, and therefore replacing the action (7) into equation (6) gives the

null geodesic equation

ρ2dt

dλ=

[a2 + r (r + rα)] [a2E − aL+ r(r + rα)E]

Δ+ aL− a2E sin2 θ (8)

ρ2dϕ

dλ=

a [a2E − aL+ r (r + rα)E]

Δ+ L csc2 θ − aE (9)

ρ2dr

dλ= ±

√R (r) (10)

ρ2dθ

dλ= ±

√Θ(θ) (11)

where we have defined

R (r) = −Δ[K + (L− aE)2

]+

[a2E − aL+ Er (r + rα)

]2(12)

and

Θ (θ) = K +

[a2E2 − L2

sin2 θ

]cos2 θ, (13)

and K is a separation constant. Note that taking rα = 0, we recover the null geodesic

equations for the Kerr black hole [10].

In order to obtain the boundary of the shadow of the black hole, we need to study

the radial motion. Equation (10) can be rewritten as

ρ4r2 + Veff = 0, (14)

where the effective potential Veff is

Veff

E2=

[a2 + r (r + rα)− aξ

]2 − [r2 + a2 + 2r

(rα2− 1

)] [η + (ξ − a)2

], (15)

with the impact parameters defined as ξ = LE

and η = KE2 . The motion of the photon

is parameterized only by ξ and η which are conserved quantities according to the null


geodesics. Since the boundary of the shadow of the black hole is determined by the

circular photon orbits, we impose the conditions defining these,{Veff = 0dVeff

dr= 0.

(16)

3.1 The non-rotating case

For a non-rotating black hole (a = 0) and using equation (15), the parameters ξ and η

satisfy the condition

ξ2 + η =r30 + 2rαr

20 + r2αr0

r0 + rα − 2, (17)

where r0 is the solution of equations (16),

r0 =1

2

[3− 3rα

2+

√(rα2

)2

− 5rα + 9

]. (18)

Note that when rα = 0, the radius of the circular orbit becomes the well known radius

of the Schwarzschild’s photon sphere, r0 = 3.

3.2 The rotating case

For the rotating black hole (a �= 0), we obtain the parameters

ξ = −(r + rα

2+ 1

)a2 + r

[Δ− a2 +

(rα2− 1

)(r + rα)

]a

(r + rα

2− 1

) (19)

η = r24a2

(r + rα

2

)−

[Δ− a2 +

(rα2− 1

)(r + rα)

]2a2

(r + rα

2− 1

)2 . (20)

Taking rα = 0 in this equation reproduces the result for the Kerr black hole obtained

by Hioki and Maeda [10].

4. The Shadow of the Black Hole

Now we will analyze the above equations in order to determine the shadow of the Sen black

hole. As discussed in the first section, the existence of the event horizon is determined by

the condition a ≤ 12|2− rα|. Since our interest goes into the astrophysical observation,

we introduce the celestial coordinates using the functions (8) and (11),

α = limr→∞

(− r2 sin θ0

dϕ

dr

∣∣∣∣θ→θ0

)= −ξ csc θ0, (21)

β = limr→∞

(r2dθ

dr

∣∣∣∣θ→θ0

)= ±

√η + a2 cos2 θ0 − ξ2 cot2 θ0, (22)


where θ0 is the angle between the rotation axis of the black hole and the line of sight of

the observer, i. e. the observer is located at infinity with angular coordinate θ0. The

coordinate α corresponds to the apparent perpendicular distance of the image as seen

from the axis of symmetry and β is the apparent perpendicular distance of the image

from its projection on the equatorial plane (a detailed derivation of such coordinates for

the Kerr metric can be found in [27]). These coordinates give the apparent position of the

image in the plane that passes through the center of the black hole and that is orthogonal

to the line joining the observer and the black hole.

The shadow of the black hole corresponds to the region in the parameter space (α, β)

not illuminated by the photon sources located at infinity and distributed uniformly in

all directions. From the dependence of equations (21) and (22), it can be seen that

the gravitational effects on the shadow are larger when the observer is situated in the

equatorial plane. Also, it is clear that for the supermassive black hole Sagittarius A* the

inclination is expected to lie close to θ0 =π2, as seen from the Earth. Hence, the celestial

coordinates take the simpler form

α = −ξ (23)

β = ±√η (24)

for equatorial obervers. Note that from this equation it is clear that α and β are inde-

pendent of the angular coordinate ϕ (because the black hole is axisymmetric).

The shadow of the non-rotating black hole, a = 0, is determined by the radius of

the photon sphere. Hence, the condition in equation (17) for the parameters ξ and η

describing the photon sphere gives

α2 + β2 =r30 + 2rαr

20 + r2αr0

r0 + rα − 2= R2

C . (25)

This expression implies that the shadow of the black hole in the (α, β) parameter

space is a circle with the radius RC =√

r30+2rαr20+r2αr0r0+rα−2 depending on the electric charge

of the Sen black hole rα. In Figure 1(a) we plot this shadow for the electric charge

rα = 0 (continuous line) which corresponds to the circle with radius RC = 3√3. For non-

vanishing electric charge, the radius of the shadow decreases until it shrinks to a point,

RC = 0, when the extremal black hole is reached, at the value rα = 2. This behavior

distinguishes the GMGHS black hole from the Reissner-Nordstrom solution because in

the latter case it reaches a minimum radius of Rs = 4 in the extremal case. However

this behavior of the GMGHS black hole is similar to the obtained for the Kaluza-Klein

rotating dilaton black hole by Amarilla and Eiroa [14].

For the rotating black hole, a �= 0, we replace the parameters ξ and η from equations

(19) and (20) into the parameters α and β. The result is a shadow that will be a deformed

circle, as plotted in Figure 1 (b) and (c). In all cases, the size of the shadow reduces

when there is a non-vanishing electric charge rα for a fixed spin a. The behavior with the

spin is such that when a increases, the shadow will shift to the right.

It is important to note that when the black hole approaches the extremal case, the

shadow will be distorted more away from a circle and it will reach different sizes for a


Figure 1 Shadows of Sen black hole situated at the origin of coordinates with inclination angleθ0 = π/2. (a) Spin a = 0 and electric charge rα = 0 (full line), rα = 0.5 (dashed line) andrα = 0.8 (dashed-dotted line). (b) Spin a = 0.5 and electric charge rα = 0 (full line), rα = 0.2(dashed line) and rα = 0.5 (dashed-dotted line). (c) Spin a = 0.8 and electric charge rα = 0(full line), rα = 0.1 (dashed line) and rα = 0.2 .

fixed value of a when compared with those obtained with the Kerr-Newman solution.

Even more, the maximum allowed charge for a fixed mass is larger for the Sen black

hole than for Kerr-Newman ones. This means that the Sen black hole can harbor larger

amounts of electric charge before becoming naked singularities than their general relativ-

ity counterparts.

5. Astronomical Observables

In this section we will introduce two observables in order to extract the information

of an astronomical object from this shadow. The size and the form of the shadow are

characterized by using the two astronomical observables introduced in [10]:

The first observable Rs is defined as the radius of a reference circle passing by three

points: the top position and the bottom position of the shadow, (αt, βt) and (αb, βb)

respectively; and the point (αr, 0) corresponding to the unstable retrograde circular orbit

seen from an observer located in the equatorial plane.

The second observable is the distortion parameter, defined as δs =DRs, where D is the

difference between the endpoints of the circle and of the shadow, both of them correspond

to the prograde circular orbit and locate at the opposite side of the point (αr, 0).

As the name of the two observables indicate, Rs measures the approximate size of

the shadow while δs measures its deformation with respect to the reference circle. By

knowing the inclination angle θ0, and with a precise enough measurements of Rs and δs,

one can obtain the value of the physical rotation parameter a and the electric charge rαof the black hole. This information can be obtained by plotting the contour curves with

constant Rs and δs in the plane (a, rα). Hence, the intersection point in this plane gives

the corresponding rotation and charge.

The geometry of the shadow let us write the observable Rs as (see [10] for a detailed

analysis)

Rs =(αt − αr)

2 + β2t

2(αr − αt), (26)

where we have used the relations αb = αt and βb = −βt. Similarly, the distortion


Figure 2 Astronomical observables as function of the electric charge of the black hole. (a) Rs

for the black hole with zero spin, a = 0, as function of rα. See text for details about the non-zerospin case. (b) δs as a function of rα for spin a = 0 (full line), a = 0.5 (dashed line), and a = 0.8(dashed-dotted line).

parameter δs is

δs =(αp − αp)

Rs

, (27)

with (αp, 0) and (αp, 0) the points where the reference circle and the contour of the shadow

cut the horizontal axis at the opposite side of (αr, 0), respectively. Writing αp = αr−2Rs,

the parameter δs can be expressed as

δs = 2− Ds

Rs

, (28)

where we defined the diameter of the shadow along β = 0 as Ds = αr − αp. Note that

this equation says that for smaller Ds and larger Rs we obtain larger values of δs. Also,

if Ds = 2Rs (i.e. in the non-rotating case) we get δs = 0, as expected.

For a non-rotating black hole, the behavior of the coordinate Rs as function of the

electric charge rα is shown in Figure 2 (a). As it can be seen, Rs decreases with rα,

which means that the approximate size of the shadow decreases with the existence of

electric charge. When the black hole is rotating, the behavior of Rs is almost the same

for different values of the rotation parameter a. In fact, for a fixed value of the electric

charge, the difference of Rs between a = 0 and a = 1 is of order 10−3, which causes such

a small variation in the size of the shadow that the curves are indistinguishable (and

therefore in the plot we see just one curve). The same behavior is obtained for the Kerr

black hole [10] as well as for the Einstein-Maxwell-Dilaton-Axion black hole and some

braneworld [12] and Kaluza-Klein solutions [14].

On the other hand, the distortion parameter δs is plotted as a function of the electric

charge for different values of the spin of the black hole in Figure 2 (b). For the non-

rotating black hole (continuous curve), a = 0, the distortion is always zero, which means

that the contour of the shadow is a perfect circle with no deformation as expected . In


Figure 3 Contour plot of the astronomical observables in the plane (a, rα). Rs is described bythe full lines (taking the values 4.5, 4.75 and 5 from top to bottom) while δs is described bythe dashed lines (taking the values 0.002, 0.003, 0.006, 0.010, 0.025, 0.050, 0.1, and 0.2 from leftto right). The gray zone at the right of the figure represents naked singularities which are notconsidered here.

the rotating case, δs as function of the electric charge is a monotonically increasing curve

which comes to a maximal value when the black hole approaches the extremal case. For

a fixed electric charge, the distortion of the shadow increases with the spin a.

Finally, we plot the contour curves of constant Rs and δs in the parameter (a, rα)

space in Figure 3.The grey zone represents naked singularities, which are outside the

scope of this work. In the figure, each point is characterized by four values, (a, rα, Rs,

and δs) and therefore, from the observational point of view, if we measure and fix the

values of Rs and δs, we are allowed to directly read out the spin and electric charge of

the black hole through the corresponding point of intersection in Figure 3.

5.1 Angular Size of the Shadow for Sagittarius A*

The observable Rs can be used to estimate the angular radius of the shadow as θs =RsMD0

where D0 corresponds to the distance between the observer and the black hole. As

observers, we are located in the equatorial plane of the Milky way, at a distance of

D0 = 8.3 kpc from the supermassive black hole Sagittarius A∗ which have an estimated

mass of M = 4.3× 106M� [28]. This data gives the function θs = 5.11388Rs μas which,

together with the values of the observable Rs, let us calculate the angular size of the

shadow for different values of spin and electric charge. Note that using the values shown

in Figure 2 (a), the resolution of 1 μas will be enough to extract the information of the

electric charge from further observations, while for the spin a, the resolution of less than

1 μas is needed. These are out of the capacity of the current astronomical observations

but can be likely to be observed with the Event Horizon Telescope at wavelengths around

1mm based on VLBI, and with the space-based radio telescopes RadioAstron [19, 20].


6. Conclusions

In this paper, we studied the size and shape of the shadow cast by a charged rotating

black hole from the low-energy effective field theory describing heterotic string theory,

which have properties which are qualitatively different from those that appear in Einstein

gravity. We have found that, for a fixed mass and rotation parameter, the presence of of

electric charge leads to a shadow that is slightly smaller and more deformed.

Supposing that the black hole is situated at the origin of the coordinate system and the

observer is located in the equatorial plane, we use the null geodesics to obtain the celestial

coordinates α and β in order to visualize the black hole’s shadow. To analyze the shadow

in detail, we define two astronomical observables: Rs that describes the approximate size

of the shadow and δs that measures its deformation. These quantities are functions of

the electric charge and the spin of the black hole. Our study shows that, for a fixed value

of the spin, the size of the shadow described by Rs decreases when increasing the electric

charge but its value is always smaller than that of the Kerr black hole. On the other hand,

the observable δs is a monotonically increasing function of the electric charge and takes

its maximal value for the extremal black hole. Also, for a fixed value of rα, an increase

in the value of the spin results in a greater distortion of the shadow. Finally, we plot the

contour curves of constant Rs and δs in the (a, rα) plane because careful measurements

of the astronomical observables allow to read out the spin and the electric charge of the

black hole from this figure.

It is expected that direct observation of the shadow of black holes will be possible

in the near future by using the very long baseline interferometer technique in (sub)

millimeter wavelengths [29, 30]. The Event Horizon Telescope will reach a resolution

of 15 μas at 345 GHz, while the RadioAstron mission will be capable to resolve1 − 10

μas and the Millimetron mission may provide a resolution of 0.3 μas or less at 0.4 mm.

Following our estimates for the angular size of the shadow, these instruments will be

capable of observing the shadow of the supermassive galactic black hole Sagittarius A*

to distinguish the presence of electric charge as described by the Sen black hole and

maybe to discriminate the string theory solution from the corresponding general relativity

counterpart.

Acknowledgements

This work was supported by the Universidad Nacional de Colombia. Hermes Project

Code 18140.

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Scattering of Waves in the Phase Space, QuantumMechanics, and Irreversibility

E. M. Beniaminov∗†

Russian State University for the Humanities, Miusskaya sq. 6, Moscow, GSP-3,125993, Russia

Received 29 November 2014, Accepted 19 December 2014, Published 10 January 2015

Abstract: We give an example of a mathematical model describing quantum mechanical

processes interacting with medium. As a model, we consider the process of heat scattering

of a wave function defined on the phase space. We consider the case when the heat diffusion

takes place only with respect to momenta. We state and study the corresponding modified

Kramers equation for this process. We consider the consequent approximations to this equation

in powers of the quantity inverse to the medium resistance per unit of mass of the particle in

the process. The approximations are constructed similarly to statistical physics, where from

the usual Kramers equation for the evolution of probability density of the Brownian motion of

a particle in the phase space, one deduces an approximate description of this process by the

Fokker–Planck equation for the density of probability distribution in the configuration space. We

prove that the zero (invertible) approximation to our model with respect to the large parameter

of the medium resistance, yields the usual quantum mechanical description by the Schrodinger

equation with the standard Hamilton operator. We deduce the next approximation to the model

with respect to the negative power of the medium resistance coefficient. As a result we obtain

the modified Schrodinger equation taking into account dissipation of the process in the initial

model, and explaining the decoherence of the wave function.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Quantum Mechanics; Kramers Equation; Waves in the Phase Space; Diffusion

Scattering; Asymptotic Solutions; Decoherence

PACS (2010): 03.65.-w; 03.65.Nk; 66.30.Ma; 03.65.Yz; 02.60.-x

1. Introduction

In this paper we continue the study of the generalized Kramers equation introduced in

the paper [1].

∗ Email: [email protected]† Tel:+79162231515


In [1], the generalized Kramers equation arose as a mathematical model of the scat-

tering process of waves in the phase space under the action of medium in the heat equi-

librium. This process is irreversible. In [1] it has been shown that, for certain parameters

of the model, the process described by the generalized Kramers equation, is the compo-

sition of a rapid transitional process and a slow process. The slow process, as proved in

loc. cit., is approximately described by the Schrodinger equation used for description of

quantum processes. The obtained approximation of the slow process is reversible. Thus,

we have given an example of an irreversible process (heat scattering of waves in the phase

space), for which it has been shown that the standard reversible quantum mechanical

description of the motion of a particle arises as the asymptotic description of this process

in the zeroth approximation.

The purpose of the present paper is to construct an approximate equation describing

the slow part of the process given by the generalized Kramers equation, with the precision

showing dissipative effects in this process. As a result, we derive the modified Schrodinger

equation taking into account dissipation of the process.

The paper consists of four Sections and the Appendix.

In the next Section we present the setting of the problem, the generalized Kramers

equation for the heat scattering process of waves on the phase space, and the properties

of the operators involved in the generalized Kramers equation.

In Section 3 we recall our earlier results describing the process given by the generalized

Kramers equation. We also give a method for the approximate description of this process

in negative powers of the medium resistance coefficient, and state the main result of

the paper, Theorem 4. In this Theorem we write down an equation describing the slow

component of the studied process, taking into account dissipation. This equation is the

modified Schrodinger equation, in which the process is not invertible. We show that

consequences of the modified Schrodinger equation are the effects of decoherence and

spontaneous jumps between the levels.

In Section 4 we state some further possible directions of research and possibilities of

comparison of the presented model with experiment.

Proof of the main Theorem 4 of the paper is given in the Appendix.

2. Mathematical Setting of the Problem, and the Main Prop-

erties of the Operators in the Equation

Thus, we consider the following mathematical model of a process. A state of the process

at each moment of time t ∈ R is given by a complex valued function ϕ[x, p, t] on the phase

space (x, p) ∈ R2n, where n is the dimension of the configuration space. Coordinates in

the configuration space are given by a tuple of numbers x ∈ Rn, and momenta by a tuple

of numbers p ∈ Rn. (Below everywhere we will write the arguments of a function in the

square brackets, in order to distinguish an argument of a function from multiplication by

a variable.)

The generalized Kramers equation for a function ϕ[x, p, t] is defined as the following


equation:∂ϕ

∂t= Aϕ+ γBϕ, (1)

where Aϕ =n∑

j=1

(∂V

∂xj

∂ϕ

∂pj− pj

m

∂ϕ

∂xj

)− i

�

(V −

n∑j=1

p2j2m

)ϕ (2)

and Bϕ =n∑

j=1

∂

∂pj

((pj + i�

∂

∂xj

)ϕ+ kBTm

∂ϕ

∂pj

);

m is the mass of the particle; V [x] is the potential function of external forces acting on

the particle; i is the imaginary unit; � is the Planck constant; γ = β/m is the medium

resistance coefficient β per unit of mass of the particle; kB is the Boltzmann constant; T

is the temperature of the medium.

If we pass in equation (1) to the following dimensionless variables:

p′ =p√

kBTm, x′ =

√kBTm

�x, V ′[x] =

V [x]

kBT, (3)

then in the new variables equation (1) takes the following form:

∂ϕ

∂t= A′ϕ+ γB′ϕ, (4)

where A′ϕ =kBT

�

( n∑j=1

(∂V ′

∂x′j

∂ϕ

∂p′j− p′j

∂ϕ

∂x′j

)− i

(V ′ −

n∑j=1

(p′j)2

2

)ϕ

)(5)

and B′ϕ =n∑

j=1

∂

∂p′j

((p′j + i

∂

∂x′j

)ϕ+

∂ϕ

∂p′j

).

Note that the operator A′ is skew Hermitian, and the operator B′ is neither skew

Hermitian neither self-adjoint. The operator B′ determines the scattering process of the

wave function with respect to momenta and hence non-invertibility of the process. In this

paper we consider the case when γ is a large quantity, i. e. the impact of the operator B′

on the general evolution process of the wave function is large. Properties of the operator

B′ are presented in the following Theorem.

Theorem 1. The operator B′ given by expression (5) has a full set of eigenfunc-

tions (in the class of functions ϕ[x, p] tending to zero at infinity) with the eigenvalues

0,−1,−2, . . . . Respectively, the operator B′ is presented in the following form:

B′ = −∞∑k=0

kPk, (6)

where Pk are the projection operators onto the eigenspaces of the operator B′ with eigen-

values −k.The projection operators Pk satisfy the relations

PkPk = Pk, PkPk′ = 0 for k �= k′, PkB′ = B′Pk = −kPk (7)


and E =∞∑k=0

Pk, (8)

where E is the identity operator.

Proof of this Theorem will be given simultaneously with the proof of the following

Theorem describing the form of the projection operators Pk.

Denote by Hkk1...kn

[p]def= Hk1 [p1]...Hkn [pn] the product of Hermite polynomials [7]

of the corresponding variables, where k = k1 + ... + kn is the sum of degrees of the

Hermite polynomials in the product. By definition, the Hermite polynomial is given by

the expression

Hkj [pj]def= exp

(p2

2

)(− ∂

∂pj

)kj

exp

(−p2

2

). (9)

Let Sk1,...,kn and Ik1,...,kn be the operators given by the expressions

ψk1,...,kn = Sk1,...,kn [ϕ]def=

∫Rn

Hkk1...kn

[p′′ + i

∂

∂x′′

]ϕ[x′′, p′′]dp′′, (10)

ϕk1,...,kn = Ik1,...,kn [ψk1,...,kn ]def= (11)

1

(2π)3n/21

k1!...

1

kn!

∫R2n

ψk1,...,kn [x′′]Hk

k1...kn[p′ − s′]e−

(p′−s′)22 eis

′(x′−x′′)ds′dx′′.

Theorem 2. The projection operators Pk have the form

Pk =k∑

k1,...,kn=0

k1+...+kn=k

Ik1,...,knSk1,...,kn , (12)

and the operators Sk′1,...,k′n, Ik1,...,kn satisfy the relations

Sk′1,...,k′nIk1,...,knψk1,...,kn = δk′1,k1 ...δk′n,knψk1,...,kn , (13)

where δk′iki equals 0 if k′i �= ki, and equals 1 if k′i = ki.

In particular, formulas (10), (11) and Theorem 2 imply that

ψ[x′] = S0[ϕ]def=

∫Rn

ϕ[x′, p′]dp′, (14)

ϕ0[x′, p′] = I0[ψ]

def=

1

(2π)3n/2

∫R2n

ψ[x′′]e−(p′−s′)2

2 eis′(x′−x′′)ds′dx′′, (15)

P0ϕ =1

(2π)3n/2

∫R3n

ϕ[x′′, p′′]dp′′e−(p′−s′)2

2 eis′(x′−x′′)ds′dx′′, (16)

P1ϕ =n∑

j=1

1

(2π)3n/2

∫R3n

(p′′j + i

∂

∂x′′j

)ϕ[x′′, p′′]dp′′ ×

(p′j − s′j)e− (p′−s′)2

2 eis′(x′−x′′)ds′dx′′. (17)


Note that the operators I0 and S0 given by formulas (15) and (14) make a bijection

between the set of functions ψ[x′] and the set of eigenfunctions ϕ0[x′, p′] of the operator

B′ with eigenvalue 0. We shall call the function ψ[x′] = S0[ϕ0[x′, p′]] by the presentation

of the eigenfunction ϕ0[x′, p′].

Proof of Theorems 1 and 2. Let us substitute into expression (5) the presentation

ϕ[x′, p′, t′] in the form of composition of the Fourier integral with respect to x′ and the

inverse Fourier transform:

ϕ[x′, p′, t′]=1

(2π)n/2

∫Rn

ϕ[s′, p′, t′]eis′x′ds′, (18)

where ϕ[s′, p′, t′]def=

1

(2π)n/2

∫Rn

ϕ[x′′, p′, t′]e−is′x′′

dx′′, (19)

and we denoted by s′x′ the expression s′x′def=

∑nj=1 s

′jx′j.

We obtain that the operator B′ has the form

B′[ϕ[x′′, p′]] =1

(2π)n

∫R2n

n∑j=1

∂

∂p′j

((p′j − s′j)ϕ+

∂ϕ

∂p′j

)eis

′(x′−x′′)ds′dx′′. (20)

Computing the integral over x′′ in the left hand side of the obtained expression, taking

into account equality (19), we obtain:

B′[ϕ[x′′, p′]] =1

(2π)n/2

∫Rn

n∑j=1

∂

∂p′j

((p′j − s′j)ϕ+

∂ϕ

∂p′j

)eis

′x′ds′. (21)

The operatorn∑

j=1

∂

∂p′j

((p′j − s′j)ϕ+

∂ϕ

∂p′j

), (22)

under the sign of integral in the previous expression, is well known (see, for example, [6]).

This operator has a full set of eigenfunctions in the space of functions tending to zero as

|p′ − s′| tends to infinity. The eigenvalues of this operator are the non-positive integers.

The eigenvalue 0 corresponds to eigenfunctions of the form

ϕ0[s′, p′] = ψ[s′]e−

(p′−s′)22 ,

where ψ[s′] is an arbitrary complex valued function of s′ ∈ Rn.

The remaining eigenfunctions are obtained (as it is easy to check) by differentiation

of the functions ϕ0[s′, p′] with respect to p′j, j = 1, . . . , n, and have the eigenvalues

equal respectively to −1,−2, . . ., depending on the degree of the derivative. Thus, the

eigenfunctions with eigenvalues −k = −(k1 + ...+ kn) are the functions of the form

ψk1...kn [s′](−1)k ∂k1

∂p′k11

. . .∂kn

∂p′knn

e−(p′−s′)2

2 = ψk1...kn [s′]Hk

k1...kn[p′ − s′]e−

(p′−s′)22 ,

where ψk1...kn [s′] are arbitrary complex valued functions of s′ ∈ Rn, and Hk

k1...kn[p′] =

Hk1 [p′1]...Hkn [p

′n] is the product of Hermite polynomials of the corresponding variables,


and k = k1 + ...+ kn is the sum of degrees of the polynomials. The Hermite polynomials

of small degrees have the form

H0 = 1, H1[pj] = pj, H2[pj] = p2j − 1. (23)

Let us represent also the functions ψk1...kn [s′] in the form of Fourier integrals:

ψk1...kn [s′] =

1

(2π)n/2

∫Rn

ψk1...kn [x′′]e−is

′x′′dx′′.

From this formula, taking into account representation (18) of the function ϕ[x′, p′, t′]through ϕ[s′, p′, t′], we obtain that the eigenfunctions ϕk1...kn [x

′, p′] of the operator B′ havethe form (differing only by inessential constant factors from formula (11) of Theorem 1):

ϕk1...kn [x′, p′] =

1

(2π)n

∫R2n

ψk1...kn [x′′]Hk

k1...kn[p′ − s′]e−

(p′−s′)22 eis

′(x′−x′′)ds′dx′′.

It is known [7] that the Hermite polynomials form a complete system of functions and

satisfy the following orthogonality relations:

1

(2π)n/21

k1!...

1

kn!

∫Rn

Hk′k′1...k′n

[p′]Hkk1...kn

[p′]e−p′22 dp′ = δk′1k1 ...δk′nkn , (24)

where δk′iki are equal to 0 if k′i �= ki and equal to 1 if k′i = ki (in these formulas it is

assumed that 0!=1).

This directly implies the statements of Theorems 1 and 2. In particular, the latter

equality of Theorem 1 follows from completeness of the set of eigenspaces of the operator

B′.Note that in representation of the eigenfunctions of the operator B′ in the form (11)

one can compute the integral over s′. To this end, let us substitute into this formula the

expression defining the Hermite polynomials (9). We obtain

ϕk1,...,kn = Ik1,...,kn [ψk1,...,kn ]= (25)

1

(2π)3n/21

k1!...

1

kn!

(−1)k∂k

∂p′1k1 ...∂p′n

kn

∫R2n

ψk1,...,kn [x′′]e−

(p′−s′)22 eis

′(x′−x′′)ds′dx′′.

Further, let us make the change of variables s′ = s′′+ p′ under the integral, and compute

the integral over s′′, using the well known equality that the Fourier transform of the

function exp(−s′′2/2) is the function of the same form. We obtain

ϕk1,...,kn = Ik1,...,kn [ψk1,...,kn ]=

1

(2π)n1

k1!...

1

kn!

(−1)k∂k

∂p′1k1 ...∂p′n

kn

∫Rn

ψk1,...,kn [x′′]e−

(x′−x′′)22 eip

′(x′−x′′)dx′′ =

(−i)k(2π)n

1

k1!...

1

kn!

∫Rn

ψk1,...,kn [x′′]

n∏j=1

(x′ − x′′)kje−(x′−x′′)2

2 eip′(x′−x′′)dx′′. (26)


The latter equality is obtained after differentiation with respect to p′ required in the

formula, but under the sign of the integral.

The latter equality, in the particular case of eigenfunctions of the operator B′ withthe zero eigenvalue, implies the following expression:

ϕ0 = I0[ψ0]=1

(2π)n

∫Rn

ψ0[x′′]e−

(x′−x′′)22 eip

′(x′−x′′)dx′′. (27)

3. The Schrodinger Equation for the Scattering Process of Waves

and its Refinement

In the papers [1, 3] the following Theorem has been proved.

Theorem 3. The motion described by equation (1) asymptotically splits for large γ

into rapid motion and slow motion.

1) After the rapid motion, an arbitrary wave function ϕ[x, p, 0] approximately goes at

the time of order 1/γ to the function ϕ0 = P0ϕ which, after normalization and in the

initial coordinates, has the following form:

ϕ0[x, p] =1

(2π�)n/2

∫Rn

ψ[y]χ[x− y]eip(x−y)/�dy, (28)

where ψ[x] =

(1

4πkBTm

)n/4∫Rn

ϕ[x, p, 0]dp and (29)

χ[x− y] =

(kBTm

π�2

)n/4

e−kBTm(x−y)2/(2�2). (30)

The wave functions of kind (28) form the linear subspace of eigenfunctions of the operator

B given by (2) with eigenvalue zero. The elements of this subspace are parameterized by

wave functions ψ[y, 0] =∫Rn ϕ[y, p]dp depending only on coordinates y ∈ Rn.

2) The slow motion starting at the moment t = 0 with the function ϕ0[x, p] of the form

(28) with nonzero function ψ[y] = ψ[y, t]|t=0, goes along the subspace of such functions

and is parameterized by the wave function ψ[y, t] depending on coordinates and time. The

function ψ[y, t] satisfies the following Schrodinger equation: i�∂ψ/∂t = Hψ, where the

action of the operator H for γ →∞ has the form

Hψ = − �2

2m

( n∑k=1

∂2ψ

∂y2k

)+ V [y]ψ − kBT

2nψ +O(γ−1). (31)

The coefficient in formula (30) is chosen so that the mapping given by formula (28)

preserves the standard scalar product in the Hilbert spaces of functions ψ and ϕ0.

Proof of the first part of Theorem 3 is given in [3]. Proof of the second part of this

Theorem is given in [1]. (This formula will be also obtained in the Appendix to the

present paper in the course of proof of Theorem 4.)

Theorem 3 describes a solution of equation (1) and respectively of equation (4) in the

zero deterministic approximation in the parameter 1/γ after the transitional process in


time of order 1/γ. The purpose of the present paper is proof of a Theorem refining the

result of the part of Theorem 3 describing the slow motion. To this end, we construct

the next approximation of equation (4) with respect to the parameter 1/γ.

Thus, let us pass to approximate description of the generalized Kramers equation (4)

for large γ by means of systematic decomposition over powers of γ−1. The method used

here is similar to the method given in the book by van Kampen [5]. In this book, from the

Kramers equation describing the Brownian motion of a particle in the phase space, one

deduces the Fokker–Planck equation describing approximately the same process, but in

the form of the Brownian motion of the particle in configuration space after certain time

of transitional process. During this process the distribution with respect to momenta

becomes the Maxwell distribution.

Let us rewrite equation (4) in the following form:

B′ϕ =1

γ

(∂ϕ

∂t− A′ϕ

). (32)

Let us look for solution of this equation in the form

ϕ = ϕ0 + γ−1ϕ1 + γ−2ϕ2 + ... (33)

Let us substitute expression (33) into equation (32), and write out the equations for

coefficients before equal powers of γ−1. We obtain:

for γ0 : B′ϕ0 = 0; (34)

for γ−1 : B′ϕ1 =∂ϕ0

∂t− A′ϕ0; (35)

for γ−2 : B′ϕ2 =∂ϕ1

∂t− A′ϕ1; . . . . (36)

Equation (34) implies that ϕ0 belongs to the subspace of eigenfunctions of the operator

B′ with eigenvalue 0, i. e. ϕ0 = P0ϕ0, where P0 is the projection operator onto the

eigenspace of the operator B′ with eigenvalue 0.

Let us apply to both parts of equality (35) the projection operator P0 from the left.

Taking into account equalities P0B′ = 0 and ϕ0 = P0ϕ0, we obtain:

0 =∂ϕ0

∂t− P0A

′P0ϕ0, so∂ϕ0

∂t= P0A

′P0ϕ0. (37)

(Note that the operator P0A′P0 corresponds to the Schrodinger operator H ′ in Theo-

rem 3.)

Provided equality (37) holds, equation (35) has a solution which we represent in the

form

ϕ1 = ϕ1,0 + f1, where P0f1 = 0, ϕ1,0 = P0ϕ1 = P0ϕ1,0,

f1 = B′−1(P0A

′P0ϕ0 − A′P0ϕ0), (38)

and B′−1 is the inverse operator to B′ on the subspace spanned by eigenfunctions of the

operator B′ with nonzero eigenvalues. Formula (6) for B′ implies that the operator B′−1


has the form

B′−1 = −

∞∑k=1

k−1Pk. (39)

This and the properties (7) of projection operators imply that P0B′−1 = 0 and PkB

′−1 =−k−1Pk.

By the equality B′−1P0 = 0 the expression for f1 in formula (38) takes the form

f1 = −B′−1A′P0ϕ0. (40)

Let us substitute the expression ϕ1 = ϕ1,0 + f1 from formula (38) into equality (36), and

apply to both parts of equality (36) the projection operator P0 from the left. Using the

equalities P0B′ = 0, P0f1 = 0, and ϕ1,0 = P0ϕ1,0, and equality (40), after the substitutions

mentioned above and opening the brackets we obtain:

0 = P0

(∂ϕ1

∂t− A′ϕ1

)or

0 =∂ϕ1,0

∂t− P0A

′P0ϕ1,0 + P0A′B

′−1A′P0ϕ0. (41)

Let us now sum up equations (37) and (41) multiplied respectively by 1 and γ−1. Then

for the function ϕ≤1,0, defined by the equality

ϕ≤1,0def= ϕ0 + γ−1ϕ1,0, (42)

we obtain the following equation up to summands of order γ−1:

0=∂ϕ≤1,0∂t

− P0A′P0ϕ≤1,0 + γ−1P0A

′B′−1A′P0ϕ≤1,0 +O[γ−2]. (43)

Respectively, for function ϕ≤1def= ϕ0 + γ−1ϕ1, which takes into account only first two

summands in the decomposition (33) for ϕ, using the equality ϕ1 = ϕ1,0+f1 in expression

(38) and substituting into it expression (40) for f1, we obtain the following equality up

to summands of order γ−1:

ϕ≤1 = ϕ≤1,0 − γ−1B′−1A′P0ϕ≤1,0 +O[γ−2]. (44)

Since ϕ≤1,0 = P0ϕ≤1,0, that is, ϕ≤1,0 is an eigenfunction of the operator B′ witheigenvalue 0, then, in accordance with formulas (14) and (15) and Theorem 2, this function

is completely determined by the function ψdef= S0[ϕ≤1,0] by the formula ϕ≤1,0 = I0[ψ],

where S0 and I0 are the operators defined by formulas (14) and (15). As above, let us

call the function ψ by the presentation of the eigenfunction ϕ≤1,0.In order to obtain the ψ-presentation of equation (43), let us substitute into it, instead

of function ϕ≤1,0 the equal expression I0[ψ], let us act on both parts of equation by the

operator S0, and let us use the following equalities deduced from relations of Theorem 2:

P0ϕ≤1,0 = I0S0[ϕ≤1,0] = I0[ψ], S0P0 = S0, and ψ = S0I0[ψ].


We obtain:

0 =∂ψ

∂t− S0A

′I0ψ + γ−1S0A′B

′−1A′I0ψ +O[γ−2]. (45)

Respectively, expression (44) for function ϕ≤1[x′, p′], presented through the function

ψ[x′], without account of summands of order γ−2, reads

ϕ≤1 = I0[ψ]− γ−1B′−1A′I0[ψ] +O[γ−2]. (46)

Note that this equality implies S0ϕ≤1 = ψ. Thus, expression (46) and the operator

S0 yield mutually inverse bijections between the set of functions ϕγ1[x′, p′] and the set of

functions ψ[x′]. Thus, function ψ[x′] is also a presentation of the function ϕ≤1[x′, p′] byformula (46). And the function ψ[x′] evolves in time according to equation (45).

Thus, from equation (45) and relation P0 = I0S0 of Theorem 2 we obtain the following

approximate equation, with account of summands up to order γ−1, for the slow subprocess

in the process described by the modified Kramers equation (4):

∂ψ

∂t= S0A

′I0ψ − γ−1S0A′B

′−1A′I0ψ +O[γ−2]. (47)

(The form of the first summand in the right hand side of the equation, namely, the

operator S0A′I0, is known to us from Theorem 3.) The complete description of the right

hand side of this equation is given by the following Theorem.

Theorem 4. The slow motion mentioned in Theorem 3, starting from the wave

function ϕ0[x, p] of the form (28) with nonzero function ψ[y, 0], and parameterized by the

wave function ψ[y, t], satisfies the modified Schrodinger equation of the form i�∂ψ/∂t =

H1ψ, where action of the operator H1 is the following:

H1ψ=− �2

2m

n∑k=1

∂2ψ

∂y2k+ V ψ − kBTn

2ψ +

iγ−1

4

( n∑j=1

�

m

∂2V

∂y2j− (kBT )

2n

�

)ψ +

+O[γ−2]. (48)

Proof of Theorem 4 is given in the Appendix.

Since for description of behavior of the system, only the differences between the

eigenvalues of the operator are essential, the constant summands in operators H and

H1 of Theorems 3 and 4 can be neglected.

Using the standard method of perturbation theory [9], let us compute the corrections

to the eigenvalues and eigenfunctions of the Hamilton operator for the operator H1.

Let E(0)n be the eigenvalues of the Hamilton operator H, and ψ

(0)n be the corresponding

eigenfunctions. Let En and ψn be the eigenvalues and eigenfunctions of the operator

H1. Then by definition of eigenfunctions, one has the equalities Hψ(0)n = E

(0)n ψ

(0)n and

H1ψn = Enψn. Let us look for En and ψn in the form

En = E(0)n + γ−1E(1)

n +O(γ−2) (49)

ψn = ψ(0)n + γ−1

∑k

k =n

cnkψ(0)k +O(γ−2). (50)


Let us substitute these expressions into equality (48). In the obtained expression, let us

equate coefficients before the corresponding powers of γ−1, and take the scalar products

of both parts of the obtained equalities with ψ(0)n or ψ

(0)k . Using the orthonormality of

the system of eigenfunctions ψ(0)k with respect to the scalar product 〈 ; 〉 given by

〈ψk;ψm〉 =∫Rn

ψk[y]ψ∗m[y]dy, (51)

we obtain:

E(1)n =

i�

4m〈Δ2V ψ(0)

n ;ψ(0)n 〉 (52)

cnk =i�

4m

〈Δ2V ψ(0)n ;ψ

(0)k 〉

E(0)n − E

(0)k

, (53)

where Δ2V =n∑

k=1

∂2V

∂y2k. (54)

These equalities and the modified Schrodinger equation i�∂ψ/∂t = H1ψ imply that the

eigenfunction ψn evolves in time according to the following expression:

ψn[t] = ψn[0] exp

(− iEnt

�

)= ψn[0] exp

(− iE

(0)n t

�− iγ−1E(1)

n t

�

),

where − iγ−1E(1)n

�=

γ−1

4m〈Δ2V ψ(0)

n ;ψ(0)n 〉 is real.

Therefore, the absolute value of the eigenfunction ψn changes in time exponentially with

the exponent (γ−1/4m)〈Δ2V ψ(0)n ;ψ

(0)n 〉t. Thus, if the system at the initial moment of time

is in the state ψ[0], where ψ[0] =∑∞

n anψn is a certain superposition of eigenstates of

the operator H1, then after the time t the system will be in the state ψ[t]/|ψ[t]|, whereψ[t] =

∑∞n anψn[t]. Since the absolute values of the eigenstates ψn[t] change exponentially

with different velocities, then after large enough time the state ψ[t]/|ψ[t]| will be close tocertain eigenstate ψk, for which the value −iE(1)

k is maximal among the values −iE(1)n for

n with nonzero an in the sum∑∞

n anψn. This phenomenon has been called decoherence

and has been studied in a series of papers [10, 11, 12]. The time of decoherence in our

model can be estimated as the time t for which akexp(−iE(1)k t/�) >> ak1 exp(−iE

(1)k1

t/�),

where −iE(1)k1

is the next value after the maximal value −iE(1)k among the numbers −iE(1)

n

for n with nonzero an in the sum∑∞

n anψn.

Besides that, formulas (50) and (53) imply that the eigenfunctions ψn of the operator

H1 are in general case not orthogonal to each other. We have:

〈ψn;ψk〉 = γ−1(cnk + c∗kn) +O[γ−2] = 2γ−1cnk +O[γ−2].

Let ϕn[x, p] and ϕk[x, p] be wave functions on the phase space corresponding to func-

tions ψn and ψk by formula (46). According to the assumptions of the model, the square

of the absolute value of the scalar product of normalized functions ϕn[x, p] and ϕk[x, p]

yields the probability to find the system in the state ψk, if it is in the state ψn. The latter

equality and formula (46) imply that this probability is nonzero if cnk �= 0.


4. Conclusion

In this paper we have constructed an approximate description of the slow phase of the

scattering process of the wave function up to γ−1, where γ is the resistance of the medium

per unit of mass of a particle–wave. The obtained approximation is described by the

Schrodinger equation, supplemented with a summand with coefficient γ−1. In this ap-

proximation one has effects of decoherence and spontaneous jumps from one level to

another.

Note that for a free particle, when V = 0, and for harmonic oscillator, when the

second derivatives of the potential are constant, the summand of the operator H1 with

the factor γ−1 in Theorem 4 is a constant. Therefore, in the approximation up to γ−1,due to this summand, in these cases all wave functions decrease in amplitude with the

same velocity, and dissipation is non-observable. Hence, to take into account dissipation

in this model either for free particle or for harmonic oscillator, one should consider the

summands with factors γ−2 and γ−3. Note also that the method of construction of the γ−1

summand in the modified Schrodinger equation used in this paper, allows, in principle,

to construct also the summands with factors γ−2 and γ−3.The next phenomenon which can appear in this model, is the occurrence of nonzero

width of spectral lines. The interaction of a quantum particle with medium should

cause occurrence of width of energy levels for the energy operator. That is, if ψj is

an eigenfunction of the Hamilton operator with eigenvalue Ej, then the corresponding

wave function ϕj[x′, p′] in the phase space is given by the formula (46). According to the

assumption of the model, the function ϕj[x′, p′] defines the density function of probability

distribution in the phase space in the form ϕj[x′, p′]ϕ∗j [x

′, p′]. Respectively, the average

value of the energy function H[x′, p′] in this case is given by the following expression:

Ej =

∫R2n

H[x′, p′]ϕj[x′, p′]ϕ∗j [x

′, p′]dx′dp′. (55)

Then (ΔEj)2, the average of square of deviation from the average energy, is computed

by the formula

(ΔEj)2 =

∫R2n

(H[x′, p′]− Ej)2ϕj[x

′, p′]ϕ∗j [x′, p′]dx′dp′. (56)

These computations can be completed for concrete quantum systems, to obtain the

dependence of ΔEj on T and γ. After that, the obtained data can be compared with the

experimental data on the width of spectral lines.

Other directions of research on this subject can be found in [13].

Appendix. Proof of Theorem 4

To prove Theorem 4 one should compute the right hand side of equation (47), and pass

to the initial coordinates.


The first summand in the right hand side of this equation reads as follows:

S0A′I0ψ = −ikBT

�

(−1

2

n∑j=1

∂2

∂(x′j)2+ V ′ − n

2

)ψ.

We are going to compute the operator S0A′B

′−1A′I0 which stands in equation (47) with

the factor γ−1.Let us first transform this expression using the equality

B′−1 = −

∞∑k=1

k−1Pk

obtained above (39), the relations Pk =∑

k1+...+kn=k Ik1,...,knSk1,...,kn from Theorem 2, and

PkP0 = 0 for k > 0 .

We obtain:

S0A′B

′−1A′I0 =−∞∑k=1

k−1S0A′PkA

′I0 =−∞∑

k1,...,kn=0

k1+...+kn=k≥1

k−1S0A′Ik1,...,knSk1,...,knA

′I0. (57)

Let us now compute the operator A′Ik1,...,kn , where the operator A′ is given by expres-

sion (5), and the operator Ik1,...,kn is defined by expression (11).

We have

A′Ik1,...,knψ=kBT

�

( n∑j=1

(∂V ′

∂x′j

∂

∂p′j− p′j

∂

∂x′j

)− i

(V ′ −

n∑j=1

p′2j

2

))◦ (58)

1

(2π)3n/21

k1!...

1

kn!

∫R2n

ψ[x′′]Hkk1...kn

[p′ − s′]e−(p′−s′)2

2 eis′(x′−x′′)ds′dx′′.

Let us put the summands of the operator A′ in this expression under the sign of

integral, and let us decompose the functions of (p′ − s′) under the sign of integral over

Hermite polynomials H ll1...ln

[p′ − s′]. Using the equalities

∂

∂p′j

(Hk

k1...kn[p′ − s′]e−

(p′−s′)22

)def= −Hk+1

k1...kj+1...kn[p′ − s′]e−

(p′−s′)22 , (59)

∂

∂x′jeis

′(x′−x′′) = is′jeis′(x′−x′′), (60)

p′2j = ((p′j − s′j)

2 − 1) + 2p′js′j − s

′2j + 1 = H2[p

′j − s′j] + 2p′js

′j − s

′2j + 1, (61)

we obtain:


�

1

(2π)3n/21

k1!...

1

kn!

∫R2n

(−

n∑j=1

∂V ′

∂x′jHk+1

k1...kj+1...kn[p′ − s′]

−in∑

j=1

p′js′jH

kk1...kn

[p′ − s′]− iV ′[x′]Hkk1...kn

[p′ − s′]

+i

2

n∑j=1

(H2[p′j − s′j] + 2p′js

′j − s

′2j + 1)Hk

k1...kn[p′ − s′]

)×ψ[x′′]e−

(p′−s′)22 eis

′(x′−x′′)ds′dx′′. (62)


After opening the brackets in the latter summand and summing up the terms similar to

the second summand, we obtain:


�

1

(2π)3n/21

k1!...

1

kn!

∫R2n

(−

n∑j=1

∂V ′

∂x′jHk+1

k1...kj+1...kn[p′ − s′]

−iV ′[x′]Hkk1...kn

[p′ − s′]+i

2

n∑j=1

H2[p′j − s′j]H

kk1...kn

[p′ − s′]

− i

2

n∑j=1

(s′2j − 1)Hk

k1...kn[p′ − s′]

)ψ[x′′]e−

(p′−s′)22 eis

′(x′−x′′)ds′dx′′. (63)

Lemma 1. The operators S0A′Ik1,...,kn read as follows:

S0A′I0[ψ] =

kBT

�

(−iV ′ψ +

i

2

n∑j=1

∂2ψ

∂x′2j

+in

2ψ

); (64)

S0A′Ik1,...,kn [ψ] =

i

2

kBT

�ψ, when k = k1 + ...+ kn = 2

and only one of k1, ..., kn equals 2; (65)

In the remaining cases S0A′Ik1,...,kn = 0.

Proof. According to formula (14) the operator S0 is an integration over p. Hence for

proof of the formulas of Lemma 1 one should compute integral over p of expression (63)

for the operator A′Ik1,...,kn . To this end, let us use the orthogonality formula (24) for

Hermite polynomials and the fact that H00 = 1. This implies that integral over p of the

first summand of expression (63) equals 0. Integral of Hkk1...kn

[p′−s′] in the second and the

fourth summand with respect to 1/(2π)(n/2) exp[−(p′ − s′)2/2]dp equals 1 only for k = 0,

and in the other cases also equals 0. Integral of H2[p′j − s′j]H

kk1...kn

[p′ − s′] in the third

summand with respect to 1/(2π)(n/2) exp[−(p′−s′)2/2]dp differs from 0 and equals 2 only

when k = 2, kj = 2, and the rest of k1, ..., kn equal 0. Then in the obtained expression we

compute integrals over s′ and x′′, using the fact that integral of 1/(2π)neis′(x′−x′′) over s′

is the delta function at the point x′, and integral of expression 1/(2π)ns′2eis′(x′−x′′) over

s′ is the delta function at the point x′ with −∂2/∂x′′2. As a result of these computations

we get the statement of Lemma 1.

Since by Lemma 1 the operators S0A′Ik1,...,kn with k = k1 + ... + kn > 0 are not

equal to 0 only when k = 2, kj = 2 (and therefore the remaining k1, ..., kn equal 0), then

for computation of operator S0A′B

′−1A′I0 by formula (57) it remains to compute the

operators Sk1,...,knA′I0 for the same values of k1, ..., kn.

Lemma 2. The operators Sk1,...,knA′I0, in the case when kj = 2 and k1 + ...+ kn = 2,

read as follows:

Sk1,...,knA′I0[ψ] =

kBT

�

(−i∂

2V ′

∂x′2j

ψ + iψ

). (66)


Proof. Under the conditions of Lemma 2, when k = kj = 2, the operator Sk1,...,kn

reads, by formula (10), as follows:

Sk1,...,kn [ϕ]def=

∫Rn

Hkk1...kn

[p′ + i

∂

∂x′

]ϕ[x′, p′]dp′

=

∫Rn

H2

(p′j + i

∂

∂x′j

)ϕ[x′, p′]dp′. (67)

This and formula (63) for A′I0 imply that

Sk1,...,knA′I0[ψ]=

kBT

�

1

(2π)3n/2

∫R3n

H2

(p′j + i

∂

∂x′j

)◦

◦(−

n∑j′=1

∂V ′[x′]∂x′j′

H1[p′j′ − s′j′ ]− iV ′[x′]+

i

2

n∑j′=1

H2[p′j′ − s′j′ ]

− i

2

n∑j′=1

(s′2j′ − 1)

)ψ[x′′]e−

(p′−s′)22 eis

′(x′−x′′)ds′dx′′dp′. (68)

Taking into account that by definition of Hermite polynomials we have

H2

(p′j + i

∂

∂x′j

)=

((p′j + i

∂

∂x′j

)2

− 1

),

and by the equality i∂

∂x′jeis

′(x′−x′′) = −s′jeis′(x′−x′′), we obtain:

Sk1,...,knA′I0[ψ]=

kBT

�

1

(2π)3n/2

∫R3n

H2

(p′j − s′j

)×

(−

n∑j′=1

∂V ′[x′]∂x′j′

H1[p′j′ − s′j′ ]− iV ′[x′]+

i

2

n∑j′=1

H2[p′j′ − s′j′ ]

− i

2

n∑j′=1

(s′2j′ − 1)

)ψ[x′′]e−

(p′−s′)22 eis

′(x′−x′′)ds′dx′′dp′ (69)

+kBT

�

1

(2π)3n/2

∫R3n

(−2i(p′j − s′j)

n∑j′=1

∂2V ′[x′]∂x′j∂x

′j′H1[p

′j′ − s′j′ ]

+n∑

j′=1

∂3V ′[x′]∂2x′j∂x

′j′H1[p

′j′ − s′j′ ] + 2(p′j − s′j)

∂V ′[x′]∂x′j

+ i∂2V ′[x′]

∂x′j2

)×ψ[x′′]e−

(p′−s′)22 eis

′(x′−x′′)ds′dx′′dp′. (70)

Further, let us integrate over p′ in each integral in the obtained expression, using the

orthogonality relation (24) for Hermite polynomials and the equalities 1 = H0(p′j − s′j)


and (p′j − s′j) = H1(p′j − s′j). We obtain:

Sk1,...,knA′I0[ψ] =

kBT

�

1

(2π)n

∫R2n

(−0− 0 +

i

22− 0

)ψ[x′′]eis

′(x′−x′′)ds′dx′′

+kBT

�

1

(2π)n

∫R2n

(−2i∂

2V ′[x′]

∂x′j2 + 0 + 0 + i

∂2V ′[x′]

∂x′j2

)ψ[x′′]eis

′(x′−x′′)ds′dx′′. (71)

After computing the integrals over s′ and x′′ in the obtained expression and summing

up similar terms, we obtain the equality required in Lemma 2:

Sk1,...,knA′I0[ψ] =

kBT

�

(−i∂

2V ′[x′]∂x

′2j

ψ[x′] + iψ[x′]). (72)

Now we are ready to compute S0A′B

′−1A′I0 using Lemmas 1 and 2, by formula (57).

By Lemma 1 the summands in formula (57) can be nonzero only if kj = 2 for some

j ∈ {1, ..., n}, and the rest of k1, ..., kn equal 0. Lemmas 1 and 2 yield expressions for

operators S0A′Ik1,...,kn and Sk1,...,knA

′I0 in this case. So, by formula (57) and Lemmas 1, 2

we have:

S0A′B

′−1A′I0[ψ] = −∞∑

k1,...,kn=0

k1+...+kn=k≥1

k−1S0A′Ik1,...,knSk1,...,knA

′I0[ψ]

= −∑

k1,...,kn∈{0;2}k1+...+kn=2

2−1S0A′Ik1,...,knSk1,...,knA

′I0[ψ]

= −n∑

j=1

2−1i

2

kBT

�

kBT

�

(−i∂

2V ′

∂x′2j

ψ + iψ

)

= −1

4

(kBT

�

)2( n∑j=1

∂2V ′

∂x′2j

ψ − nψ

). (73)

If in this expression we return to the initial coordinates by formula (3), then we obtain

the equality

S0A′B

′−1A′I0)[ψ] = −1

4

(1

m

n∑j=1

∂2V

∂x2j

−(kBT

�

)2

n

)ψ. (74)

Thus, we have computed the operator in the second summand in equation (47). This

operator gives, after multiplying by −ihγ−1, the second summand in the operator H1

from Theorem 4.

Acknowledgments. The author is thankful to Professor G. L. Litvinov for support

of the work in this direction, and mourns for loss of this wonderful person and friend.

References

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Frequency Distribution of Spontaneous Emission

Saul. M. Bergmann∗†

Potomac, Maryland USA

Received 13 October 2014, Accepted 20 December 2014, Published 15 September 2014

Abstract: A two-level system, originally in the excited state, is irradiated by an incident beam

whose frequency bandwidth is narrow compared with the natural linewidth. A generalised

Weisskopf-Wigner treatment is developed by means of which the probability distribution of the

frequency of spontaneous emission is found. It is shown under which circumstances the beam

can influence this probability distribution. A broadening of the natural line occurs.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Quantum Mechanics, Fock Space; Weisskopf-Wigner Treatment; Spontaneous

Emission Spectrum; Induced Emission

PACS (2010): 03.65.-w; 03.67.a; 03.67.Bg; 42.50.Ct; 32.30.-r; 32.50.+d

1. Introduction

In ordinary perturbation theory, for a two-level system irradiated by a feeble incident

beam, the probability of induced emission and spontaneous emission, as a first order

process is being made to be proportional to n + 1, where n is the number of photons

in the incident Fock state, and “1” is regarded as an immutable contribution of spon-

taneous emission. What is striking in this result is that no matter how high n may be,

(for the range of validity of this perturbation method), the beam has no influence on the

probability of spontaneous emission. The drawback of perturbation theory, historically

speaking, has been its inability to describe neither the natural line shape nor the expo-

nential decay of a two-level system. Weisskopf and Wigner remedied this situation and

were able to describe both features by using a non-perturbational method. In retrospect,

their task may be viewed as an “exercise”, since they knew what the final result should

be. However, they treated only spontaneous emission. Since then, it has been tempting

to generalize their first-order treatment to the case of the presence of an incident beam.

∗ Email:[email protected]† Tel:3013401655


2. Equations in Interaction Representation

The subject of interest is a two-level atomic system having an excited state |e〉 and a

ground state |g〉. One assumes that the system is originally in the excited state and is

irradiated by an incident beam whose frequency bandwidth is narrow compared with the

natural linewidth of the atomic system. Since decay is involved, the exact equations of

motion in the interaction representation is given by

ih∂

∂tU = HU (1a)

with the subsidiary boundary condition

U(t = 0) = 1 (1b)

See e.g. Kallen. The latter presents a modern version of the Weisskopf-Wigner ap-

proximation for electromagnetic interactions. Refs. 3-5 deal with the same subject for

weak and strong interactions. Further referenes are given in refs. (9-12). Here H is the

Hamiltonian, and U is the unitary time displacement operator. The latter is defined by

|t〉 = U(t) |t = 0〉 (2)

3. Generalised Weisskopf-Wigner Approximation

The generalised Weisskopf-Wigner approximation includes the rule that in the matrix

multiplication, HU , only Fock states are retained whose number of photons are at most

one more than are contained in the Fock state of the incident beam of radiation.

The Hamiltonian H is here described as a scalar product of the quantised vector

potential �A(�r, t) and an atomic vector. Subsequently, for the vector potential, we are

mostly concerned with a plane-wave expansion, appropriate to a cubical volume V , such

as

�A(�r, t) = c∑�k,α

(h

2ωkV

) 12

�ε�k,α

[a�k,αe

i(�k·�r−ωkt) + a†�k,αe−i(�k·�r−ωkt)

](3)

where �ε�k,α is the unit polarization vector and a�k,α and a†�k,α are the usual photon

annihilation and creation operators, respectively; ωk = ck = c|�k| is the angular frequency.The Fock state of the incident beam is given by

∏k

a†nk

k√nk!|0〉 (4)

where nk is the photon occupation number for the mode k.


4. The Weisskopf-Wigner Approximation

The first dynamical equation in the Weisskopf-Wigner approximation to Eqs. (1a) and

(1b) is

ih∂

∂t〈0|

∏k

ankk√nk!〈e|U |e〉

∏k

a†nk

k√nk!|0〉 =

∑k

〈0| ankk√nk!

∏λ =k

anλλ√nλ!〈e|H |g〉

∏λ =k

a†nk

λ√nλ!

a†nk+1k√nk + 1

|0〉×

〈0| ank+1k√

(nk + 1)!

∏λ =k

anλλ√nλ!〈g|U |e〉

∏k

a†nk

k√nk!|0〉+

∑k′ =k

〈0| 〈e|H |g〉 a†k′ |0〉〈0| ak′ 〈g|U |e〉 |0〉 .

(5)

Here

〈0|∏k

ankk√nk!〈e|U |e〉

∏k

a†nk

k√nk!|0〉 ≡ a(t) (6)

is the probability amplitude for the system to remain in its original state,

〈0| ank+1k√

(nk + 1)!

∏λ =k

anλλ√nλ!〈g|U |e〉

∏k

a†nk

k√nk!|0〉 ≡ bnk+1 (7)

is the probability amplitude for induced or spontaneous emission in the forward di-

rection of the incident beam, and

〈0| ak′ 〈g|U |e〉 |0〉 ≡ bk(t) (8)

is the probability amplitude for spontaneous emission in a direction other than the

forward direction. The second dynamical equation is

ih∂

∂tbnk+1(t) =

ank+1k√

(nk + 1)!

∏λ =k

anλλ√nλ!〈g|H |e〉

∏λ =k

a†nλ

λ√nλ!

a†nk

k√nk!|0〉 a(t) , (9)

and the third dynamical equation is

ih∂

∂tbk′(t) = 〈0| ak′ 〈g|H |e〉 a(t) . (10)

5. Evaluation of the Matrix Elements Containing H in Eqs. (5),

(9) and (10)

After evaluating Eq. (5), it becomes:

ih∂

∂ta(t) =

∑k

√h

2ωkVc∗ke

−i(ωk−ωo)t√nk + 1 bnk+1(t) +

∑k′ =k

√h

2ωk′Vc∗ke

−i(ωk′−ωo)tbk′(t)

(11)


where

ck = −ieh

m

∫d3r Ψ∗g(r)e

i�k·�rtαk · ∇ Ψl(r) , (12)

is the atomic matrix element, and ωo is the resonant angular frequency 2πνo of the

two-level system. Similar processing is done to Eqs. (9) and (10). After ∂∂tbnk+1 and

∂∂tbk′ , are integrated from 0 to t, we have

bnk+1(t) =1

ih

√h

2ωkVc∗k√nk + 1

∫ t

0

dt′ei(ωk−ωo)t′a(t′) , (13)

and

bk′(t) =1

ih

√h

2ωkVck′

∫ t

0

dt′ei(ωk−ωo)t′a(t′) . (14)

After inserting Eqs. (13) and (14) into Eq. (11), the latter becomes

∂

∂ta(t) =− 1

h2

∑k

h

2ωkV|ck|2(nk + 1)e−i(ωk−ωo)t

∫ t

0

dt′ei(ωk−ωo)t′a(t′)

− 1

h2

∑k′ =k

h

2ωkV|ck′ |2e−i(ωk′−ωo)t

∫ t

0

dt′ei(ωk′−ωo)t′a(t′) .

(15)

Please note that the spontaneous emission in the forward direction appearing as “1”

in nk + 1 can be fused together with the spontaneous emission in all other directions, so

that Eq. (15) takes the form

∂

∂ta(t) =− 1

h2

∑k

h

2ωkV|ck|2nke

−i(ωk−ωo)t

∫ t

0


− 1

h2

∑k1

h

2ωk1V|ck|2e−i(ωk1

−ωo)t

∫ t

0

dt′ei(ωk1−ωo)t′a(t′) .

(16)

Eq. (16) is the fundamental integral equation governing the generalised Weisskopf-

Wigner Equation. Note that for nk = 0 the equation reverts to the conventional Weisskopf-

Wigner equation for emission in the absence of an incident beam. This is reassuring, albeit

not compelling. Eq. (16) must be solved for a(t).

6. Solution of Eq. (16)

In order to solve Eq. (16), we postulate the solution

a(t) = e−12Γt (17)

After inserting a(t) into Eq. (16), the time function on the r.h.s. becomes


e−Γ2t(1− e−i[(ωk−ωo)t+iΓ

2t])

i(ωk − ωo) + iΓ2

≡ e−Γ2tf(ωk) (18)

With this abbreviation Eq. (16) becomes

−Γ

2e−

Γ2t = − 1

h2

∑k

h

2ωkVnk|ck|2e−

Γ2tf(ωk)−

1

h2

∑k1

h

2ωk1V|ck1 |2e−

Γ2tf(ωk1) . (19)

Hence

−Γ

2= − 1

h2

∑k

h

2ωkVnk|ck|2f(ωk)−

1

h2

∑k1

h

2ωk1V|ck|2f(ωk1) . (20)

7. Transition of summation in k to a continuum in ω

In order to achieve a continuum in ω, the r.h.s of Eq. (20) is written as

−∫ ∞

0

1

h2

h

2ωV|ck|2n

ν2V

c3f(ω)dν −

∫ ∞

0

1

h2

h

2ωV

4πν2

c3|ck|2f(ω)dν =

−∫ ∞

0

I(ν)|ck|2

2(hν)2cf(ω)dν −

∫ ∞

0

2πν

hc3f(ω)dν .

(21)

where I(ν) is the spectral density of the intensity of the incident radiation beam. We

rewrite f(ω) as

f(ω) =1− e−i[(ω−ωo)t+iΓ

2t]

i[(ω − ωo) + iΓ2]

. (22)

We neglect Γ which is small compared with νo, and only retain the “i sin” term. The

real term only contributes to a phase shift, which is of no interest here.

On the other hand, we have the well-known relationship

t→∞ sin(ω − ωo)t

ω − ωo

=1

2δ(ν − νo) (23)

After inserting Eq. (23) into Eqs. (20, 21) and some manipulation one finds

Γ =

∫ ∞

0

I(ν)|ck|2

2(hν)2cδ(ν − νo)dν +

∫ ∞

0

|ck|2δ(ν − νo)dν

= I(νo)|ck|2

2(hνo)2c+

2πνohc3|ck|2

≡ γi + γ

(24)

Here, γ is the conventional natural line damping term, and is the inverse of the lifetime

of the excited state |e〉; whereas γi is the damping term due to induced emission.


8. Probability of Induced Emission and Spontaneous Emission

From Eq. (13) we can deduce that the probability of induced emission Wi is

Wi =∑k

1

h2

h

2ωkV|ck|2nk

∣∣∣∣∫ t

0


∣∣∣∣2 (25)

and the probability of spontaneous emission Ws is

Ws =∑k1

h

2ωk1V|ck1 |2nk

∣∣∣∣∫ t

0

dt′ei(ωk1−ωo)t′a(t′)

∣∣∣∣2 (26)

The time function in (25) and (26) yields, for large t

1

(ω − ωo)2 +14Γ2≡ [g] . (27)

After moving to the continuous ω, and integrating with respect to ν, one has

Wi +Ws =

∫ ∞

0

dνI(ν)|ck|2[g]2(hν)2c

+

∫ ∞

0

dν2πν

hc3|ck|2[g] . (28)

For (28) to be normalized

I(ν)

2(hν)2must become

I(νo)

2(hνo)2(29)

and

2πν must become 2πνo . (30)

Consequently, the probability Pi for induced emission per unit frequency, and the

probability Ps for spontaneous emission per unit frequency, both taken at a given fre-

quency ν is, respectively

Pi =γi

(ω − ωo)2 +14(γi + γ)2

(31)

Ps =γ

(ω − ωo)2 +14(γi + γ)2

(32)

9. Line Broadening

A glance at Eq. (32) reveals that the line has broadened by an amount of γi. The total

breadth of the line is Δω = γ + γi. From Eq. (24) it follows that the bandwidth Δν is

δν =δω

2π=

γ

2π+

πI(νo)

h2νo2c(33)

This simple result, arrived at after all of the above, could be “reasoned away” by

arguing that it is in agreement with the uncertainty principle, owing to the shortened


lifetime of the state |e〉 caused by the incident beam. This argument could not, however,

serve as a substitute for the development set forth above.

10. Frequency Domain Two-Level First Order Emission Exper-

iment

The preceding treatment describes a process in which a two-level system, originally in the

excited state, is illuminated by a resonant external beam. A photon is emitted and the

system transitions to the ground state, thereby terminating the process. It is important

to compare this process with the extensive literature on two-level systems as given by

refs. (6-8). It deals with the theory and experiments of resonance fluorescence and higher

order processes. In e.g. resonance fluorescence, assuming the initial state is in the ground

state, the incident beam excites the system into the upper level with subsequent emission

activated by the beam, and a return to the ground state, thereby terminating the process.

Consequently, such a process is a second order process.

Now, returning to the first order process just mentioned, how is such a process exper-

imentally implemented?

In a frequency domain two-level emission experiment, the population must be in-

verted, so that the upper level is largely over-populated with respect to the lower level.

Once a population inversion has been generated and maintained, no atoms are available,

statistically speaking, in the ground state to be excited into the upper level by the beam

as occurs in the resonance fluorescence experiment. Here the process includes an excited

upper level, and incident beam followed by and emission and transition to the ground

state, thereby terminating the process. Such an experimental set-up corresponds to a

frequency domain two-level first order emission process.

References

[1] D. Ito, K. Mori, E. Carriere, An example of dynamical systems with linear trajectory,Nuovo Cimento A 51 (1967) 1119-1121.


Spatially Homogeneous Cosmological Models inf (R, T ) Theory of Gravity

S. Chandel and Shri Ram∗†

Department of Applied Mathematics, Indian Institute of Technology,(Banaras Hindu University), Varanasi 221 005, India

Received 8 January 2014, Accepted 27 October 2014, Published 10 January 2015

Abstract: In this paper, we study a general spatially homogeneous and anisotropic Bianchi

space-time model in the presence of a perfect fluid within the framework of f(R, T ) gravity

theory. To obtain deterministic solutions of the field equations, we choose the average scale

factor of the model as a(t) =√tnet, which yields a time-dependent deceleration parameter

(DP). We find that solutions represent a class of models which generate a transition of the

universe from the early decelerating phase to recent accelerating phase in conformity with the

present day observations. For different positive values of n, we can generate a class of physically

viable perfect fluid models of the universe in f(R, T ) gravity theory. We discuss the stability

of the model by cosmological perturbations and find that it is completely stable for describing

a model of the universe. The physical and geometrical aspects of the cosmological model are

discussed.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Gravity ; Bianchi Space ; Variable Deceleration Parameter ; Perfect Fluid

PACS (2010): 98.80.-k; 95.36.+x; 98.80.Cq; 04.20.-q

1. Introduction

General Relativistic cosmological models provide a framework for the investigation of

evolution of the universe. Present cosmology is based on the Friedmann-Robertson-

walker(FRW) model. In this model, the universe is completely homogeneous and isotropic

which is in good agreement with the observational data about the large scale structure

of the universe. The adequacy of a FRW model for describing the present state of the

universe is no basis for expecting that it is equally suitable for describing the early stages

of evolution of the universe. There are theoretical arguments (Misner [1], Chimento [2])

∗ Corresponding Author† Email:[email protected]


and recent experimental data of the cosmic microwave background radiation which sup-

port the existence of an anisotropic phase that approaches an isotropic one (Land and

Maguejo [3]). This stipulates search for anisotropic cosmologically acceptable models of

the universe at least in its early stages of evolution.

It is believed that the early universe evolved through some phase transitions thereby

yielding a vacuum energy density which is at present is at least 118 orders of magnitudes

smaller than in plank time (Weinberg [4]). Such a discrepancy between theoretical ex-

pectations and empirical observations constitutes a fundamental problem in the interface

uniting astrophysics, particles physics and cosmology is the cosmological constant prob-

lem. The recent cosmological observations obtained by type Ia supernova (Riess et al.

[5, 6], Perlmutter et al. [7], Tonry et al. [8]), large scale structure (Tegmark et al. [9],

Seljak et al. [10], Percival et al. [11], Kamatsu et al. [12]), baryon oscillation (Eisenstein

et al. [13]) and weak lens (Jain and Tayler [14]) have suggested that the expansion of

the universe is accelerating. These observations seem to change the entire picture of our

matter filled universe. It has been observed that a fluid known as dark energy with large

negative pressure is responsible for this acceleration. Many dark energy models have

been proposed to explain the cosmic accelerated expansion (Copeland et al. [15]). The

cosmological constant Λ, responsible for cosmic accelerated expansion, is the simplest

candidate of dark energy (Sahni and Starobinsky [16], Padmnabhan [17]).

In recent years, modifications of general relativity are attracting more and more atten-

tion to explain the present time acceleration of the universe and dark energy. Noteworthy

amongst them is f(R) theory of gravity formulated by Nojiri and Odintsov [18].This the-

ory provides a natural gravitational alternative to dark energy. It has been suggested

that cosmic acceleration can be achieved by replacing the Einstein-Hilbert action of gen-

eral relativity with a general function of Ricci scalar f(R). The f(R) theory of gravity

has been shown equivalent to scalar-tensor theory of gravity that is incompatible with

solar system tests of general relativity as long as the scalar field propagates over solar

system scales. The f(R) model is a modified gravity model constructed by replacing the

gravitational Lagrangian with a general function of the Ricci scalar R. It provides a very

natural unification of early -time inflation and late-time acceleration. It describes the

transition from deceleration to acceleration in the evolution of the universe(Nojiri and

Odintsov [19,20]). Sharif and Shamir [21] have studied the vacuum solutions of Bianchi

types-I and V space-times in the framework of f(R) gravity by using the variation law

of Hubble parameter. Shamir [22] studied the exact vacuum solutions of Bianchi Type I,

III and Kantowski-Sachs space-times in modified f(R) gravity theory. Recently, Singh

and Singh [23] have obtained an Bianchi type-I space time in f(R) theory of gravity in

the presence of a perfect fluid distribution and have shown that the function f(R) is

completely stable for describing the decelerating phase of the universe.

Harko et al.[24] developed a generalized f(R, T ) gravity theory where the gravita-

tional Lagrangian is given in terms of any arbitrary function of the Ricci scalar R and

the trace T of the energy-momentum tensor Tij and obtained field equations in metric

formalism from Einstein- Hilbert type variational principle. They have presented several


models corresponding to three explicit forms of the function f(R, T ). Adhav [25] has

obtained LRS Bianchi type-I cosmological model in the presence of a perfect fluid in

f(R, T ) gravity theory. Reddy et al. [26] have also presented a five dimensional Kaluza-

Klein cosmological model with perfect fluid. Also, Reddy et al. [27] have investigated a

Bianchi type-III dark energy cosmological model in f(R, T ) theory in the presence of a

perfect fluid with the assumption that the shear scalar is proportional to the expansion

scalar. Chaubey and Shukla [28] have studied a new class of Bianchi types III, V V I0and V Ih cosmological models filled with a perfect fluid in f(R, T ) gravity theory. Shri

Ram et al. [29] have presented a class of non-singular bouncing cosmological models for

a general Bianchi spaces filled with perfect fluid in this theory. Chandel and Shri Ram

[30] have derived an algorithm for generating new class of solutions of Einstein’s field

equations with perfect fluid of Bianchi type III.

Motivated by these works, we investigate, in this paper a general Bianchi space-time

model filled with a perfect fluid in f(R, T ) gravity theory. Exact solutions of the field

equations are obtained explicitly by choosing the average factor a(t) =√tnet, where n is a

positive constant.The paper is organized as follows: In Sect. 2, we present the space-time

metric and the field equations for a perfect fluid distribution in f(R, T ) gravity theory for

the particular form of f(R, T ) = R+ 2λT , where λ is a constant. We obtain a new class

of exact solutions of the field equations in Sect. 3. In Sect. 4, we discuss some physical

and dynamical properties of the model. In sect.5 we study the stability of the solution

by invoking a cosmological perturbative approach. Finally, conclusions are summarized

in the last Sect.6.

2. The Metric and Field Equations

We consider the diagonal form of the metric of general class of Bianchi cosmological

models given by

ds2 = dt2 − a21dx2 − a22e

−2xdy2 − a23e−2mxdz2 (1)

where a1(t), a2(t), a3(t) are cosmic scale factors and m is a constant. The metric (1)

corresponds to a Bianchi type-III model for m = 0, type-V model for m = 1, type-VI0model for m = −1 and type- VIh model for all other m = h− 1.

The field equations in f(R, T ) theory of gravity for the function f(R, T ) = R+2f(T )

when the matter source is perfect fluid and given by (Harko et al. [24]);

Rij −1

2Rgij = 8πTij + 2f ′(T )Tij + [2pf ′(T ) + f(T )]gij (2)

where the prime indicates differentiation with respect to argument. The matter tensor

for perfect fluid is

Tij = (ρ+ p)vivj − pgij (3)

where ρ is the stress energy density of matter, p is the pressure and vi is the 4-velocity

vector. Here we take

f(T ) = λT (4)


where λ is a constant.

Now choosing comoving coordinates, the field equations (2), with the help of equation

(3) and (4) for the metric (1), can be written as:

a1a2a1a2

+a2a3a2a3

+a3a1a3a1

− m2 +m+ 1

a21= λp− (8π + 3λ)ρ, (5)

a2a2

+a3a3

+a2a3a2a3

− m

a21= (8π + 3λ)p− λρ, (6)

a1a1

+a3a3

+a1a3a1a3

− m2

a21= (8π + 3λ)p− λρ, (7)

a1a1

+a2a2

+a1a2a1a2

− 1

a21= (8π + 3λ)p− λρ, (8)

(m+ 1)a1a1− a2

a2−m

a3a3

= 0 (9)

where an over dot stands for ordinary derivative with respect to t.

Equations (5)− (9) are highly nonlinear differential equations with five unknowns a1,

a2, a3, ρ and p. To obtain deterministic solutions of these equations, we shall need two

extra conditions based on physical and mathematical grounds.

The average scale factor a(t) is defined by

a(t) = (a1a2a3)13 . (10)

The spatial volume is given by

V = a3. (11)

The mean Hubble parameter H for the metric (1)is given by

H =a

a=

1

3(H1 +H2 +H3) (12)

where H1, H2 and H3 are the directional Hubble parameters in the x, y and z directions

respectively defined as

H1 =a1a1

, H2 =a2a2

, H3 =a3a3

. (13)

The dynamical scalars such as expansion scalar (θ), shear scalar (σ) and the mean

anisotropy parameter (Am) defined as usual

θ = 3H =a1a1

+a2a2

+a3a3

, (14)

σ2 =1

2

(a21a21

+a22a22

+a23a23

)− 1

6θ2, (15)

Am =1

3

3∑i=1

(ΔHi

H

)2

(16)


where ΔHi = Hi −H, i=1,2,3.

An observational quantity is the deceleration parameter q, which is defined as

q = −V V

V 2= − a

aH2. (17)

For constructing physically relevant cosmological models, the Hubble parameter and the

deceleration parameter (DP) play important roles. It has been the common practice to

use a constant DP. Berman [31], Berman and Gomide [32] proposed a law of variation of

Hubble parameter in FRW model that yields a constant value of DP, which subsequently

leads to power-law and exponential forms of the average scale factor. The recent obser-

vations of SNe Ia (Riess et al. [5], Perlmutter et al. [7] etc.) indicate that universe is

accelerating at present while there was decelerated expansion in the past, and the universe

undergoes transition from decelerated expansion to accelerated expansion at present. So,

in general, DP is not a constant but rather a function of time. Some authors proposed

the time-dependent forms of the DP and then derived different forms of the scale factor

(Abdussattar and Prajapati [33], Akarsu and Dereli [34] etc.). However, some authors

first choose the average scale factor and then derive the time-dependent DP.

Now, we take the following ansatz for the average scale factor where the increase in

term of time evolution is

a(t) =√tnet, (18)

n being a positive constant. Pradhan and Amirhashchi [35] and Saha et al. [36] exam-

ined this form of the scale factor to study accelerating dark energy models in Bianchi

type−V space-time and a two-fluid scenario for dark energy models in an FRW universe

respectively. Pradhan [37] assumed this form of a(t) to discuss some features of Bianchi

type-V I0 models in the presence of a perfect fluid that has an anisotropic equation of

state parameter in general relativity. This choice of average scale factor yields a time- de-

pendent deceleration parameter such that before the DE era, the corresponding solution

gives the inflation and radiation/matter dominated era, with subsequent transition from

deceleration to acceleration. For n = 0, this choice of scale factor gives an exponential law

of variation for the scale factor. The choice (18) of the average scale factor is physically

acceptable. From equations (17) and (18), the time-dependent q(t) is obtained as

q(t) =2n

(n+ t)2− 1. (19)

From equation (19), it is clear that q > 0 for t <√2n−n and q < 0 for t >

√2n−n. For

0 < n < 2, the model is evolving from deceleration phase to acceleration phase.Recent

observations of SNe Ia have shown that the present universe is accelerating and q lies

in the range −1 < q < 0. Thus, the model has accelerated expansion at present epoch

which in consistent with recent observations of Type Ia supernova .It follows that in our

derived model, one can choose the value of DP consistent with the observation.Figure 1

depicts the deceleration parameter (q) versus time which gives the behavior of q from

decelerating to accelerating phase for different values of n.


3. Cosmological Solutions

We now obtain physically realistic cosmological models to describe the decelerating and

accelerating phases of the universe. We assume that a3 = V b where b be any constant

number. Then, from equations (9), (10) and (18), we obtain the explicit solutions of a1,

a2 and a3 as follows:

a1(t) = (tnet)3(1+mb−b)

2(m+2) , (20)

a2(t) = (tnet)3(1+m−b−2mb)

2(m+2) , (21)

a3(t) = (tnet)3b2 . (22)

Thus,the metric (1) can be written in the form

ds2 = dt2 − (tnet)3(1+mb−b)

(m+2) dx2 − (tnet)3(1+m−b−2mb)

(m+2) e−2xdy2 − (tnet)3be−2mxdz2. (23)

4. Physical and Geometrical Behaviors of the Model

The Hubble parameters H1, H2 and H3 have values given by

H1 =3(1− b+mb)

2(m+ 2)

(1 +

n

t

), (24)

H2 =3(1 +m− b− 2mb)

2(m+ 2)

(1 +

n

t

), (25)

H3 =3b

2

(1 +

n

t

). (26)

From (12), the average Hubble’s parameter H has the value given by

H =1

2

(1 +

n

t

). (27)

The dynamical scalars σ, θ and anisotropy parameter Am are given by

σ2 =3[18m2b2 + 18mb2 − 12m2b− 12mb+ 2m2 + 18b2 + 2m− 12b+ 2]

8(m+ 2)2

(1 +

n

t

)2

,

(28)

θ =3

2

(1 +

n

t

), (29)

Am =2[9m2b2 + 9mb2 − 6m2b− 6mb+m2 + 9b2 +m− 6b+ 1]

(m+ 2)2. (30)

The expressions for energy density and isotropic pressure of the model are obtained as

ρ =1

8(λ2 + 6πλ+ 8π2)

[3

4t2(m+ 2)2{3(n+ t)2(λA1 − 8πA2 − 3λA2)

+2nλ(m2b+mb−m2 − 3m− 2b− 2)}+{(8π + 3λ)(m2 +m+ 1)−mλ}(ettn)

−3(1+mb−b)(m+2)

], (31)


p =1

8(λ2 + 6πλ+ 8π2)

[3

4t2(m+ 2)2{3(n+ t)2(8πA1 + 3λA1 − λA2)

+(16nπ + 6nλ)(m2b+mb−m2 − 3m− 2b− 2)}+{m(8π + 2λ)− (m2 + 1)λ}(ettn)

−3(1+mb−b)(m+2)

](32)

where

A1 = 3b2(m2 +m+ 1)− 3b(m2 +m) + (m2 + 2m+ 1),

A2 = −3b2(m2 +m+ 1) + 2b(m2 +m+ 1) + (1 +m).

The scalar curvature R for the model is given by

R = 2

[a1a1

+a2a2

+a3a3

+a1a2a1a2

+a2a3a2a3

+a3a1a3a1

− (m2 +m+ 1)

a21

]

=9(n+ t)2

2t2(m+ 2)2{(1+mb−b)2+(1+m−b−2mb)2+b2(m+2)2+(1+mb−b)(1+m−b−2mb)

+b(m+ 2)(2−mb− 2b+m)} − 3n

t2− (m2 +m+ 1)(tnet)

−3(1+mb−b)(m+2) . (33)

From the above results, we observe that the spatial volume is zero at t = 0 and it

increases with increase of t. The expansion scalar is infinite at t = 0. These show that

the universe starts evolving with zero volume at t = 0 and expands with cosmic time t.

All the three directional Hubble’s parameters and the average Hubble parameter diverge

at t = 0. These indicate that the model has a point-type singularity at t = 0. All the

other physical parameters p, ρ and σ diverge at t = 0. As t → ∞, the scale factors and

volume become infinite where ρ, p approach to zero. Since the anisotropy parameter Am

is constant throughout the passage of time, the model is anisotropic for all time. We also

find that scaler curvature R is positive throughout the whole evolution of the universe

and R→ 0 as t→∞ and R→∞ when t→ 0 showing initial singularity at t = 0.

It is interesting to note that in this model for n = 0, we get q = −1, indicating that

the universe is accelerating which is consistent with the present day observations that

universe is undergoing the accelerated expansion.

For the physical acceptability of the solutions, firstly it is required that the velocity of

the sound vs = dpdρ

should be less than velocity of light c. As we are working in the

gravitational units with unit speed of light, the velocity of sound must exist within the

range 0 ≤ vs ≤ 1. Here the speed of sound is obtained as

vs =P (t)

Q(t), (34)

where

P (t) = 18(8πA1 + 3λA1 − λA2)(n2 + nt) + (16nπ + 6nλ)(m2b+mb−m2 − 3m− 2b− 2)

+12(nt2 + t3)(m+ 2)(1 +mb− b){m(8π + 2λ)− (m2 + 1)λ}(tnet)−3(1+mb−b)

(m+2) (35)


and

Q(t) = 18(λA1 − 8πA2 − 3λA2)(n2 + nt) + 12nλ(m2b+mb−m2 − 3m− 2b− 2)

+12(nt2 + t3)(m+ 2)(1 +mb− b){m(8π + 3λ)(m2 +m+ 1)−mλ}(tnet)−3(1+mb−b)

(m+2) (36)

It is clear from fig. (6) that vs < 1 throughout the evolution of the universe.

Secondly, the weak energy conditions (WEC) and dominant energy conditions (DEC)

are given by (i)ρ ≥ 0, (ii)ρ+p ≥ 0 and (iii)ρ−p ≥ 0.The strong energy condition

(SEC) is given by ρ+ 3p ≥ 0.

From figures (3), (7), (8) and (9), it can be seen that these energy conditions are identically

satisfied throughout the evolution of universe.

From the above discussion we find that our model is physically acceptable.

5. Stability of the Model

In this section we study the stability of the solution by invoking a perturbative approach.

The perturbations of the fields in a gravitational system against the background evolu-

tionary solution should be checked to ensure the stability of the exact or approximate

background solution (Chan and Kao [38]). We follow Saha et al. [36] and Pradhan [37]

to study the stability of the background solution with respect to the perturbations of the

metric. Perturbations are considered for three expansion factors ai via.,

ai → aBi+ δai = aBi

(1 + δbi). (37)

We focus our attentions on the variables δbi instead of δai. Accordingly, the perturbations

of the volume scalar VB = Π3i=1ai, directional Hubble factors θi =

aiai, the mean Hubble

parameter θ = Σ3i=1

θi3= V

3Vare shown as follows:

V → VB + VBΣiδbi, θi → θBi+ Σiδbi, θ → θB +

1

3Σiδbi (38)

It can be shown that the metric perturbations δbi, to the linear order in δbi, obey the

following equations

Σiδbi + 2ΣθBiδbi = 0, (39)

δbi +VB

VB

δbi + ΣjδbjθBi= 0, (40)

Σδbi = 0. (41)

From Equations (39), (40) and (41), it can easily be seen that

δbi +VB

VB

δbi = 0, (42)


where VB is the background volume scalar. Here

VB = t3n2 e

3t2 (43)

Substituting for VB in equation (42) and integrating, we obtain

δbi = cit−3n4 e

−3t4 WittakerM

(−3n4

,−3n4

+1

2,3t

2

), (44)

where ci is a constant of integration. Therefore, the actual fluctuations, for each expansion

factor δai = aBiδbi,are given by

δai = cit−3n4 e

−3t4 WittakerM

(−3n4

,−3n4

+1

2,3t

2

)(45)

From Equation (45), we observe that for n ≥ 1, δai approaches zero. Consequently, the

background solution is stable against the perturbation of the graviton field.

6. Conclusion

In this paper, we have studied a general spatially homogeneous and anisotropic Bianchi

space-time model in f(R, T ) theory of gravity in the presence of a perfect fluid source

having initial singularity at t = 0. Einstein’s field equations have been solved by consid-

ering the average scale factor a(t) =√tnet, which yields a time-dependent deceleration

parameter. The derived model represents expanding, shearing and non-rotating universe

which does not tend isotropy for all large time t. We have discussed the physical and ge-

ometrical behaviors of the cosmological model. The variation of the physical parameters

have been shown graphically. It is shown that model starts expanding from a decelerating

phase to an accelerating phase. By cosmological perturbation method, we have shown

that our model is stable. Also, the cosmological model is physically acceptable in con-

cordance with the fulfillment of energy conditions WEC, DEC and SEC. Exact solutions

presented in this paper may be useful for better understanding the characteristics in the

evolution of the universe within the framework of f(R, T ) theory of gravitation.

Acknowledgment

S. Chandel is extremely thankful to CSIR, India for providing financial support.

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0 1 2 3 4 5 6 7 8 9 101

0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

cosmic time (t)

dece

lera

tion

para

met

er (q

)

n1=0.5n2=1n3=1.5

Figure 1 The plot of deceleration parameter q vs. cosmic time t

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7

8

9

10

cosmic time (t)

Shea

r sca

lar (

σ)

Figure 2 The plot of shear scalar σ vs. cosmic time t for m=1.5, n= 0.5, b=1.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

18

20

cosmic time (t)

mat

ter d

ensit

y (ρ

)

Figure 3 The plot of matter densityρ vs cosmic time t for m=1.5, b=1.5, n=0.5 and λ = 0.05


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

8

9

10

cosmic time (t)

pres

urre

(p)

Figure 4 The plot of pressure p vs. cosmic time t for m=1.5, b=1.5, n=0.5 and λ=0.05

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

2000

4000

6000

8000

10000

12000

cosmic time (t)

Ric

ci s

cala

r(R)

Figure 5 The plot of Ricci scaler R vs. cosmic time t for m=1.5, b=1.5, n=0.5 and λ=0.05

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cosmic time (t)

v s

Figure 6 The plot of speed velocity vs vs. cosmic time t for m=1.5, b=1.5, n=0.5 and λ=0.05


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

18

20

cosmic time (t)

ρ+p

Figure 7 The plot of WEC vs. cosmic time t for m=1.5, b=1.5, n=0.5 and λ=0.05

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

18

20

cosmic time (t)

ρp

Figure 8 The plot of DEC vs. cosmic time t for m=1.5, b=1.5, n=0.5 and λ=0.05

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

18

20

cosmic time (t)

ρ+3p

Figure 9 The plot of SEC vs. cosmic time t for m=1.5, b=1.5, n=0.5 and λ=0.05


Relativistic Pentametric Coordinates fromRelativistic Localizing Systems and the Projective

Geometry of the Spacetime Manifold

Jacques L. Rubin∗

Universite de Nice–Sophia Antipolis, UFR SciencesInstitut du Non-Lineaire de Nice, UMR7335

1361 route des Lucioles, F-06560 Valbonne, France

Received 5 November 2014, Accepted 20 December 2014, Published 10 September 2015

Abstract: Relativistic pentametric coordinates supplied by relativistic auto-locating

positioning systems made up of four satellites supplemented by a fifth one are defined in

addition to the well-known emission and reception coordinates. Such a constellation of five

satellites defines a so-called relativistic localizing system. The determination of such systems

is motivated by the need to not only locate (within a grid) users utilizing receivers but, more

generally, to localize any spacetime event. The angles measured on the celestial spheres of the

five satellites enter into the definition. Therefore, there are, up to scalings, intrinsic physical

coordinates related to the underlying conformal structure of spacetime. Moreover, they indicate

that spacetime must be endowed everywhere with a local projective geometry characteristic of

a so-called generalized Cartan space locally modeled on four-dimensional, real projective space.

The particular process of localization providing the relativistic pentametric coordinates is based,

in a way, on an enhanced notion of parallax in space and time generalizing the usual parallax

restricted to space only.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Emission Coordinates; Generalized Cartan Space; Projective Geometry; Relativistic

Positioning Systems

PACS (2010): 04.20.Cv; 02.40.Dr; 04.50.-h; 04.20.Gz

1. A Protocol Implemented by Users to Localize Events

Almost simultaneously, Bahder [1], Blagojevic et al. [2], Coll [3] and Rovelli [4] laid, from

different approaches, the foundations of the relativistic positioning systems (RPS) and,



in particular, following Coll’s terminology,2 “primary” RPSs, i.e., RPSs which satisfy the

three following criteria: they are 1) “generic,” i.e., the system of coordinates they provide

must exist independently of the spacetime geometry for each given class of spacetime, 2)

they are “free,” i.e., their structures do not need the knowledge of the gravitational field,

and 3) they are “immediate,” i.e., the users know their positions without delay at the

instant they receive the four “time stamps τα” sent by the four emitting satellites of the

RPS satellite constellation.

Among this set of primary RPSs, there exists the sub-class of the so-called “auto-

locating RPSs,” i.e., those RPSs in which each satellite broadcasts its own time stamp

but also the time stamps it receives from its neighboring satellites. The SYPOR system

(“SYsteme de POsitionnement Relativiste”), developed by Coll and Tarantola [5], belongs

to this category, but we ask, more generally, for an enhanced RPS and a supplementary

protocol to allow any located user to localize any event in the spacetime region covered

by this particular enhanced RPS.

We make the following strict distinction between location and localization. To locate

an event, a protocol (of location) is needed to build a coordinate grid, and then, to position

this event in this grid once the coordinates of this event are known. To localize an event, a

protocol (of localization) is needed that effectively obtains the coordinates of the event to

be then, only, located in a given coordinate grid. Auto-locating positioning systems only

allow building the coordinate grids from the users’ knowledge of the satellites’ worldlines,

and then, to position the users in these grids, but they do not supply the coordinates of

events. Upstream, non auto-locating systems only allow knowing the users’ coordinates

but without location and, more generally, without localization of events in the users’

surroundings.

Furthermore, downstream, the sub-class of the so-called “autonomous systems,” con-

tained in the sub-class of auto-locating systems, includes those auto-locating systems

allowing, from “autonomous data,” the users to draw (from Coll’s definition [3]) the

satellites’ worldlines in the spacetime where these users are living. Beside, we consider

rather another sub-class contained in the sub-class of auto-locating systems, namely, the

sub-class of “relativistic localizing systems” of which the satellites broadcast also, in ad-

dition to their time stamps, data to localize events. In the present paper, we define such a

relativistic localizing system made up of four satellites constituting an auto-locating sys-

tem supplemented by an ancillary fifth satellite providing data (actually, supplementary

time stamps) to localize events. These five satellites can define five different auto-locating

systems connected by ten changes of coordinate grids but only one of the five is required

to operate.

Besides, the goal for seeking such an enhanced RPS, viz., a relativistic localizing

system, provided with a tracking, localizing protocol is also to find a process to break

the underlying arbitrariness in scaling that is due, in a way, to the arbitrary choice

2 Throughout the present paper, we use terms such as primary, local, intrinsic, location system, reference

system, positioning system, auto-locating system, autonomous system or data, laws of physics, emission

and reception coordinates, etc., as defined in [3]


of time parameterizations of the satellites’ worldlines. Indeed, the satellites of a given

RPS satellite constellation can broadcast time stamps defined not only by their own

proper times given by on-board clocks, but, more generally, by any “numbered events

generator” (such as proper time clocks) not necessarily synchronized with their proper

times. Thus, any time parameterization can be defined, in particular, affinely from any

other given time parameterization. In other words, the links between, on the one hand,

the conformal structure of spacetime with, behind it, the time parameterization scalings

and, on the other hand, the time parameterizations of the satellites’ worldlines must be

questioned, technologically unveiled, and then fixed by a particular enhanced RPS. By

“fixed,” we mean that the enhanced RPS should be “sensitive” to the conformal structure

of spacetime and then, in particular, sensitive to any scaling change of the Lorentzian

metric defined on spacetime. But, it should also provide a univocal linkage with the

conformal structure and, in addition, this linkage must be unaffected by the changes in

the time parameterization along the satellites’ worldlines.

Furthermore, the conformal structure of spacetime can be deduced from the causal

axiomatics as shown, historically, for instance, by Ehlers et al. [6], Hawking et al. [7],

Kronheimer et al. [8], Malament [9], or Woodhouse [10]. As a consequence, the chrono-

logical order, i.e., the history in spacetime, is not affected by scalings of the Lorentzian

metric. Hence, the changes of coordinates in spacetime which are compatible with the

chronological order transform the Lorentzian metric up to scalings, i.e., up to functional

conformal factors. Then, the Lorentzian metric is said to be “conformally equivariant.”

As a consequence of this conformal structure, only the generators of the null cones and

not their constitutive sets of points (events) are then the intrinsic, hybrid, and causal ob-

jets intertwining physics and geometry that should be used in the geometrical statements

of the laws of physics. And then the events should be only considered as the intersection

points of congruences of such generators.

Hence, intrinsic (physical) observables and “genuine, causal processes” such as the

location protocols must be unaffected by metric scalings, i.e., metric scalings are not in-

trinsic. Also, coordinate systems such as emission or reception coordinates which can be

subjected to scalings due to changes of time parameterization of, for instance, the satel-

lites’ worldlines, are then also not intrinsic. Therefore, we must, somehow, discriminate

in any given coordinate system its intrinsic part from its “scaling sensitive,” non-intrinsic

part. Actually, an auto-locating system cannot provide such a discrimination, as will be

shown in what follows. A fifth satellite must be attached to this positioning system. Us-

ing a metaphor, this fifth satellite is a sort of cursor indicating the scale of the positioning

system from which an intrinsic part alone can be excerpted. Moreover, this intrinsic part

cannot provide by itself a complete, functional coordinate system.

Angles on a celestial sphere are such intrinsic observables compatible with the con-

formal structure of spacetime. But, their evaluations from a causal (intrinsic) process

of measurement need a particular protocol if an auto-locating system only is involved.

Such a protocol is presented in the next sections using emission coordinates with a fifth

satellite. In return, we obtain, from the emission coordinates provided by this particular


five-satellite constellation, a local relativistic localizing system defined with new coordi-

nates, namely, the “relativistic pentametric coordinates.” As a result, we gain much more

than a “mere” auto-locating system with a fifth satellite since not only location is then

available but localization, in addition, becomes available. Also, a “pentametric grid” is

obtained and linked to the emission grid provided by the auto-locating sub-system. Fur-

thermore, it appears that spacetime must be embedded in a five-dimensional, intermedi-

ate manifold in which spacetime must be considered locally as a four-dimensional, real

projective space, i.e., spacetime is then a generalized Cartan space “modeled” on a pro-

jective space. Thus, we obtain a local, projective description of the spacetime geometry.

Nevertheless, we have, in return, access to the “genuine” Riemannian four-dimensional

spacetime structure without the need for any autonomous sub-system unless considering

that the five-satellite constellation constitutes a sort of “enhanced autonomous system.”

This kind of protocol can be called a relativistic pentametric protocol [3].

In the next sections, we present such a complete protocol. It has two major flaws

which we nevertheless think are unavoidable: its implementation is complicated and may

be immediate only in some very particular situations or regions covered by the RPS

depending on the localized events. In full generality, obviously, it cannot be immediate,

because the satellites of any constellation must “wait” for the signals coming from the

source event which will be later localized. Nevertheless, it really breaks the scaling

arbitrariness and provides access to the spacetimeM as expected.

The results presented in the next sections are given when increasing successively

the dimension of spacetime. Thus, in Section 2., the relativistic localizing protocol is

applied in a two-dimensional spacetime. In this particular case only, the relativistic

localizing system essentially reduces to the relativistic positioning system itself with a

relativistic stereometric protocol. In Section 3., all of the basic grounds and principles

of the localizing process with a relativistic quadrometric protocol are presented in a

three-dimensional spacetime. Then, they are naturally generalized in Section 4. to a

four-dimensional spacetime and a relativistic pentametric protocol before ending with

the conclusion in Section 5..

2. The Protocol of Localization in a (1 + 1)-dimensional Space-

time M

In this situation, the protocol is rather simple. We recall, first, the principles for relativis-

tic positioning with a two-dimensional auto-locating system. We consider two emitters,

namely, E1 and E2 and a user U with their respective (time-like) worldlines W1, W2 and

WU . The two emitters broadcast emission coordinates which are two time stamps τ1and τ2 generated by on-board clocks, and then the two-dimensional emission grid can be

constructed from this RPS.

More precisely, the principles of positioning and construction of the emission grid

are the following [1, 11, 12, 4, 13, 14]. Firstly, the two emitters not only broadcast

their own time stamps, viz., τi for Ei (i = 1, 2), but also the time stamps they receive


from the other. Hence, each time stamp received by an emitter along his worldline is

immediately broadcast again with the time stamp of reception. For instance, from Fig. 1,

the emitter E1 receives at E1 on W1, the time stamp τ−2 from E2, and also, simultaneously,

it broadcast τ−2 with its own time stamp τ+1 ; both received by the user at U2. Secondly,

the user receives at this event U2 four time stamps: two emitted at E1 by E1, viz., (τ+1 , τ−2 ),and two others emitted at E2 by E2, viz., (τ−1 , τ+2 ). As a result, the user can deduce his

own spacetime position (τ+1 , τ+2 ) at the event U2, but also, the spacetime positions of the

two emitters at E1 and E2, viz., (τ+1 , τ

−2 ) and (τ−1 , τ

+2 ) respectively. Then, the user can

continuously deduce his spacetime position in the emission grid all along his worldline

and the worldlines of the two emitters as well (Fig. 2). Consequently, if the trajectories

of the emitters are known in a given system of reference, then the trajectory and the

positions of the user are also known in this reference frame.

This process of positioning can be easily generalized in a four-dimensional spacetime.

In this case, there are four emitters. Each emitter broadcast four time stamps of which

three of them are received from the three other emitters. As a result, the user receives

16 times stamps emitted at four events on the worldlines of the emitters. Then, if the

system of reference is the CTRS (Conventional Terrestrial Reference System) or the ITRF

(International Terrestrial Reference Frame) for instance, the user can know his position

on earth, and the positions of the emitters as well with respect to the Earth surface or

geoid.

Now, the process of localization of an event e is the following. From a system of

echoes (Fig. 1), the user at the events U1 ∈ WU and U2 ∈ WU receives respectively four

numbers: (τ •1 , τ•2 ) from E• and (τ−1 , τ

+2 ) from E2. In addition, from this RPS, the user

can also know in this emission grid the two events E• and E2 at which the two emitters

sent these four time stamps viz, E• ≡ (τ •1 , τ•2 ) and E2 ≡ (τ−1 , τ

+2 ).

Then, let e be an event in the domain covered by the RPS (mainly between the two

worldlines W1 and W2). This event can be at the intersection point of the two light rays

received by E1 and E2 at the events E• and E2 (see Fig. 1). Hence, the position of e in

the emission grid is easily deduced by U if 1) U records (τ •1 , τ•2 ) and (τ−1 , τ

+2 ) along WU ,

and 2) a physical identifier for e is added at E• and E2 to each pair of time stamps to

be matched by U . Thus, in the particular case of two dimensions, the emission grid is

identified with the stereometric grid and, as a consequence, the stereometric coordinates

are also identified with the emission coordinates.


time M Modeled on RP 3

In this case, the complexity of the protocol of localization increases “dramatically.” Again,

we consider three emitters E , E and E transmiting three sets of time stamps denoted,

respectively, by τ , τ and τ . Then, the emission grid is the Euclidean space R3 with the

system of Cartesian emission coordinates (τ, τ , τ). Then, we consider, first, the system

of echoes from E to the user U . This system can be outlined as indicated in Fig. 3.


Fig. 1 The system of echoes in a two-dimensional spacetime.

Fig. 2 The two-dimensional emission/stereometric grid.

In this figure, the four past null cones of the four events E, E ′, E ′ and U are repre-

sented and the time axis is oriented vertically from the bottom to the top of the figure.

Also, we denote by UW , W , W and W the worldlines of, respectively, the user U and the

emitters E , E and E .Then, the user receives at the reception event U ∈ UW seven time stamps sent by E


Fig. 3 The system of echoes with four past null cones.

and emitted at the event of emission E ∈ W : (τ1, (ττ1 , τ

τ2 , τ

τ3 ), (τ

τ1 , τ

τ2 , τ

τ3 )). In addition,

the emitter E receives at E six time stamps from the other two emitters E and E , viz,p˜E′ ≡ (τ τ1 , τ

τ2 , τ

τ3 ) emitted at E ′ ∈ W from E , and p

E′ ≡ (τ τ1 , ττ2 , τ

τ3 ) emitted at E ′ ∈ W

from E . Actually, p˜E′ and p

E′ are the 3-positions of, respectively, E ′ and E ′ in the three-

dimensional emission grid. Moreover, E sends at E the time stamp τ1 received at U by

the user U .In addition, two of the three time stamps received at E ′ are sent by E at E ′: τ τ1 , and

by E at E ′′: τ τ3 ; and we have a similar situation for E ′ (see Fig. 3).

The user can then deduce the 3-position pE of the event E in the emission grid:

pE ≡ (τ1, τ2, τ3) ≡ (τ1, ττ2 , τ

τ3 ), and the two 3-positions p

˜E′ and pE′ of the two events E ′

and E ′ respectively. In addition, τ τ2 is emitted by E at E ′, and τ τ3 is emitted by E at E ′.Also, these two 3-positions are obtained from four time stamps emitted from four events,

namely, E ′ and E ′′ for E ′, and E ′′ and E ′′ for E ′ (see Fig. 3).

Actually, the user receives 3× 7 time stamps, i.e., three sets of data, namely, dE, d ˜E

and dE such that

• dE is received at U ∈ UW : dE ≡ (τ1, (ττ1 , τ

τ2 , τ

τ3 ), (τ

τ1 , τ

τ2 , τ

τ3 ), idE) ,


• d˜E is received at U ∈ UW : d

˜E ≡ (τ2, (ττ1 , τ

τ2 , τ

τ3 ), (τ

τ1 , τ

τ2 , τ

τ3 ), id˜E) ,

• dE is received at U ∈ UW : d

E ≡ (τ3, (ττ1 , τ

τ2 , τ

τ3 ), (τ

τ1 , τ

τ2 , τ

τ3 ), idE) ,

where idE , id˜E and idE are identifiers of the emitters (see Fig. 4).

Fig. 4 The sets of time stamps defining, with the three identifiers idE , id˜E and idE , the three

data dE , d ˜E and dE received and recorded by the user U at, respectively, U , U and U on the

worldline UW of U .

From now on, we consider only the sets of events represented in Fig. 3.

3.1 A first procedure of localization without quadrometric grid

The user can, then, also deduce three future light-like vectors generating the future null

cone at E, namely, kE, kE and kUE , such that

kE ≡ pE − pE′ ≡

−−→EPE ,

kE ≡ pE − p˜E′ ≡

−−→EPE ,

kUE ≡ pU − pE ≡

−−−→EPU

E ,

where PUE ≡ U and pU is the 3-position of U in the emission grid. The three endpoints PE,

PE and PUE define an affine plane AE in the emission grid. Then, a unique circumcircle

in AE contains these three endpoints from which the unique circumcenter C ∈ AE can


be deduced by standard formulas.3

Now, let e be an event to be localized in the emission grid (in this first procedure,

the quadrometric grid is not defined). It is featured and identified by a set se of physical,

non-geometrical characteristics such as, for instance, its spectrum, its shape, its temper-

ature, etc. We assume also that this event e can be detected and almost instantaneously

physically analyzed by the emitters at the events E, E and E from signals carried by

light rays (for instance) coming from e. Also, we assume that these light rays (which

carry this various physical information) manifest themselves in “bright points” on their

respective “celestial circles” of the emitters at the events E, E and E. For the sake of

illustration, we consider only the celestial circle C � S1 of the emitter E at the event E.

Also, we provide E with an apparatus made of an optical device and a compass to locate

the event e on the celestial circle C.4 For this, we need also to define a projective frame

for C. For this purpose, the two other satellites E and E manifest themselves in “bright

points” on C ascribed to the two events E ′ and E ′ in the past null cone of E. Then, the

projective point [0]E ∈ C is ascribed to E ′ and kE, and the projective point [∞]E ∈ C is

ascribed to E ′ and kE:

E ′ ←→ [0]E ←→ kE ,

E ′ ←→ [∞]E ←→ kE .

Then, we assume that RP 1 � C = S1. Note that we cannot ascribe to kUE and U a

projective point [1]E ∈ C since U is in the future null cone of E, and thus, no corresponding

“bright points” exists on C. Therefore, we need a fourth satellite, namely, S, in addition

to E , E and E . A priori, S does not need to broadcast a supplementary time stamp, but

it must be clearly identified with an identifier idS .

3 That is, we define the two relative vectors with origin U : r = kE − kUE and r = kE − kUE . Then, in R3,

the circumcenter C is the point C ∈ AE such that

−−→UC =

(‖r‖2 r − ‖r‖2 r) ∧ (r ∧ r)

2 ‖r ∧ r‖2 .

4 The only remaining step utilizing material objects is the angle measurement by compasses. Their use

implies that the angles remain invariant regardless of the size of the compass. And then, this also implies

that there is an absolute notion of angle in contrast to the notions of time and length which depend

on frames. This has historically been considered by Weyl and Godel with their concepts of “inertial

compass” or “star compass” in objection to Mach’s principle. This absolute feature cannot come from

any geometry of space-time. It is therefore possible that it comes from a different physics, such as

quantum mechanics. Thus, a true compass would be based on the use of a quantum phenomenon of

angle measurement, i.e., a quantum compass. This can be done with a Michelson interferometer (see for

example [15, 16]) or interferences in cold atomic gases. Nevertheless, we think that the compass should

be rather graduated by fractional numbers, for instance, such as those appearing in the fractional Hall

effect.


Then, another fourth “bright point” ascribed to the third projective point [1]E ∈ C is

observable on C due to S sending its identifier idS from the event S (see Fig. 3):

S ←→ [1]E .

Now, e can be localized in the emission grid by applying the following procedure.

From the “bright points” [∞]E, [0]E and [1]E, and the optical device and compass

embarked on E , the optical observation of e on C provides a projective point [tanα]E ∈ Cwith α clearly, numerically evaluated from the projective frame FE ≡ {[∞]E, [0]E, [1]E}.5Moreover, to [tanα]E there correspond two vectors �v+

E and �v−E such that

�v±E ≡−−−→EV ±E ≡

−−→EC ±

(−−→CPE + tanα

−−→CPE

),

where C is the circumcenter in AE and, in addition,−−→CPE and

−−→CPE are ascribed to the

following projective points:

−−→CPE ←→ [0]E ,−−→CPE ←→ [∞]E .

Now, the two vectors �v±E define a two dimensional affine plane Pe containing e such that

−→Ee = a+ �v+

E + a− �v−E ∈ Pe

for two reals a± to be determined by applying the same procedure with the two emitters

E and E at, respectively, E and E. Indeed, we deduce the two other analogous affine

planes Pe and Pe and two relations as

−→Ee = a+ �v+

E + a− �v−E ∈ Pe ,−→Ee = a+ �v+

E + a− �v−E ∈ Pe .

Then, e is the intersection point of P , Pe and Pe. Therefore, we obtain six algebraic

linear equations determining completely the a’s and then e in the emission grid. Neither

quadrometric coordinates nor, a fortiori, a quadrometric grid need to be defined. But,

this procedure cannot be generalized to higher dimensional spacetime manifolds: it is

specific to the three dimensional case. Indeed, the intersection point of three, two by two

non-parallel planes always exists in R3 whereas four, two by two parallel, two-dimensional

hyperplanes do not always have intersection points in R4.

5 In this definition of [tanα]E , the angles α vary over the interval [−π/2, π/2] of range π on the celestial

circle rather than over the usual interval [0, 2π] of range 2π. Another viewpoint is to consider the “angles”

to vary within the extended set of real numbers R ≡ [+∞,−∞], and then, to write [α]E with α ∈ R

instead of [tanα]E with α ∈ [−π/2, π/2] .


3.2 The intrinsic procedure of localization

A second, simpler, intrinsic and more effective procedure can be applied using again

optical devices and compasses. It is based on a change of projective frame in C. More

precisely, in the previous procedure with the projective frame FE at E, the three projective

points [∞]E, [0]E and [1]E defining FE were ascribed to, respectively, E ′, E ′ and S. Now,

we consider another projective frame F′E ≡ {[∞]′E, [0]′E, [1]

′E} such that

E ′ ←→ [τ τ1 ]′E ,

E ′ ←→ [τ τ1 ]′E ,

S ←→ [τS]′E ,

assuming now that S broadcasts also a fourth emission coordinate τ in addition to the

three emission coordinates τ , τ and τ . Then, in particular, S sends at the event S the

fourth time stamp τS received by E at the event E (see Fig. 3). Moreover, in a similar

way, each other emitter E and E receives, respectively, at E, the time stamp τ˜S and, at

E, the time stamp τS, from S at two events, respectively, S and S in SW differing in

full generality from the event S ∈ SW . Hence, there are three corresponding emission

events on the worldline of S for these three supplementary time stamps τS, τ˜S and τS.

Then, there corresponds also to e another projective point [τe]′E with respect to this new

projective frame F′E. As a consequence, the following correspondences

[0]E ←→ [τ τ1 ]′E ,

[∞]E ←→ [τ τ1 ]′E ,

[1]E ←→ [τS]′E ,

[tanαe]E ←→ [τe]′E

define the change of projective frame and, consequently, the projective point [τe]′E (see

Fig. 5).

Fig. 5 The change of projective frame at E.

In homogeneous (projective) coordinates, this change of projective frame is defined

by a matrix K ∈ GL(2,R) such that

K ≡

⎛⎜⎝a b

c d

⎞⎟⎠ ,


and satisfying the four following additional correspondences:

[0]E ≡

⎛⎜⎝0

1

⎞⎟⎠ K−−−−−−−−→

⎛⎜⎝a

c

⎞⎟⎠ ≡ [τ τ1 ]′E where τ τ1 = a/c ,

[∞]E ≡

⎛⎜⎝1

0

⎞⎟⎠ K−−−−−−−−→

⎛⎜⎝b

d

⎞⎟⎠ ≡ [τ τ1 ]′E where τ τ1 = b/d ,

[1]E ≡

⎛⎜⎝1

1

⎞⎟⎠ K−−−−−−−−→

⎛⎜⎝a+ b

c+ d

⎞⎟⎠ ≡ [τS]′E where τS =

(a+ b

c+ d

),

[tanαe]E ≡

⎛⎜⎝tanαe

1

⎞⎟⎠ K−−−−−−−−→

⎛⎜⎝a tanαe + b

c tanαe + d

⎞⎟⎠ ≡ [τe]′E where τe =

(a tanαe + b

c tanαe + d

).

Therefore, we obtain⎧⎪⎪⎨⎪⎪⎩a = − τ τ1 [ τ τ1 : τ τ1 : τS ] d ,

b = τ τ1 d ,

c = [ τ τ1 : τ τ1 : τS ] d ,

where [ τ τ1 : τ τ1 : τS ] is such that

[ τ τ1 : τ τ1 : τS ] ≡(τ τ1 − τSτ τ1 − τS

).

Then, we deduce τe such that

τe ≡(

τ τ1 − τ τ1 [ τ τ1 : τ τ1 : τS ] tanαe

1− [ τ τ1 : τ τ1 : τS ] tanαe

). (1)

This is a birational continuous function, and thus bijective. In particular, we obtain the

following important property:

If tanαe = 0, 1 or ∞, then we find that τe = τ τ1 , τS or τ τ1 .

Also, from the other emitters at E and E, the user can compute the three time stamps

pe ≡ (τe, τe, τe) ascribed to the 3-position pe of the event e; therefore localized as expected.

However, it is important to note that the event e is not strictly located in the emission

grid but in a new grid, namely, the quadrometric grid. More precisely, the quadrometric

grid shares with the emission grid 1) the three Cartesian time axes associated with the

three emission coordinates τ , τ and τ , and also, from the property above, 2) the three

emitter worldlines only which are therefore common, point-to-point, to the two grids.

Therefore, rigorously, if e is not a point on an emitter worldline, then, pe must not be

positioned (located) in the emission grid. Moreover, we begin the procedure with time

stamps associated with events located in the emission grid, and we produce sets of time


stamps to localize events. Then, because the process of location strictly differs from the

process of localization, we must consider that any triplet (τe, τe, τe) constitutes the three

quadrometric coordinates ascribed to the event e positioned in the quadrometric grid

only. This procedure can be outlined with the following diagram:

Emission

coordinates−−−→ Intrinsic angles

+ non-intrinsic time stamps−−−→ quadro/pentametric

coordinates

Also, it is important to note that given E, E and E, the event e is unique since it is the

intersection point of three two-dimensional past null cones. Moreover, we can say that

there exists a unique set of three events E, E and E “attached” to e, i.e., we have a

fibered product of past null cones (over the set of localized events e inM) homeomorphic

toM.

Hence, we need four satellites E , E , E and S with their four emission coordinates to

localize an event in the quadrometric grid, and thus, the three dimensional spacetimeMmust be embedded in R4. For instance, we have the following coordinates in R4:

E ←→ (τ1, ττ2 , τ

τ3 , τS) , (2a)

E ←→ (τ τ1 , τ2, ττ3 , τ˜S) , (2b)

E ←→ (τ τ1 , ττ2 , τ3, τS) . (2c)

Also, the data sent by the satellites E , E and E to the user U are reduced. We just need

the following reduced data:

dE ≡ ((τ1, ττ2 , τ

τ3 , τS), idE , αe, se) ,

d˜E ≡ ((τ τ1 , τ2, τ

τ3 , τ˜S), id˜E , αe, se) ,

dE ≡ ((τ τ1 , τ

τ2 , τ3, τS), idE , αe, se) ,

where se allows matching the three first data sets dE, d ˜E and dE ascribed to e.

Besides, the question arises to know if a fourth coordinate τe can be ascribed also

to the event e as for the three events E, E and E. A coordinate τe could be easily

obtained from the 3-position of e in the quadrometric grid if 1) e is in the future horismos

[8, 17] of a point p on the worldline of S, and then, τp ≡ τe, and 2) S broadcasts also,

in particular to the user, the coordinates of p in the quadrometric grid obtained from

the three other emitters E , E and E . The first condition cannot always be physically

or technologically satisfied since there necessarily exists an origin event o at which the

fourth satellite S begins to run. Hence, we can expect to know the positions of S in the

quadrometric grid only beyond this starting point o on the future worldline SW+o ≡ {o�

p,where p is an emission event of S} of S contained in the chronological future of o (the

symbol � denotes the chronological order. See [8, 17] for instance).

Nevertheless, it is easy to circumvent this difficulty, assuming that we define the

prolongation SW−o of the worldline of S in the causal past of o by a given, arbitrary,

nevertheless well-defined by geometric conventions, curve in the quadrometric grid. Now,


from a given time parameterization of SW−o , we can also ascribe to any event e a fourth

time stamp τe from the message function f−SW−o: e −→ τe [6]. Then, the worldline

SW of

S is such that SW = SW−o ∪ {o} ∪ SW+

o and we obtain the complete message function

f−SW : e ∈ M −→ τe ∈ R � SW . As a consequence, from f−SW , we obtain an embedding

ofM in R4. This embedding is explicit since we cannot localize events without giving a

fourth time stamp such as, for instance, τ˜S.

Furthermore, we recall that we have a local chart μ : ([tanαe], [tan αe], [tan αe]) ∈(RP 1)3 −→ pe = (τe, τe, τe) ∈ R3, and we consider now the action of PGL(4,R) on the

triplets (tanαe, tan αe, tan αe). Before, we denote by αi (i = 1, 2, 3) the three angles such

that αe ≡ α1, αe ≡ α2 and αe ≡ α3, and by τj (j = 1, 2, 3) the three time stamps such

that τe ≡ τ1, τe ≡ τ2 and τe ≡ τ3. We put below the list of formulas we start with.

In particular, we have a first set of formulas from the formulas such as (1) at E ≡ E1,

E ≡ E2 and E ≡ E3:

τi =

(uQi tanαi + vQiw�

i tanαi + k�i

)at Ei , (3)

where we assume w�i �= 0 and where the superscripts Q and � indicate, respectively,

that uQi , v

Qi , w

�i and k�

i are homogeneous polynomials of degree 2 (Q ≡ quadratic) and

homogeneous polynomials of degree 1 (� ≡ linear) with respect to the set of time stamps

collected at the three events Ei for the localization of e. Also, we consider that any

element [P ] ∈ PGL(4,R) acts on the three tangents tanαi of the angles αi to give three

other tangents of angles tanα′j such that

tanαi =

( ∑3j=1 P

ji tanα′j + P 4

i∑3k=1 P

k4 tanα′k + P 4

4

), (4)

where P ≡ (P ab ) ∈ GL(4,R) and a, b = 1, . . . , 4 . Then, replacing the three tangents

tanαi in the formulas (3) by the three tangents tanαi given in the formulas (4), we

obtain the following second set of formulas:

τi =

( ∑3j=1 K

ji tanα′j +K4

i∑3k=1 H

ki tanα′k +H4

i

), (5)

where the coefficientsKab andHa

b (a, b = 1, . . . , 4) are linear with respect to the coefficients

of P ≡ (P ab ). But, we can easily verify that these formulas can be rewritten in the

following general form:

τi =

(pQi tanα′i + qQir�i tanα

′i + s�i

), (6)

which are of the same form as (3) where pQi , qQi , r

�i and s�i depend on the remaining angles

differing from α′i . In other words, any projective transformation [P ] provides admissible

changes of projective frames from the given projective frames FEito other projective


frames F•Eion the celestial circles at the events Ei. These changes of projective frames

are defined from the whole of the time stamps collected at the three events Ei and not

only at a given particular one. Thus, these changes differ from those from which we

obtained, for instance, the formulas (3). As a consequence, the coefficients pQi , qQi , r

�i and

s�i depend on all of the time stamps and not only of those collected at the event Ei. In

addition, because we obtain admissible changes of projective frames, then any [P ] is an

admissible projective transformation which can be, therefore, applied to the complete set

of tangents, viz, the set of tangents ([tanα′1], [tanα′2], [tanα

′3]) in the present case or the

set of tangents ([tanα1], [tanα2], [tanα3]) as well.

Besides, remarkably, the (non-unique) element P ∈ GL(4,R) such that, for instance,

P aa = P i

4 = P 43 = 1 , (7a)

P 41 = P 3

1 , P 42 = P 3

2 , (7b)

P ji =

1

w�i

(w�j + k�

j − k�i ) , (7c)

where a = 1, . . . , 4, i �= j and i, j = 1, 2, 3, gives the formulas (5) with the same denomi-

nator for all the τi, i.e., we have

3∑k=1

Hk1 tanα

′k +H4

1 =3∑

k=1

Hk2 tanα

′k +H4

2

=3∑

k=1

Hk3 tanα

′k +H4

3 .

(8)

More precisely, we obtain

Hki = w�

k + k�k , H4

i = w�3 + k�

3 , (9)

for all i, j = 1, 2, 3, and

Kai =

1

w�i

Lai (10)

for all i = 1, 2, 3 and a = 1, . . . 4, where the L’s are homogeneous polynomials of degree 2

with respect to the coefficients wQi , u

Qi , v

�i and k�

i . The element P is not unique and we

can obtain from other elements in GL(4,R) such a common denominator for the τ ’s.

Beside, from this admissible definition of P , we define the virtual time stamps τ vpi to

be the limits obtained when the tangents tanα′i go to infinity. Then, we get quadromet-

ric points denoted by τ vp which are “aligned” in (element of) a two-dimensional affine

subspace in the three-dimensional quadrometric grid. We call such points τ vp vanishing

points or, equivalently, points at infinity. In addition, this subspace of vanishing points

is (locally) homeomorphic to the two-dimensional projective real space RP 2. It is impor-

tant to note that any set of parallel infinite lines in the three-dimensional space (locally

only homeomorphic to RP 3) of the “3-tangents” ([tanα′1], [tanα′2], [tanα

′3]) ∈ (RP 1)3 are


transformed by any [P ] into a congruence of infinite lines all crossing at a unique com-

mon vanishing point τ vp in the quadrometric grid. Hence, we can say, somehow, that

each vanishing point is virtually “spangled” by a congruence of crossing lines defining

the extended notions of spatio-temporal perspective or spatio-temporal parallax. In addi-

tion, it is the so-called vanishing point of the projective geometry well-known by painters

drawing perspectives on their canvas; hence the terminology. We suggest the existence of

a sort of “Big-Bang (visual) effect” due to the “spatio-temporal perspective” relative to

these vanishing points. We can note also, for example, that the particular event E ′ canbe possibly identified by localization with a vanishing point because one of its projective

coordinates is [∞]E.

3.3 Remarks and consequences

From all these preliminary results, we can now deduce the following.

(1) We have shown that any projective transformation [P ] ∈ PGL(4,R) applied on the

3-tangents ([tanαi])i=1,2,3 is compatible with changes of projective frames on the

celestial circles of the three events, viz., E, E and E (see Fig. 4), attached to any

localized event e.

(2) There always exists a particular projective transformation [P ] equalizing the denom-

inators of the relations (5) and such that these relations express another projective

transformation (PT) in PGL(4,R) from the space of 3-tangents to the space of

localized events. This has two consequences:

(a) The relations (5) with the denominators equalized are the defining relations of a

soldering map from the projective space RP 3 of 3-tangents to the quadrometric

grid of localized events in the spacetime manifoldM. This soldering is a bira-

tional local map from RP 3 to the quadrometric grid ofM. From (5), it is only

a local map because

(i) if the 3-tangent θe ≡ ([tanα′1], [tanα′2], [tanα

′3]) is considered as an ele-

ment of (RP 1)3, i.e., θe ∈ (RP 1)3 and θe goes to the unique limit θ∞ ≡([∞], [∞], [∞]) in (RP 1)3, then, there corresponds to θ∞ only one event

e∞ ∈M, and, on the contrary,

(ii) if θe is considered as an element of RP 3, then θe has an infinite set of

possible limits θ∞ ≡ ([λ], [ρ]) ∈ RP 2.

Hence, assuming the soldering map to be non-local would involve 1) the wrong

equivalence (RP 1)3 � RP 3, and 2), that any direction θ∞ ∈ RP 2 is completely

identified with a unique corresponding spacetime event e∞ ∈ M. Thus, we

would go wrong in identifying a spacetime direction (i.e., a topological set of

“parallel” lines inM) with a particular (unique) event in spacetime.

(b) If e� is another localized event attached to three other events E�, E� and E�,

then, there exists a PT from the quadrometric coordinates τ �i of e� to the

quadrometric coordinates τi of e. Thus, M is a so-called generalized Cartan

space “modeled” (locally) on the projective space RP 3 (and not modeled on


the vector space Rn usually associated with any tangent vector space defined at

every point of a differentiable manifold) [18, 19].

(3) The PTs (5) with (8) can be recast within the framework of the Lie groupoid

structures. For, we define, first, the data-point Te to be the set of all of the time

stamps collected at the events E, E and E to localize e, and, moreover, we denote

by T the set of all such data-points Te as the localized event e varies. We assume Tto be locally a smooth manifold. We shown that given two data-points Te and Te� ,

then, the 3-position pe� is obtained from the 3-position pe by a PT defined explicitly

and univocally from Te and Te� . Hence, we can define the Lie groupoid G ⇒ Ts×Ttof PTs such that πs : G −→ Ts ≡ T is the source map and πt : G −→ Tt ≡ T is the

target map of the groupoid. Then, the PTs deduced from any pair (Te, Te�) ∈ Ts×Ttdefine sections of G. We can say that the translations from the source Te ∈ Ts to

the target Te� ∈ Tt are in one-to-one correspondence with a PT defining pe� from

pe. In other words, the projective structure given by this set of PTs is not, a priori,

strictly defined onM but rather on the data manifold T . Nevertheless, to any data-

point Te there corresponds a unique localized event e relative to the given RPS. The

reciprocal is less obvious but it is also true. Indeed, e is the unique intersection point

of three past null cones and only one triplet of such null cones have their apexes E,

E and E on the worldlines of the three emitters E , E and E . Therefore, once the

worldlines of E , E and E , S are known from this given RPS, then all the data needed

to localize e can be reached, and thus, Te. Hence, we can say also that we have a

Lie groupoid structure onM meaning that given pe and pe� only we can deduce the

unique PT compatible with the localization process to pass from pe to pe� . This PT

is not applied to the whole of the events in the quadrometric grid. It is not a PT of

the quadrometric grid.

Also, we can say that a mere translation from pe to pe� in the quadrometric grid

is, somehow, “converted” to a PT “compatible” with the localization process. By

“compatible,” we mean that the translations, for instance, in the quadrometric grid

cannot be directly and physically observed by the use of an explicit causal proto-

col, unlike the admissible PTs on the celestial circles. And, moreover, assuming

that we are not permanently drunk, “lucidly” looking at two simultaneous realities

hierarchized according to our degree of consciousness into an “appearance” and a

“reality,” then, if we see only one “manifest image” [20, 21] on each celestial circle,

then, this is just “the” reality... Thus, those transformations, such as the transla-

tions or any transformation in the affine group, must be interpreted or, somehow,

“converted” into a manifest PT. But, we can avoid such conversion or interpretation

considering that the grid has the structure of a projective space onto which trans-

formations in the affine group, for instance, are forbidden, useless or not physical

because physically not manifest or obervable via a causal protocol.

From a more mathematical viewpoint, if, on the one hand, the (finite) local PTs

are defined as elements of a Lie groupoid G overM×M, then, on the other hand,

from the present particular groupoid structure, the corresponding Lie algebroid is


just identified with the module of vector fields onM. In other words, the tensorial

calculus must be a projective tensorial calculus over M. As a consequence, the

connections onM must be projective Cartan connections.

Moreover, the latter can be restricted to reduced projective connections on each

celestial circle in accordance with a mathematical procedure/computation analogous

to the one giving the transformation formulas (6) on each celestial circle from the

general transformation formulas (5) on the whole ofM.

Hence, because the data space T is locally homeomorphic toM (we assume that

it is, actually, diffeomorphic), we can make the geometrical computations on Min an abstract way, i.e., avoid considering the full set of time stamps of Te and

considering only the restricted set of time stamps directly identified with pe as much

as only infinitesimal, tensorial computations are carried out; and thus, the origin of

the “infinitesimal” projective geometry ofM (but the finite projective geometry on

M×M via the groupoid G).Lastly, we call the worldline SW of the emitter S an anchoring worldline, and we call

the event a ∈ SW such that the time stamp τa emitted by S at a is such that τa = f−SW (e)

and τa ≡ τe the anchor a of e.


time M Modeled on RP 4

The generalization of the previous protocol follows a similar process with five emitters

E , E , E , E and E associated with five emission coordinates, respectively, τ , τ , τ , τ and

τ . They constitute five RPSs made up, each, of four emitters among these five with

the fifth one used for the localization of spacetime events denoted by e. Also, as in the

preceding sections, we denote the user by U and the celestial spheres of the five emitters

by, respectively, C, C, C, C and C. The five emission grids of these five RPSs are Euclidean

spaces R4. The passage from any emission grid to another one among the four others is

a change of chart which is well-defined once the dated trajectories of the five emitters in

the grids are obtained from each RPS and recorded.

For the sake of argument, we consider only the RPS made with the first four emitters,

namely, E , E , E and E and its associated emission grid with the four time stamps τ , τ , τ

and τ defining the so-called 4-positions of the events in this emission grid. Then, the fifth

emitter E ≡ S is used to complement this, for the localization process. Consequently, the

worldline W of E is the anchoring worldline of the relativistic localization system.

Now, we consider only the set of particular events represented in Figs. 6, 7 and 8 with

their corresponding tables of 4-positions.

Fig. 6 shows the different events, namely, E on the worldline W of E , E on the

worldline W of E , E on the worldline W of E and E on the worldline W of E , at whichthe event e is manifest on their respective celestial spheres. We assume that the data of

localization for e collected at the events E, E, E and E are sent to the user and they are

received at the events, respectively, U , U , U and U on the worldline UW of U .


Fig. 6 The event e in the four past null cones of the four events E, E, E and E. This event e

is observed on their respective celestial spheres C, C, C and C.

Fig. 7 indicates, first, the events E′, E ′ and E ′ from which the 4-position of the event

E can be known in the emission grid (see Table 1) and, second, two other events, namely,

E ′ and e, which are observed on the celestial sphere C of the emitter E at E. Obviously,

e is the event to be localized and E ′ is a particular event on the worldline of E which

broadcasts the time stamp τ ′5 to E used for the localization process. Similar figures could

be indicated concerning the three other events E, E and E on Fig. 6, but they are not

really necessary for the description of the localization process presented below. These

unnecessary supplementary figures would indicate supplementary events on the worldline

of E , such as, for instance, E• from which (see Fig. 8) the time stamp τ •5 is transmitted

to the event E of Fig. 7. These particular time stamps are denoted by τ5 (with different

superscripts) and they are sent from different events on the worldline of E to the other

four emitters.

Also, angles are evaluated on each celestial sphere from optical devices and compasses

providing pairs of angles, namely, (α, β) ascribed to each “bright point” observed and

tracked on any given celestial sphere. Actually, each celestial sphere (homeomorphic to

S2) is considered as the union of a circle and two hemispheres. They are topological sets

of which the first one is a closed set and also the common boundary of the others, which

are two open sets in S2. In addition, each hemisphere is embedded in an open, connected

and simply connected set in RP 2 and, moreover, each hemisphere is supplied with a given

projective frame made of four particular points to be specified in the sequel.


Fig. 7 The event E in the five future null cones of the five events e, E′, E′, E′ and E′.

Table 1 The 4-positions of the events in Fig. 7.

Event 4-position

E′

(τ1, τ2, τ3, τ4)

E′ (τ1, τ2, τ3, τ4)

E′ (τ1, τ2, τ3, τ4)

E (τ1, τ2, τ3, τ4)

E′ (τ ′1, τ ′2, τ ′3, τ ′4)

One hemisphere is made of a little spherical cap, as small as possible, and the other is

its complementary hemisphere in S2 with their common boundary to be, for instance, a

polar circle. This choice is motivated from metrological considerations. Indeed, we want

the probability of passage from one hemisphere to the other to be as small as possible

when tracking trajectories of moving points on the celestial spheres. Nevertheless, we

provide each celestial sphere with a computing device ensuring, on the polar circle, the

change of projective frame from one hemisphere to the other and, for each moving point,

recording the signature of its passage, viz, a plus or minus sign. As a consequence, we can

track more completely moving “bright points,” and then, we can position these points in

only one specified, given system of projective coordinates common to the two hemispheres

minus a point (the north pole for instance) to which is ascribed an identifying symbol

instead of two angles. Then, we can establish the correspondences between the pairs of

angles in the two hemispheres and on the polar circle.


Fig. 8 An example of successive events E′, E•, E◦ and E∗ on the anchoring worldline of Etransmiting their coordinates τ5 towards the four events E, E, E and E.

We usually represent one hemisphere embedded in RP 2 by a two-dimensional disk

in R2 to which is added one-half of the polar circle. Then, we have projective frames

made of the four projective points [∞, 0], [0,∞], [0, 0] and [1, 1] with the first two on

the polar circle (see Fig. 9). Also, a projective point [tanαe, tan βe] is ascribed to the

event e observed on each celestial sphere. More precisely, one of the two projective

spaces RP 2 attached to the celestial sphere C of E at the event E is represented in

Fig. 9. Also, a first projective frame FE ≡ {[∞, 0]E, [0,∞]E, [0, 0]E, [1, 1]E} attached to

this projective space is represented providing the projective coordinates [tanα, tan β]E.

Also, a second projective frame F′E ≡ {[∞, 0]′E, [0,∞]′E, [0, 0]′E, [1, 1]

′E} is defined from a

change of projective frame from FE to F′E. This change of frame is based on pairs of

numerical values given, for instance, by the first pair of time stamps, namely, (τ1, τ2)

obtained from the first emitters E and E .

More precisely, we define the first four correspondences:

e ←→ [tanαe, tan βe]E ←→ [τEe , τEe ]′E ,

E′ ←→ [∞, 0]E ←→ [τ1, τ2]

′E ,

E ′ ←→ [0,∞]E ←→ [τ1, τ2]′E ,

E ′ ←→ [0, 0]E ←→ [τ1, τ2]′E ,


Fig. 9 The projective disk at the event E associated to the celestial sphere C of the emitter E .

but with the additional correspondence

E ′ ←→ [1, 1]E ←→ [τ ′5, λ]′E,

where λ is a time value free to vary at this step of the process. Other correspondences

can be chosen. All can be brought back to any fixed, given one once the changes of charts

between the five possible emission grids are known. Thus, one correspondence only can

be used to present the localization protocol.

Also, it is important to note that τ ′5 can be one of the four other time stamps received

at E ′ by E from the four other satellites, i.e., it can be equal to τ ′1, τ′2, τ

′3 or τ

′4. But, these

four values are clearly independent on the whole of the other time stamps such as, for

instance, τ1, τ3, τ4, etc., involved in the localization process, all the more so since these

time stamps τ ′i depend on the worldline of E . Hence, τ ′5 is considered as an independent

time variable in the process—so, a fifth supplementary time stamp indexed by the number

5. In addition, the parameter λ is, actually, well-defined, as shown in the sequel, from

the complete description of the process of localization.

Furthermore, we can set the Table 2 of attributions based on the following pairs of

time stamps: τ1 and τ2 for E, τ2 and τ3 for E, τ3 and τ4 for E, and τ4 and τ1 for E (only

the correspondences [angles]←→ [time stamps] are indicated in this table; the others are

not need for the explanations given below and they are indicated by the marks “∗ ∗ ∗”).Then, we determine the change of projective frame in RP 2 on the celestial sphere C


Table 2 Attributions of time stamps, angles and events.

E E E E E eventpair of

time stamps

— E′

E′ E′ E′ e

E — [∞, 0]E [0,∞]E [0, 0]E [1, 1]E [tanαe, tanβe]E (τ1, τ2)

— [τ1, τ2]′E [τ1, τ2]

′E [τ1, τ2]

′E [τ ′5, λ]′E [τEe , τEe ]′E

E• — E• E• E• e

E [∞, 0]E — [0, 0]E [0,∞]E [1, 1]E [tan αe, tan βe]E (τ2, τ3)

*** — *** *** [τ•5 , λ]′E [τEe , τEe ]′E

E∗ E∗

— E∗ E∗ e

E [0,∞]˜E

[0, 0]˜E

— [∞, 0]˜E

[1, 1]˜E

[tan αe, tan βe] ˜E (τ3, τ4)

*** *** — *** [τ∗5 , λ]′˜E [τ˜Ee , τ

˜Ee ]′

˜E

E◦ E◦

E◦ — E◦ e

E [0, 0]E

[0,∞]E

[∞, 0]E

— [1, 1]E

[tan αe, tan βe] E (τ4, τ1)

*** *** *** — [τ◦5 , λ]′E [τEe , τ

Ee ]′

E

of E at E. For this, we must compute the matrix K as

K =

⎛⎜⎜⎜⎜⎝a d g

b e h

c f k

⎞⎟⎟⎟⎟⎠ (11)

associated with this change of frame. This matrix K is defined from the following corre-


spondences in R3:

E′: [∞, 0]E ≡

⎛⎜⎜⎜⎜⎝1

0

0

⎞⎟⎟⎟⎟⎠ K−−−−−−−−→ [τ1, τ2]′E ≡

⎛⎜⎜⎜⎜⎝a

b

c

⎞⎟⎟⎟⎟⎠ where

{τ1 = a/c

τ2 = b/c

E ′ : [0,∞]E ≡

⎛⎜⎜⎜⎜⎝0

1

0

⎞⎟⎟⎟⎟⎠ K−−−−−−−−→ [τ1, τ2]′E ≡

⎛⎜⎜⎜⎜⎝d

e

f


{τ1 = d/f

τ2 = e/f

E ′ : [0, 0]E ≡

⎛⎜⎜⎜⎜⎝0

0

1

⎞⎟⎟⎟⎟⎠ K−−−−−−−−→ [τ1, τ2]′E ≡

⎛⎜⎜⎜⎜⎝g

h

k


{τ1 = g/k

τ2 = h/k

E ′ : [1, 1]E ≡

⎛⎜⎜⎜⎜⎝1

1

1

⎞⎟⎟⎟⎟⎠ K−−−−−−−−→ [τ ′5, λ]′E ≡

⎛⎜⎜⎜⎜⎝a+ d+ g

b+ e+ h

c+ f + k


⎧⎨⎩τ ′5 =(

a+d+gc+f+k

)λ =

(b+e+hc+f+k

)

e : [tanαe, tan βe]E ≡

⎛⎜⎜⎜⎜⎝tanαe

tan βe

1

⎞⎟⎟⎟⎟⎠ K−−−−−−−−→ [τEe , τEe ]′E ≡

⎛⎜⎜⎜⎜⎝u

v

w


{τEe = u/w

τEe = v/w

and

u = a tanαe + d tan βe + g ,

v = b tanαe + e tan βe + h ,

w = c tanαe + f tan βe + k .

From the above, we deduce the four following linear equations:

(τ1 − τ ′5) x+ (τ1 − τ ′5) y + (τ1 − τ ′5) = 0 , (12a)

(τ2 − λ) x+ (τ2 − λ) y + (τ2 − λ) = 0 , (12b)

(τ1 − τEe ) x tanαe + (τ1 − τEe ) y tan βe + (τ1 − τEe ) = 0 , (13a)

(τ2 − τEe ) x tanαe + (τ2 − τEe ) y tan βe + (τ2 − τEe ) = 0 , (13b)

where x ≡ c/k and y ≡ f/k, and where x, y, λ, τEe and τEe are the unknowns. From the

system (12), we obtain, first, the values for x and y, and second, from (13), we obtain


the pentametric coordinates τEe and τEe such that

τEe =P (λ, τ ′5, tanαe, tan βe)

P0(λ, τ ′5, tanαe, tan βe), (14a)

τEe =P (λ, τ ′5, tanαe, tan βe)

P0(λ, τ ′5, tanαe, tan βe), (14b)

where P , P and P0 are polynomials of degree one with respect to λ and τ ′5 of which the

coefficients are polynomials of degree one with respect to tanαe and tan βe.

We also compute the four other pairs of time stamps ascribed to the event e, i.e.,

(τEe , τEe ), (τ˜Ee , τ

˜Ee ) and (τ

Ee , τ

Ee ) (see Table 2), respectively, obtained at the events E, E

and E. We obtain expressions similar to (14) with respect to the other λ’s, τ5’s, tanα’s

and tan β’s . And then, we set the following constraints:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩τEe = τ

Ee ,

τEe = τEe ,

τEe = τ˜Ee ,

τ˜Ee = τ

Ee .

(15)

These constraints are well-justified because any event e has only one 4-position. Then,

we deduce four equations of the form

λ1 =

(uλ2 + w

w λ2 + r

), (16)

for any pair (λ1, λ2) of distinct λ in the set {λ, λ, λ, λ} from which we deduce one quadratic

equation for each λ with coefficients independent of the other λ’s but, nevertheless, de-

pending on the angles and the various time stamps τ . Therefore, we have proved that

each λ has a value which is independent on the other λ’s. But, in addition, the λ’s must

also be independent of the angles because they are ascribed to the projective points [1, 1]

independently of the events such as e. Hence, we can arbitrarily fix the values for the

λ’s. The natural choice is to set the following:

λ ≡ τ ′5 , λ ≡ τ •5 , λ ≡ τ ∗5 , λ ≡ τ ◦5 . (17)

In return, from (16) with (17), we deduce also four fractional relations between, on the

one hand, the α’s, and, on the other hand, the β’s. The general form of these relations

is the following. For instance, for tan βe, we have:

tan βe =

(u tanαe + u tan αe + u tan αe + u tan αe + r

w tanαe + w tan αe + w tan αe + w tan αe + s

), (18)

where the coefficients u, u, etc., depend on the time stamps except those ascribed to the

localized event e.


We then obtain the 4-position pe ≡ (τe, τe, τe, τe) for e in the grid such that τe ≡ τEe ,

τe ≡ τEe , τe ≡ τ˜Ee and τe ≡ τ

˜Ee depending on the four angles αe, αe, αe and αe and the

time stamps. For instance, the pentametric coordinate τe satisfies

τe =

(p tanαe + p tan αe + p tan αe + p tan αe + q

m tanαe + m tan αe + m tan αe + m tan αe + n

). (19)

As a result, from 1) the form of this expression which is the same for each pentametric

coordinate of the 4-position of e, and 2) following the same reasoning as in the preced-

ing section for a (2 + 1)-dimensional spacetime, the group PGL(5,R) acts on M via a

projective transformation applied to the four tangents tanαe, tan αe, tan αe and tan αe.

Now, we can almost completely paraphrase what we described from p. 96 in the

preceding section, adding just one time stamp τ and another supplementary angle α.

And then, following the same reasoning, we deduce thatM is modeled on RP 4 and that

it is embedded in R5. Finally, we denote by τ the fifth pentametric coordinate of the fibers

of the submersion R5 to M. This supplementary pentametric coordinate τ is, actually,

defined from the anchoring worldline W following similarly the method indicated at the

end of the last section.

Lastly, the present protocol is based on the particular class of pairs of time stamps

specified in the last right column of Table 2. The pentametric coordinates ascribed to

each event e would differ for a different class of pairs. Hence, we can obtain different,

possible localizations for the same event e: a result which can be baffling only if we assume

that localization is an absolute, intrinsic property of each spacetime event independent

of any process. But, after all, we are already faced with this situation when producing

atlases of charts for manifolds. In the same way, we just need to know the changes of

localization charts (pentametric grids) which are, actually, deduced naturally from the

changes of charts defined by the changes of emission grids. Therefore, localization and

location as well cannot be intrinsic processes.

5. Conclusion

Even though spacetime is represented by a four-dimensional manifold, the localization

processes show that spacetime cannot be physically apprehended if its manifold counter-

part is not embedded geometrically in a five-dimensional manifold modeled locally on a

four-dimensional projective space. Then, the spacetime manifold must be considered as

a generalized Cartan manifold endowed necessarily, as a consequence, with a projective

Cartan connexion [19].

Also, the space and time splitting ascribed usually to the four dimensions of spacetime

should be enhanced to encompass a fifth dimension. Then, to be complete, a supplemen-

tary notion should be added to space and time.

Besides, the present pentametric protocol can be explicitly applied to the determina-

tion of stellar positions of emitting sources, standing inside or outside the Milky Way for

instance, or, more generally, at any distance in the universe. The fundamental advantage


of this protocol is that patterns of, for instance, starfields can be obtained directly with no

use of any intermediate method or model of light bending by massive stellar bodies such

as the PPN method for instance [22, 23, 24, 25, 26, 27]. With the pentametric protocol,

the evaluations of the patterns of starfields are direct (no mathematical simulations) since

the stellar positions are not obtained from measurements of spatial parallaxes followed

by a simulated model-dependent reconstruction of the light ray trajectories. Moreover,

the pentametric localization protocol provides spacetime positions which encompass and

are sensitive to the whole of physical phenomena producing light ray deflections. Hence,

these spacetime positions are, somehow, “nature-dependent” (e.g., Jupiter in [25]) mean-

ing not absolute in an abstract way and meaning also that they follow what we call the

“Whitehead’s paradigm” on the systems of measurements (see pp.196–197, Chap. IX,

“The Ultimate Physical Concepts” in [28]):

“ Furthermore the admission of stratifications of nature bears on the formulation of

the laws of nature. It has been laid down that these laws are to be expressed in dif-

ferential equations which, as expressed in any general system of measurement, should

bear no reference to any other particular measure-system. This requirement is purely

arbitrary. For a measure-system measures something inherent in nature; otherwise it

has no connexion with nature at all. And that something which is measured by a par-

ticular measure-system may have a special relation to the phenomenon whose law is

being formulated. For example the gravitational field due to a material object at rest in

a certain time-system may be expected to exhibit in its formulation particular reference

to spatial and temporal quantities of that time-system. The field can of course be ex-

pressed in any measure-systems, but the particular reference will remain as the simple

physical explanation.”

And this paradigm is, in some way, the opposite of the “Hilbert’s paradigm” inherent to

the present day general relativity (see p.61 in [29]):

“Was nun das Kausalitatsprinzip betrifft, so mogen fur die Gegenwart in irgend einem

gegebenen Koordinatensystem die physikalischen Großen und ihre zeitlichen Ableitun-

gen bekannt sein: dann wird eine Aussage nur physikalischen Sinn haben, wenn sie

gegenuber allen denjenigen Transformationen invariant ist, bei denen eben die fur

die Gegenwart benutzten Koordinaten unverandert bleiben; ich behaupte, daß die

Aussagen dieser Art fur die Zukunft samtlich eindeutig bestimmt sind d. h. das

Kausalitatsprinzip gilt in dieser Fassung:

Aus der Kenntnis der 14 physikalischen Potential gμν, qs in der Gegenwart folgen alle

Aussagen uber dieselben fur die Zukunft notwendig und eindeutig, sofern sie physikalis-

chen Sinn haben.”6

6 “As for the principle of causality, the physical quantities and their time-rates of change may be known

at the present time in any given coordinate system; a prediction will then have a physical meaning only

when it is invariant with respect to all those transformations for which exactly those coordinates used for

the present time remain unchanged. I declare that predictions of this kind for the future are all uniquely

determined; that is, that the causality principle holds in this formulation:

From the knowledge of the 14 physical potentials gμν , qs, in the present, all predictions about the same

quantities in the future follow necessarily and uniquely insofar as they have physical meaning.”(translation


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[25] M. Crosta and F. Mignard, Microarcsecond light bending by Jupiter, Class. Quant.Grav. 23(15), 4853 (2006).

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[27] S. Kopeikin and E. Fomalont, Gravimagnetism, causality, and aberration of gravity inthe gravitational light-ray deflection experiments, Gen. Rel. Grav. 39(10), 1583–1624(2008).

[28] A. N. Whitehead, The Concept of Nature, Tarner lectures, Cambridge UniversityPress, Cambridge, UK, 1920.

[29] D. Hilbert, Die Grundlagen der Physik. (Zweite Mitteilung.), Nachr. Konigl. Gesell.Wiss. Gottingen, Math.-Phys. Kl. 1917(1), 53–76 (1917).

[30] C. Misner, K. Thorne, and J. Wheeler, Gravitation, W. H. Freeman and Compagny,San Francisco, 1973.


Theoretical Calculations for PredictedStates of Heavy Quarkonium ViaNon-Relativistic Frame Work

T. A. Nahool1∗, A. M. Yasser1† and G. S. Hassan2‡

1Physics Department, Faculty of Science at Qena, South Valley University, Egypt2Physics Department, Faculty of Science, Assuit University, Egypt

Received 18 October 2014, Accepted 20 December 2014, Published 10 January 2015

Abstract: We calculate the mass spectra of heavy quarkonium by using matrix Numerov’s

method to make the predictions of F and G states for further experiments. The method gives

a very reasonable result which is in a good agreement with other methods and with recently

published theoretical data. From the yielded wave functions we calculate the root mean square

radius rms and β coefficients of heavy quarkoniumc© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Quark Model; heavy quarkonium; Numerov’s method

PACS (2010): 12.40.Yx, 14.40.Pq, 02.60.Cb, 12.39.Jh

1. Introduction

The theoretical studies of the heavy quarkonium system [1] and its applications to bot-

tomonium [2] and char-monium [3] is one of the special interest because of its relies

entirely on the first principles of quantum chromo dynamics( QCD). From the view-

point of the ”heavy quarkonium” spectra, we calculate the theoretical mass spectra via

non-relativistic frame work [4], [5], [6] by using matrix Numerov’s method [7], [8] in two

levels F and G. Many studies investigate ”heavy quarkonium” properties within the quark

model [9-15]. An essential progress has been made in the theoretical investigation of the

non-relativistic heavy quark dynamics. Our point of departure is to calculate the spectra

of heavy quarkonium and the corresponding wave functions of nn states (n = c, b) for

the predicted F and G states. Actually, we dont have any experimental data of F and G

∗ Email:[email protected]† Email:[email protected]‡ Email: [email protected]


states. Therefore, we compare the present theoretical predictions with published theoret-

ical data from [16], [17] and [18]. Moreover, the heavy-meson wave functions determined

in this work can be employed to make predictions of other properties. Besides, the main

motivation is to calculate the root mean square radius rms of different states for bot-

tomonium and the numerical values of β coefficient [19], which can be used to calculate

the decay widths [20], and differential cross sections [21] for quarkonium states. In this

work, we consider the mass spectra and some properties of heavy quarkonium systems in

the non-relativistic quark model using matrix Numerov’s method.In section 2, we review

the main formalism of the matrix Numerov’s method used in our analysis and the used

model. Besides, the analytical formula of the wave functions. Some characteristics prop-

erties of bottomonium mesons are introduced in section 3. After that, numerical results

and discussion are given. Finally in the last section, we summarize our main results and

conclusions.

2. Theoretical Basis

2.1 Matrix Numerovs Method

The Matrix Numerovs method is a method that provides us an approximate solution of

non-relativistic Schrdinger equation of the form:

ψ′′(x) = f(x)ψ(x) (1)

For the time-independent 1D Schrdinger equation, we have:

−�22m

AN,N +BN,NVNψi = EiBN,Nψi (2)

Where i run from 1 to N and define matrices

AN,N =I−1 − 2I0 + I1

d2, BN,N =

I−1 + 10I0 + I112

, VN = diag(..., Vi−1, Vi, Vi+1) WhereI−1, I0andI1represent sub-, main-, and up- diagonal unit matrices respectively.

For the 3D radial Schrdinger equation, the last equation could be written as:

−�22μ

AN,NB−1N,Nψi + [VN(r) +

l(l + 1)

r2]ψi = Eiψi (3)

Suggesting that this particle is compound and it consists of two smaller particles (meson

consist of two quarks), then the reduced mass in the non-relativistic model can be iden-

tified as

μ =mqmq

mq +mq

Wheremq = mq is the mass of quark and anti-quark for Quarkonium system. In natural

units � = c = 1 then, Eq. (3) could be written as:

−12μ

AN,NB−1N,Nψi + [VN(r) +

l(l + 1)

r2]ψi = Eiψi (4)

The first term is the Matrix Numerovs representation of the kinetic energy operator and

the second is the Matrix Numerovs representation of the potential energy operator.


2.2 Non-relativistic Potential Quark Model

The non-relativistic quark model has been highly successful not only for heavy quarko-

nium cc and bb states, but also for states involving light quarks. As a minimal model of

heavy quarkonium system, we use a non relativistic potential model, with wave functions

determined by the Schrdinger equation. We use the more realistic model of the potential

which is Coulomb plus linear plushyperfine interaction model [22], [23]

V(r) =−43

αs

r+ br +

32παs

9mqmq

δσ(r)�Sq.�Sq +l(l + 1)

2μr2(5)

where

δσ(r) = (σ/√π)3e−σ

2r2 (6)

And s is the total spin quantum number of the meson [24] And

SqSq =S(S + 1)

2− 3

4(7)

The hyperfine term is spin-dependent term which makes it possible to distinguish between

mesons which have different spins. By including the tensor operator, the potential of the

qq system has the following form for the bottomonium:

VN(r) =l(l + 1)

2μr2− 4αs

3+ br +

32παsδ(r)SbSb

9mb

+1

m2b

[(2αs

r3− b

2r)�L.�S +

4αs

r3T ] (8)

2.3 Wave functions of bottomonium mesons

Quarkonium system can be described by the wave function of the bound quark-antiquark

state which satisfies the Schrdinger equation (SE) by using the potential given in Equation

(8). Radial Schrdinger equation,ψ(r) = rR(r) is written (in natural units) as:

ψ(r) + 2μ(E − V (r))ψ(r) = 0 (9)

Where R(r) is the radial wave function, r is the inter quark distance,E is the sum of

kinetic and potential of quark-antiquark system, and V (r) and μ are defined above

through Equation (8). The matrix Numerove’s method is used to solve Eq. (8) to get

spectra of bottomonium bb as an example of heavy quarkonium mesons, the detailed of

this method could be found in Ref. [7]. In the following sections, we employ that method

to obtain the wave functions of bottomonium which in turn are used to determine some

properties of bottomonium.

3. Basic Properties of Bottomonium Meson

3.1 Bottomonium root mean square radius rms

Bottomonium root mean square radius rms is one of basic properties of bottomonium.

If the distance between the quark and anti-quark in bottomonium is r fm it may be


regarded that bottomonium has radius r/2 fm where r is the distance from the point

quark to anti-quark. rms can be derived from the meson wave function and may be

written as [25]:

r2ms =

∫ ∞

0

ψ2(r)r2dr (10)

3.2 β Coefficient

The meson wave function is characterized by a momentum width parameter that is

related to the root mean square quark-antiquark separation rms of the meson by [26]:

β =

√(2(n− 1) + (L) +

3

2)× 1

rms

(11)

where n is the principal quantum number and L is the sub- atomic energy level number.

The parameter β is typically taken as a parameter of the model. However, since we are

seeking for describing the decay of heavy quark states, it is preferable to reproduceβcoefficient

of the quark model states. These values of β are obtained for the first time. So, we suggest

using it to calculate the decay width of heavy quarkonium system.

4. Numerical results

4.1 Theoretical Predictions of Bottomonium spectrum

A non relativistic potential model is used to study bottomonium meson spectra by using

the Numerovs Method. We predict the masses of the twelve bb states shown in Table

1, where we compared the present theoretical predictions with those from [16], [17] and

[18]. Figure 1, Figure 2and Figure 3, illustrate the ratio between the obtained theoretical

predictions of bottomonium spectra and those from [16], [17] and [18] respectively. It is

seen that the ratio converges to one particularly in Figure 1.This means that the yielded

results are in a good agreement with recently published predictions.

4.2 Theoretical Calculations of Some Properties of Bottomonium

The eigenvalues and the corresponding wave functions are found by using the same

method. Then we norma-lized the wave functions and we calculate the root mean square

radius of bottomonium mesons by using Equa-tion (10). Moreover, we obtain computa-

tional values of βcoefficient by using Equation (11). The normalized radial wave functions

for bottomonium mesons are graphically represented in Figure 4and Figure 5 respective-

ly. For bottomonium mesons, our calculated masses and root mean square radius are

reported in Table 2 in case of F and G States respectively. The β values could be used to

calculate the decay constants [27], decay widths [27], and differential cross sections [28]

for quarkonium states with high accuracy as we used complicated potential model. The


predictions about these quantities are also reported in Table 2 for bottomonium F and

G States respectively.

4.3 Figures and Tables

Fig. 1 Ratios of the obtained theoretical masses of bottomonium F and G states to the theoreticaldata in [18].

Fig. 2 Ratios of the obtained theoretical masses of bottomonium F state to the theoretical datain [16].

Fig. 3 Ratios of the obtained theoretical masses of bottomonium F state to the theoretical datain [17].


Fig. 4 Bottomonium F-states reduced radial wave functions plotted together with the usedpotential .

Fig. 5 Bottomonium G-state reduced radial wave functions plotted together with the usedpotential .

5. SUMMARY AND CONCLUSION

In this paper, the mass spectra of bottomonium as an example of heavy quarkonium were

studied within the framework of non-relativistic frame work. Eigenvalues and Eigenfunc-

tions were obtained numerically for bottomonium meson using matrix Numerov’s method.

The predictions from our method are found to be in good agreement with the available

theoretical results. Our calculated masses are reported in Table 1 along with the theoret-

ical predictions of the other works. We observed that our results are in good agreement

with existing theoretically predicted values, which shows the validity of the used method.

Besides, there are an additional properties have been studied in this work since we used

the matrix Numerov’s method to obtain the radial wave functions of bottomonium me-

son to calculate the bottomonium bb root mean square rms and β coefficient. As a

remarkable result, we can point out that it is recommended to use the obtained values

of rms and β coefficient to calculate the decay widths and differential cross sections for

bottomonium system. Moreover, the matrix Numerov’s method [7] is tested again to

obtain some mesons properties. Then, the method could be safely used to solve SE.

Our calculated root mean square radius and the values of β coefficient are reported in


Table 1 Theoretical predictions of bottomonium mass spectra in [Gev].

State Theoretical Theoretical Theoretical Theoretical ratio1 ratio2 ratio3

spectra [ours] spectra[16] spectra[17] spectra[18] [our]/[16] [our]/[17] [our]/[18]

13F4 10.345 10.337 10.359 10.345 0.99864 1.0008 1

23F4 10.592 10.597 10.617 10.591 0.99764 0.9995 1.00009

13F3 10.344 10.340 10.355 10.344 0.99893 1.0004 1

23F3 10.591 10.599 10.613 10.588 0.99792 0.9993 1.0003

13F2 10.342 10.341 10.351 10.342 0.99913 1.0001 1

23F2 10.589 10.599 10.609 10.591 0.99811 0.9991 1.00009

11F3 10.344 10.339 10.355 10.344 0.99893 1.0005 1

21F3 10.591 10.598 10.613 10.591 0.99792 0.9993 1

13G5 10.5 10.501 0.99990

13G4 10.501 10.501 1

13G3 10.501 10.5 1.00009

11G4 10.5 10.5 1

Table 2 Theoretical predictions of bottomonium rmsand the values of β coefficent

State rms[fm] β coefficent

13F4 3.46755 0.611763

23F4 4.58349 0.556238

13F3 3.44305 0.616117

23F3 4.55974 0.559135

13F2 3.42368 0.619602

23F2 4.54079 0.561469

11F3 3.44926 0.615007

21F3 4.56573 0.558401

13G5 4.03734 0.58088

13G4 4.02281 0.582977

13G3 4.01083 0.584718

11G4 4.02576 0.582551

Table 2. As a side result, these theoretical results are expected to give some hints to the

forthcoming experiments. Eventually, we may notice that the calculated values of rms


and other parameters are the newer outputs where we didnt find others for comparison.

So, we are looking forward to take these data in consideration by other experimental or

theoretical researchers.

References

[1] T. Appelquist and H.D. Politzer, Phys. Rev.Lett 34, 43(1975).

[2] V.A. Novikov, et al.., Phys. Rep. C41, 1 (1978).

[3] E. S. Swanson, Physics Reports429, 243 (2006).

[4] Bhaghyesh, K. B. V. Kumar, M. Yong-Liang, Int. J. Mod. Phys. A 27, 125001 (2009).

[5] C. Quigg, J. L. Rosner, Phys. Rep 56, 167 (1979).

[6] S. S. Gershtein, V. V. Kiselev, A. K. Likhoded, A. V. Tkabladze, Phys. Rev. D 51,3613 (1995).

[7] A. M. Yasser, G. S. Hassan and T. A. Nahool, Int. J. New. Hor. Phys 2,(2014).

[8] M. Pillai, J. Goglio, and T. G. Walker, Am. J. Phys 80, 1017 (2012).

[9] S. N. Gupta, S. F. Radford and W. W. Repko, Phys.Rev. D 34, 201 (1986).

[10] W. Buchmuller and S. H. H. Tye, Phys. Rev. D 24, 132 (1981).

[11] P. Moxhay and J. L. Rosner, Phys. Rev. D 28,1132(1983).

[12] S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).

[13] F. Iachello, N. C. Mukhopadhyay and L. Zhang, Phys. Lett. B 256, 295 (1991).

[14] W. Lucha, F. F. Schoberl and D. Gromes, Phys. Rept 200, 127 (1991).

[15] L. P. Fulcher, Phys. Rev. D 42, 2337 (1990); Phys. Rev. 44, 2079 (1991).

[16] T. Wei-Zhao, CAO Lu, Y. You-Chang, C. Hong, , arXiv. nucl-[hep-ph] 1, 0960(2008).

[17] Radford S F and Repko , Phys. Rev. D 75, 074031 (2007).

[18] A. A. Aly, M.SC. Thesis, ”Heavy Meson Spectra in The Non-Relativistic Quarkmodel”, South Valley University, Egypt 2012

[19] A. M. Yasser, G. S. Hassan and T. A. Nahool, Journal of Modern Physics,5, 1938-1944. http://dx.doi.org/10.4236/jmp.2014.517188 (2014).

[20] B. Patel and P. C. Vinodkumar, J. Phys. G 36, 035003 (2009).

[21] c. H. Chaug, C. F. Qiao and J. X. Wang, , Phys. Rev. D 57, 4035 (1998)

[22] O. Lakhina and E. S. Swanson, Phys. Rev. D 74, 014012 (2006).

[23] T. Barnes, S. Godfrey and E. Swanson, Phys. Rev, D 69, 54008 (2004).

[24] N. Akbar, B. Masud, S. Noor, Eur. Phys. J. A 47, 124 (2011).

[25] F. Zolfagharpour, arXiv. nucl-th 2, 1623(2008).

[26] C.Y. Wong, arXiv. nucl-th 2, 28(2003).

[27] B. Patel and P. C. Vinodkumar, J. Phys. G 36, 035003 (2009).


The One-dimensional Thermal Properties for theRelativistic Harmonic Oscillators

Abdelmalek Boumali∗

Laboratoire de Physique Appliquée et Théorique,Université de Tébessa, 12000, W. Tébessa, Algeria

Received 19 ○ctober 2014, Accepted 20 December 2014, Published 10 January 2015

Abstract: In this paper, we want to improved the calculations of the thermodynamic quantitiesof the relativistic Harmonic oscillator using the Hurwitz zeta function. The comparison of ourresults with those obtained by a method based on the Euler-MacLaurin approach has been made.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Klein-Gordon Oscillator ;Dirac Oscillator ; Euler-MacLaurin Formula ; Hurwitz ZetaFunctionPACS (2010): 03.65.-w ; 03.65.Ge ; 03.65.Pm ; 03.65.Pm ; 02.60.-x ; 02.70.-c ; 05.70.-a

1. Introduction

The relativistic harmonic oscillator is one of the most important quantum system, asit is one of the very few that can be solved exactly.

The Dirac relativistic oscillator (DO) interaction is an important potential both fortheory and application. It was for the first time studied by Ito et al [1]. They considereda Dirac equation in which the momentum �p is replaced by �p − imβω�r, with �r beingthe position vector, m the mass of particle, and ω the frequency of the oscillator. Theinterest in the problem was revived by Moshinsky and Szczepaniak [2], who gave it thename of Dirac oscillator (DO) because, in the non-relativistic limit, it becomes a harmonicoscillator with a very strong spin-orbit coupling term. Physically, it can be shown thatthe (DO) interaction is a physical system, which can be interpreted as the interaction ofthe anomalous magnetic moment with a linear electric field [3, 4]. The electromagneticpotential associated with the DO has been found by Benitez et al[5]. The Dirac oscillatorhas attracted a lot of interest both because it provides one of the examples of the Dirac’sequation exact solvability and because of its numerous physical applications [6, 7, 8, 9].

∗. Email :[email protected]


Fortunately Franco-Villafane et al [10], in order to vibrate this oscillator, exposed theproposal of the first experimental microwave realization of the one-dimensional (DO).

The thermal properties of the one-dimensional Dirac equation in a Dirac oscillatorinteraction was at first considered by Pacheco et al [11] . The authors have been calcu-lated the all thermal quantities of the oscillator by using the Euler-MacLaurin formula.Although this method allows to obtain the all thermal properties of the system, the ex-pansion of the partition function using it could be valid only for higher temperaturesregime, but not otherwise. Also, the partition function at T = 0K reveals a total diver-gence (see Annexe A). Encouraged by the experimental realization of a Dirac oscillator,we are interested in : (i) to improve the calculations of the thermodynamics properties forthe relativistic harmonic oscillators in all range of temperatures, and (ii) to remove thedivergence appears in the partition function at T = 0K. Both objectives can be achievedby using a method based on the zeta function [12, 13]. This method has been used by [14]with the aim of calculating the partition function in the case of the graphene. We notehere that the zeta function has been applied successfully in different areas of physics, andthe examples vary from ordinary quantum and statistical mechanics to quantum fieldtheory (see for example [15]).

Thus, the main goal of this paper is the improvement of the calculations of all thermalquantities of the one-dimensional Dirac oscillator. This work is organized as follows : Insection. 2, we review the solutions of both Dirac and Klein-Gordon oscillators in onedimension. Section. 3 is devoted to our numerical results and discussions. Finally, Section.4 will be a conclusion.

2. Review of the Solutions of both Dirac and Klein-Gordon Os-cillators in One-dimension

2.1 One-dimensional Klein-Gordon Oscillator

The free Klein-Gordon oscillator is written by(p2x −

E2 −m20c

4

c2

)φ = 0. (1)

In the presence of the interaction of the type of Dirac oscillator, it becomes[c2 (px + im0ωx) · (px − im0ωx)− E2 +m2

0

]φ (x) = 0, (2)

or (p2x2m0

+m0ω

2

2x2

)φ (x) =

(m0c

2�ω + E2 −m20c

4

2m0c2

)φ (x) ≡ Eφ (x) , (3)

with

E =m0c

2�ω + E2 −m20c

4

2m0c2. (4)


The equation (3) is the standard equation of a harmonic oscillator in 1D. The energylevels are well known, and the solutions are

εn = ±m0c2√1 + 2rn (5)

with r = �ωm0c2

being a parameter which controls the non relativistic limit.The eigenfunctions may be expressed in terms of Hermite Polynomial of Degree n as

φ (x) = Nnorm(m0ω

π�

) 14H

(√m0ω

�x

)e−

m0ω2�

x2

. (6)

where the functions H is the so called Hermite polynomials, and Nnorm is a normalizingfactor [16].

2.2 One-dimensional Dirac Oscillator

The one-dimensional Dirac oscillator is{c · αx · (px − imωβ · x) + βmc2

}ψD = EψD, (7)

with ψD =

(ψ1 ψ2

)T

, αx = σx and β = σz.

From Eq. (7), we get a set of coupled equations as follows :(E −mc2

)ψ1 = c (px + imωx)ψ2, (8)(

E +mc2)ψ2 = c (px − imωx)ψ1. (9)

Using Eq. (9), we have

ψ2 (x) =c (px − imωx)

E +mc2ψ1 (x) . (10)

Putting Eq. (10) into (8), we get

[c2 (px + imωx) (px − imωx)− E2 +m2

]ψ1 (x) = 0, (11)

or (p2x2m

+mω2

2x2

)ψ1 (x) =

(�ωmc2 + E2 −m2c4

2mc2

)ψ1 (x) ≡ Eψ1. (12)

The equation (12) is the standard equation of a harmonic oscillator in 1D. The energylevels are well-known, and are given by

εn = ±mc2√1 + 2rn (13)

with r = �ωmc2

is a parameter which controls the non relativistic limit. The eigenfunctionsmay be expressed in terms of Hermite Polynomial of Degree n a

ψ1 (x) = N′norm

(mω

π�

) 14H

(√mω

�x

)e−

mω2�

x2

. (14)


with N′norm is a normalizing factor. The total associated wave function is

ψD (x) = N′norm

⎡⎢⎣ 1

c(px−imωx)E+mc2

⎤⎥⎦H

(√mω

�x

)e−

mω2�

x2

. (15)

3. Thermal Properties of the Relativistic Harmonic Oscillator

Before we study the thermodynamic properties of both oscillators, we can see thatthe form of the spectrum of energy (see Eqs. (5) and (13)), for both cases, is the same.As consequently, the numerical thermal quantities found, for both oscillators, are similar. Thus, we focus, firstly, on the study of the thermal properties of a Dirac oscillator, andthen all results obtained can be extended to the case of the one-dimensional Klein-Gordonoscillator.

3.1 Methods

In order to obtain all thermodynamic quantities of the relativistic harmonic oscillator,we concentrate, at first, on the calculation of the partition function Z. The last is definedby

Z =∑n

e−βEn =∑n

e−√

1+2rnτ (16)

With the following substitutions :

α =1

2r, γ =

√2r, (17)

it becomesZ =

∑n

e−γτ

√α+n, (18)

with τ = kBTmc2

denotes the reduce temperature.Using the formula [12, 13]

e−x =1

2πi

ˆC

dsx−sΓ (s) , (19)

the sum in Eq. (18) is transformed into∑n

e−γτ

√α+n =

1

2πi

ˆC

ds(γτ

)−s ∑n

(n+ α)−s2 Γ (s) =

1

2πi

ˆC

ds(γτ

)−sζH

(s

2, α

)Γ (s) ,

(20)with x = γ

τ

√α + n, and Γ (s) and ζH

(s2, α

)are respectively the Euler and Hurwitz zeta

function. Applying the residues theorem, for the two poles s = 0 and s = 2, the desiredpartition function is written down in terms of the Hurwitz zeta function as follows :

Z (τ) =τ 2

2r+ ζH (0, α) . (21)


Now, using that

ζH (0, α) =1

2− α, (22)

the final partition function is transformed into

Z (τ) =τ 2

2r+

1

2− 1

2r. (23)

From this definition, all thermal properties of both fermionic and bosonic oscillators canbe obtained.

3.2 Numerical Results and Discussions

Fig. 1 depicted the one-dimensional thermal properties of both Dirac and Klein-Gordon oscillators. Following the figure, all thermal quantities are plotted versus a re-duced temperature τ : here we have taken r = 1 which corresponds to the relativisticregion . From the curve of the numerical entropy function, no abrupt change, around τ0,has been identified. This means that the curvature, observed in the specific heat curve,does not exhibit or indicate an existence of a phase transition around a τ0 temperature.Now, according to the condition on α parameter which appears in the zeta function (see

-120

-100

-80

-60

-40

-20

0

20

0 5 10 15 20

F/m

c2

τ

0

5

10

15

20

25

30

35

40

0 5 10 15 20

U/m

c2

τ

0

1

2

3

4

5

6

7

8

0 5 10 15 20

S/k B

τ

τ0=1.70

0

0.5

1

1.5

2

2.5

0 5 10 15 20

Cv/

k B

τ

τ0=1.70

Figure 1 One-dimensional thermal properties for both oscillators in the case wherer = 1.

Annexe B), we can distinguish two regions : the first, defined by r ≥ 0.5, corresponds tothe relativistic regime, and the other, with r < 0.5, represents the non-relativistic regime.These observations, for both Dirac and Klein-Gordon oscillators, are shown clearly in theFig. 2. Also, following the same figure, three remarks can be made :


– The τ0 reduce temperature increases for the values of r > 1, and disappears whenr < 1.

– When r < 0.5 all curves coincide with the non-relativistic limit (r = 10−10).– For all values of r , all curves of the specific heat coincide with the limit 2kB.

0

2

4

6

8

10

0 5 10 15 20

S/k B

τ

r=2r=1

r=0.99r=0.9r=0.5

0

0.5

1

1.5

2

2.5

0 5 10 15 20

Cv/

k B

τ

r=2r=1

r=0.99r=0.9r=0.5

(a) for the values of r ≥ 0.5

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18 20

S/k B

τ

r=0.45r=0.3r=0.1

r=10-10

0

0.5

1

1.5

2

0 2 4 6 8 10 12 14 16 18 20

Cv/

k B

τ

r=0.45r=0.3r=0.1

r=10-10

(b) for the values of r < 0.5

Figure 2 The entropy and specific heat as a function of a reduced temperature τ fordifferent values of the parameter r.

Pacheco et al [11] have been studied the thermal properties of a Dirac oscillator in onedimension. They obtained all thermodynamics quantities by using the Euler-MacLaurinapproximation(see Annexe A). The formalism used by [11] is valid, only, for higher tem-peratures. So, in order to cover all range of temperatures, we have employed the Hurwitzzeta function method.

In the Fig. 3, we are focused on the curves of the numerical specific heat calculated fordifferent values of the parameter r. Then, for comparison with [11], we have inserted thenumerical calculation of the specific heat based on the Euler-MacLaurin approximation.Thus, we can see that our results can be considered as an improvements of the resultsobtained in [11]. This consideration can be argued as follows : (i) all thermal quantitiesobtained from the zeta function method are valid in all range of temperatures, and (ii) thedivergence of the partition function, which appears in Euler-MacLaurin formula, has beenremoved. All results obtained in this case can be extended to the case of the Klein-Gordonoscillator (see Fig. 4).

Finally, one may compute the vacuum expectation value of the energy defined by [15]

ε0 = lims→−1

∞∑n=0

|εn|−s . (24)

From the spectrum of energy of a Dirac oscillator (Eq. (13)), the equation (24) can beexpressed in terms of the ζH as follows :

ε0mc2

=√2rζH

(−1

2,1

2r

). (25)


0

0.5

1

1.5

2

2.5

0 5 10 15 20

Cv/

k B

τ

r=1 from Pacheko et alr=1 using ζH function

r=0.99 using ζH functionr=0.9 using ζH function

Figure 3 Comparison of our specific heat for different values of r with that obtainedby using the Euler-MacLaurin formula for a Dirac oscillator in one-dimension.

0

0.5

1

1.5

2

2.5

0 5 10 15 20

Cv/

k B

τ

r=1 using EulerMaclaurin formula r=1 using ζH function

r=0.99 using ζH function r=0.9 using ζitH function

Figure 4 Comparison of our specific heat for different values of r with that obtainedby using the Euler-MacLaurin formula for a Klein-Gordon oscillator in one-dimension.

Using the asymptotic series corresponding to the Hurwitz zeta function (see Annexe B),Eq. (25) becomes

ε0mc2

= − 1

3r+

1

2− 2

3

∞∑k=2

Bk

k!

Γ(−3

2+ k

)Γ

(−3

2

) (1

2r

)1−k. (26)

In the Fig. (5), we show the ε0mc2

as a function of the parameter r. We can see thatthe vacuum expectation value of the energy, which depend on the parameter r, canapproximate with

ε0mc2

� − 1

3r+

1

2. (27)


0.5 1.0 1.5 2.0 2.5r

−10

−8

−6

−4

−2

0

ε 0/mc2

−

13r+1

2

−

13r+1

2−0.0833r

−

13r+1

2−

23

k=6∑

k=2

Bk

k!

Γ(−1.5+k)

Γ(−1.5)( 12r)1−k

Figure 5 The reduced vacuum expectation value of the energy ε0mc2

versus a parameterr around a relativistic region.

4. Conclusion

In this work, we reviewed the relativistic harmonic oscillator for both fermionic andbosonic massive particles in one dimension. The statistical quantities of both Dirac andKlein-Gordon oscillators were investigated by employing the zeta function method. Theboth cases have been confronted with those obtained by using the Euler-MacLaurin for-mula. The vacuum expectation value of the energy, for both oscillators, has been estima-ted.

A Euler-MacLaurin Formula

The partition function Z of the Dirac oscillator at finite temperature T is obtainedthrough the Boltzmann factor [7],

Z =n∑

n=0

e−β(En−E0) = eβ√b

n∑n=0

e−β√an+b, (A.1)

where β = 1kBT

, kB is the Boltzmann constant, E0 is the ground state energy corres-pondent to n = 0.

Before entering in the calculations, let us test the convergence of the series of (A.1).For that, we apply the integral test which shows that the series and the integral convergeor diverge together. So, from (A.1), we can see that the function f(x), where

f (x) = e−β√an+b, (A.2)


is a decreasing positive function, and the integralˆ ∞

0

f (x) dx =2

aβ2e−β

√b(1 + β

√b), (A.3)

is convergent. This means that, according to the criterion of the integral test, the nume-rical partition function Z converges.

In order to evaluate this function, we use the Euler-MacLaurin formula defined asfollows ∞∑

x=0

f (x) =1

2f (0) +

ˆ ∞

0

f (x) dx−∞∑p=1

1

(2p)!B2pf

(2p−1) (0) , (A.4)

Here, B2p are the Bernoulli numbers, f (2p−1) is the derivative of order (2p− 2). In ourcase, we have taken B2 =

16

and B4 = − 130

.In our case, we have used

f (1) (0) =aβ

2√be−β

√b (A.5)

f (3) (0) =

[−3βa3

8 (b)5/2− 3β2a3

8 (b)2− β3a3

8 (b)3/2

]e−β

√b. (A.6)

Following Eqs. (A.4), (A.5) and (A.6), the partition function can be cast into

Z =1

2+

1

2rτ +

1

2rτ 2 +

(r

6− r3

60

)1

τ− 1

30

r3

τ 2− 4

45

r3

τ 3. (A.7)

B Some Properties of Zeta Function

The Riemann zeta function is defined by[15]

ζ (s) =∞∑n=0

1

ns, with s ∈ C. (B.1)

Nowadays the Riemann zeta function is just one member of a whole family of zeta func-tion’s (Hurwitz, Epstein,Selberg). The most important of them is the Hurwitz zeta func-tion ζH given by

ζH (s, α) =∞∑n=0

1

(n+ α)s, (B.2)

where 0 < α ≤ 1, is a well-defined series when �e (s) > 1, and can be analyticallycontinued to the whole complex plane with one singularity, a simple pole with residue 1at s = 1.

An integral representation is

ζH (s, α) =1

Γ (s)

ˆ ∞

0

dtts−1e−tα

1− e−t, � (s) > 1, � (α) > 0. (B.3)

It can be shown that ζH (s, α) has only one singularity –namely a simple pole at s = 1

with residue 1 and that it can be analytically continued to the rest of the complex s-plane.


Also, we can shown that ζH (s, α) have the following properties :

ζH (0, α) =1

2− α, (B.4)

ζH (−m,α) = −Bm+1 (α)

m+ 1, m ∈ N, (B.5)

Br(a) being the Bernoulli polynomials. The asymptotic series corresponding the Hurwitzzeta function is given by

ζH (1 + z, α) =1

zα−z +

1

2α−1−z +

1

z

∞∑k=2

Bk

k!

Γ (z + k)

Γ (z)α−z−k, (B.6)

with Bkare Bernoulli’s numbers.

Références

[1] D. ItÃŽ, K. Mori and E. Carriere, Nuovo Cimento A, 51, 1119 (1967). 1.

[2] M. Moshinsky and A. Szczepaniak, J. Phys. A : Math. Gen, 22, L817 (1989). 1.

[3] R. P. Martinez-y-Romero and A. L. Salas-Brito, J. Math. Phys, 33 , 1831 (1992). 1.

[4] M. Moreno and A. Zentella, J. Phys. A : Math. Gen, 22 , L821 (1989). 1.

[5] J. Benitez, P. R. Martinez y Romero , H. N. Nunez-Yepez and A. L. Salas-Brito,Phys.Rev. Lett, 64, 1643–5 (1990). 1.

[6] C. Quesne and V. M. Tkachuk, J. Phys. A : Math. Gen, 41 , 1747–65 (2005). 1.

[7] A. Boumali and H. Hassanabadi, Eur. Phys. J. Plus. 128, 124 (2013). 1., A

[8] C. Quimbay and P. Strange, arXiv:1311.2021 (2013). 1.

[9] C. Quimbay and P. Strange, arXiv:1312.5251 (2013). 1.

[10] A. Franco-Villafane, E. Sadurni, S. Barkhofen, U. Kuhl, F. Mortessagne, and T. H.Selig- man, Phys. Rev. Lett. 111, 170405 (2013). 1.

[11] M. H. Pacheco, R. R. Landim and C. A. S Almeida, Phys. Lett. A, 311, 93–96 (2003).1., 3.2

[12] Marina-Aura Dariescu and C. Dariescu, J. Phys. : Condens. Matter 19, 256203(2007). 1., 3.1

[13] Marina-Aura Dariescu and C. Dariescu, Chaos. Solitons and Fractals 33 , 776–781(2007). 1., 3.1

[14] Marina-Aura Dariescu and C. Dariescu, Rom. Journ. Phys, 56, 1043–1052, (2011).1.

[15] E. Elizalde, Ten physical applications of spectral zeta functions, Springer-VerlagBerlin Heidelberg (1995). 1., 3.2, B

[16] W. Greiner, Quantum Mechanics: An Introduction, 4th ed, Springer-Verlag, Berlin,(2001). 2.1


The Effectiveness of Hénon Map for Chaotic Optimization

Algorithms Using a Global Locally Averaged Strategy

Tayeb Hamaizia ∗

Department of Mathematics, Faculty of Sciences, University Constantine -1-, Algeria

Received 27 February 2014, Accepted 20 December 2014, Published 10 January 2015

Abstract: Chaos has three important dynamic properties : the sensitive dependence on initial conditions,the intrinsic stochastic property and ergodicity. So it was applied to optimization problems. In mostoptimization methods such as the COA [1] and [4], variables chaos generated by different typesof applications such as the Logistic, the application Tent, Lozi, Ikeda and others have shown veryinteresting results that random application. The main idea is to improve the convergence of theoptimization chaotic. In this paper, a chaotic strategy is proposed based on a Hénon application chaotic.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Chaos ; Chaos optimization ; Hénon map ; Global optimizationPACS (2010): 05.45.-a ; 05.10.-a ; 05.45.Gg ; 05.45.Pq ; 02.70.-c ; 02.60.-x ; 87.55.de

1. Introduction

Theory of chaos is one of the most important achievements of nonlinear system research.The term chaos was coined par Li and Yorke [7] are recognized as very useful in many en-gineering applications. Chaotic dynamical systems have many interesting properties, such asno periodic and complex temporal behavior, sensitive dependence on initial conditions or theso-called butterfly effect, fractal structure, and long-term unpredictability [7], [8], [9] and [11].The references included above show interactive demonstrations of these properties. With thesecharacteristics of chaos, we can apply it in optimization calculation. Chaotic optimization isa novel stochastic optimization algorithm, which directly utilizes chaotic variables to searchthe optimal solution. The ergodic, regularity and intrinsic stochastic property of chaos makechaotic optimization to obtain the global optimal solution more possible than the method ha-ving been adopted before. It can more easily escape from local minima than other stochasticalgorithms. Bing Li firstly proposed chaos optimization algorithm (Chaos Optimization Algo-

∗. Email :[email protected]


rithm, COA) in 1997. His basic idea is to map chaos variables into the ranges of optimizationvariables, and then search in the ranges of optimization variables according to the disciplineof chaos changes. The paper is organized as follows : in Section 2 we present a new strategybased on locally averaged strategy of the global search and Hénon chaotic attractor, in Section3 we analyze the effectiveness of the proposed algorithm on a benchmark suite of 4 well-knownnonlinear test functions which are optimized. Finally, we propose a conclusion.

2. The Principal of Chaos Optimization

2.1 Chaos Model

Chaos theory is recognized as very useful in many engineering applications. An essentialfeature of chaotic systems is sensitive dependence on initial condition, (i.e. small changes inthe parameters or the starting values for the data lead to drastically different future behaviours).The application of chaotic sequences can be an interesting alternative to provide the searchdiversity in an optimization procedure. Due to the non-repetition of chaos, it can carry outoverall searches at higher speeds than stochastic ergodic searches that depend on probabilities[1], [4] and [12]. In the most of COA methods [3], chaos variables generated by logistic mapand one of the most famous being a two-dimensional discrete map suggested by Lozi andstudied in detail by others [1] and [2]. It is possible to change the form of this map to obtainother chaotic attractors, but in this paper, we assume chaos variable can be generated by aHénon two-dimensional discrete map. The Hénon map is a discrete-time dynamical system [9],[10] and [11]. It is one of the most studied examples of dynamical systems that exhibit chaoticbehaviour. The Hénon map takes a point (xn, yn) in the plane and maps it to a new point map

⎧⎪⎨⎪⎩ y1(k) = 1− a(y1(k − 1))2 + by(k − 1)

y(k) = y1(k − 1)(1)

where k is the iteration number. In this work, the values of y are normalized in the range[0, 1] to each decision variable in unidimensional space of optimization problem. This transfor-mation is given by

z(k) =(y(k)− α)

β − α.

The parameters used in this work are a = 1.4 and b = 0.3, these values suggested by (1).Anexample of evolution of new map is shown in Fig(1). The properties of stochastic sensitivity toinitial value and ergodicity of the tow-dimensional (1) is expressed as Fig.1 by iterating 1000times.


FIGURE 1 Attractor and temporal series of Hénon map

FIGURE 2 Density of iterated values of the Hénon map

2.2 Design of the Algorithm

Consider the following optimization problem about the minimum of functions continuouslydifferentiable :Find X to minimize f(X);X ∈[x1; x2; ...; xn]Subject to xi ∈[Li;Ui] ; i = 1; 2; ...;n;

Where f is the objective function, and Xis the decision solution vector consisting of n variablesxi ∈ Rn bounded by lower (Li) and upper limits (Ui).Suppose

Inputs :Mg : max number of iterations of chaotic Global search.Ml : max number of iterations of chaotic Local search.Mgl1, : max number of iter of chaotic Local search in Global search.Mgl2 : max number of iter of chaotic Local search in Global search.Mg × (Mgl1 +Mgl2) +Ml : stopping criterion of chaotic optimization method in iter.λgl1 : step size in first global-local search.λgl2 : step size in second global-local search.λ : step size in chaotic local search.


Outputs :X : best solution from current run of chaotic search.f : best objective function (minimization problem).

Algorithm 1 COHM-Step 1 :Initialize the number Mg, Mgl1,Mgl2, Ml of chaotic search and initialization of va-riables and initial conditions Set k = 1, y(0), y1(0),a = 1.4 and b = 0.3 of Hénon map. Set theinitial best objective function f = +∞-Step 2 : algorithm of chaotic global search :

while k ≤Mg doxi(k) = Li + zi(k).(Ui − Li)if f(X(k)) < f then

X = X(k); f = f(x(k))end if-Step 2-1 : sub algorithm of chaotic local search :while j ≤Mgl1 do

if r ≤ 0.5 thenxi(j) = xi + λgl1zi(j).|(Ui − Li)|

elsexi(j) = xi − λgl1zi(j).|(Ui − Li)|

end ifif f(X(j)) < f then

X = X(j); f = f(x(j))end ifj = j + 1

end while-Step 2-2 : sub algorithm of chaotic local search :while s ≤Mgl2 do

if r ≤ 0.5 thenxi(s) = xi + λgl2zi(s).|(Ui − Li)|

elsexi(s) = xi − λgl2zi(s).|(Ui − Li)|

end ifif f(X(s)) < f then

X = X(s); f = f(x(s))end ifs = s+ 1

end whilek = k + 1

end while-Step 3 : algorithm of chaotic local search :while k ≤Mg × (Mgl1 +Mgl2) +Ml do

if r ≤ 0.5 thenxi(k) = xi + λzi(k).|(Ui − Li)|

elsexi(k) = xi − λzi(k).|(Ui − Li)|

end ifif f(X(k)) < f then

X = X(k); f = f(x(k))end ifk = k + 1

end while


3. Numerical Results

In order to verify the typical function of this paper to optimize the effectiveness of thealgorithm, 4 target function expression is as follows :

F1 = 100(x21 − x2)

2 + (1− x1)2. (2)

−2.048 ≤ xi ≤ 2.048

F2 = (1 + (x1 + x2 + 1)2(19− 14x1 + 3x21 − 14x2 + 6x1x2 + 3x2

2))

(30 + (2x1 − 3x2)2(18− 32x1 + 12x12 + 48x2 − 36x1x2 + 27x2

2)).(3)

−2 ≤ xi ≤ +2

F3 = −cos(x1)cos(x2)exp(−(x1 − pi)2 − (x2 − pi)2). (4)

−100 ≤ xi ≤ +100

F4 = −.5 + ((sin√

(x21 + x2

2))2 − .5)./(1 + .001(x2

1 + x22))

2; (5)

−4 ≤ xi ≤ +4

FIGURE 3 The three-dimensional maps of the four functions

For the duration of the chaotic local search, the step size λ is an important parameter inconvergence behavior of optimization method, which adjusts small ergodic ranges around X .The step size λ is employed to control the impact of the current best solution on the generatingof a new trial solution. A small λ tends to perform exploitation to refine results by local search,while a large one tends to facilitate a global exploration of search space[1, 2]. A suitable valuefor the step size usually provides balance between global and local exploration abilities and


consequently a reduction on the number of iterations required to locate the optimum solution.In this work, the tested values of step size in chaotic optimization method based on Hénon mapare described as follows :

– Mg = 600;Ml = 100λ = 0.1, λgl1 = 0.04, λgl2 = 0.01,Mgl1 = 2,Mgl2 = 2

Best Value Mean Value Std. Dev (x, y) Time

F1 0.0000 0.0003 0.0003 (0.9913, 0.9831) 33.9249s

F2 3.0031 3.0031 0.0000 (0.0032, -0.9978) 34.4000s

F3 -0.9961 -0.9961 0.0000 ( 3.1364 , 3.0906) 37.6054s

F4 -0.9993 -0.9989 0.0001 (-0.0310, 0.0080) 35.6340s

TABLE 1 Optimized function test using COHM approaches

– Mg = 600;Ml = 100, λ = 0.001, λgl1 = 0.04, λgl2 = 0.01,Mgl1 = 2,Mgl2 = 2


F1 0.0000 0.0000 0.0000 (1.0001, 1.0002) 33.8875s

F2 3.0000 3.0000 0.0000 (-0.0001, -1.0001) 33.9260s

F3 -1.0000 -0.9997 0.0001 (3.1438 , 3.1411) 37.1161s

F4 -0.9999 -0.9998 0.0001 (-0.0006, -0.0017) 35.7883s


– Mg = 600;Ml = 100, 0.001 ≤ λ ≤ 0.1, λgl1 = 0.04, λgl2 = 0.01,Mgl1 = 2,Mgl2 = 2


F1 0.0000 0.0001 0.0001 (1.0044, 1.0088) 35.2739s

F2 3.0000 3.0027 0.0010 ( 0.0024, -0.9985) 35.4629s

F3 -0.9994 -0.9963 0.0008 ( 3.1374 , 3.0952) 38.2361s

F4 -0.9996 -0.9989 0.0001 (-0.0290 , 0.0063) 36.9733s


4. Conclusion

The chaotic optimization method based on Hénon map COHM methodologies were suc-cessfully validated for testing four different function cost. From the case studies and compari-


son of the results through three tested COHM approaches, it has been show that the parameterof step size λ is essential to the good convergence profile. In this context, the parameter λ regu-lates the trade-off between the global and local exploration abilities of the chaotic local search.However, in future works will include a detailed study of self-adaptive heuristics for the stepsize design.

References

[1] T. Hamaizia, R. Lozi, An improved chaotic optimization algorithm using a new globallocally averaged strategy, Journal of Nonlinear Systems and Applications 3 (2) (2012)58-63.

[2] L. S. Coelho, Tuning of PID controller for an automatic regulator voltage system usingchaotic optimization approach, Chaos, Solitons and Fractals, 39, 2009, 1504-1514.

[3] B. Li, W.S. Jiang, Chaos optimization method and its application, Journal of ControlTheory and Application 14 (4) (1997) 613-615.

[4] R. Caponetto, L. Fortuna, S. Fazzino and M. G. Xibilia, Chaotic Sequences toImprove the Performance of Evolutionary Algorithms, IEEE Transactions on EvolutionaryComputation, Vol. 7, n 3, June 2003, 289-304.

[5] W.-H. Ho, J.-H. Chou, C.-Y. Guo, Parameter identification of chaotic systems usingimproved differential evolution algorithm. Nonlinear Dyn. 61, 2941 (2010)

[6] Z. Wei, F. Ge, Y. Lu, L. Li, Y. Yang, Chaotic ant swarm for the traveling salesman problem.Nonlinear Dyn. DOI 10.1007/s11071-010-9889-x (2010).

[7] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 , 1975,985-992.

[8] S. H. Strogatz, Nonlinear dynamics and chaos, Massachussets : Perseus Publishing, 2000.

[9] M. Hénon. A two dimensional mapping with a strange attractor Commun. Math. Phys.1976 ; 50(1), 69-77.

[10] A. Aziz-Alaoui, C. Robert, C. Grebogi, Dynamics of a Hénon-Lozi map, Chaos, SolitonsFractals 12 (11) (2001) 2323-2341.

[11] M. Misiurewicz, strange attractors for the lozi mappings. Annals of the new york academyof sciences, nonlinear dyn. 357, 348-358(1980).

[12] S. Cong, G. Li and X. Feng, An Improved Algorithm of Chaos Optimization, Proceedings2010 8th IEEE International Conference on Control and Automation

[13] Li B, Jiang WS. Optimizing complex function by chaos search. Cybern Syst 1998 ;29(4) :409-19.


Weakton Model of Elementary Particlesand Decay Mechanisms∗

Tian Ma1 and Shouhong Wang2†

1Department of Mathematics, Sichuan University, Chengdu, P. R. China2Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Received 15 April 2014, Accepted 20 December 2014, Published 10 January 2015

Abstract: Sub-atomic decays and electron radiations indicate that there must be interior

structures for charged leptons, quarks and mediators. The main objectives of this article are 1)

to propose a sub-leptons and sub-quark model, which we call weakton model, and 2) to derive

a mechanism for all sub-atomic decays and bremsstrahlung. The theory is based on 1) a theory

on weak and strong charges, 2) different levels of weak and strong interaction potentials, 3)

a new mass generation mechanism, and 4) an angular momentum rule. The weakton model

postulates that all matter particles (leptons, quarks) and mediators are made up of massless

weaktons. The weakton model offers a perfect explanation for all sub-atomic decays and all

generation/annihilation precesses of matter-antimatter. In particular, the precise constituents

of particles involved in all decays both before and after the reaction can now be precisely derived.

In addition, the bremsstrahlung phenomena can be understood using the weakton model. Also,

the weakton model offers an explanation to the baryon asymmetry problem, and provides an

alternate explanation of the solar neutrino problem.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Elementary Particles; Sub-Quark; Sub-Lepton; Sub-Mediators; Weakton Model;

Subatomic Decay; Matter And Antimatter Creation And Annihilation; Weakton Exchange; Weak

Interaction; Strong Interaction; Weak Charge; Strong Charge

PACS (2010): 12.40.-y; 12.15.-y ; 14.60.-z; 14.65.-q; 13.30.-a; 14.20.Dh; 12.39.-x; 24.85.+p;

12.15.Ff; 13.90.+i

∗ The work was supported in part by the Office of Naval Research, by the US National Science Founda-

tion, and by the Chinese National Science Foundation.† Email:[email protected], http://www.indiana.edu/ fluid


1. Introduction

The matter in the universe is made up of a number of fundamental constituents. The

current knowledge of elementary particles shows that all forms of matter are made up of

6 leptons and 6 quarks, and their antiparticles. The basic laws governing the dynamical

behavior of these elementary particles are the laws for the four interactions/forces: the

electromagnetism, the gravity, the weak and strong interactions. Great achievements and

insights have been made for last 100 years or so on the understanding of the structure

of subatomic particles and on the fundamental laws for the four interactions; see among

many others [3, 2, 5, 12, 10].

However, there are still many longstanding open questions and challenges. Here are

a few fundamental questions which are certainly related to the deepest secret of our

universe:

Q1 What is the origin of four forces?

Q2 Why do leptons not participate in strong interactions?

Q3 What is the origin of mass?

Q4 What is the mechanism of subatomic decays and reactions?

Q5 Why can massless photons produce massive particles? Or in general, why can lepton

and anti-lepton pairs produce hadron pairs?

Q6 Are leptons and quarks true elementary particles? Do leptons and quarks have

interior structure?

Q7 Why are there more matters than anti-matters? This is the classical baryon asym-

metry problem.

Q8 What are the strong and weak force formulas?

Q9 Why, in the same spatial scale, do strong and weak interactions exhibit both repelling

and attraction?

Q10 Why are the weak and strong interactions short-ranged, and what are the ranges of

the four interactions?

Q11 What is the mechanism of quark confinement?

Q12 What is the mechanism of bremsstrahlung?

Q13 Why is there a loss of solar neutrinos?

The main objectives of this article are 1) to study the mechanism of subatomic decays,

2) to propose a weakton model of elementary particles, and 3) to explain the above

questions Q1–Q13. We proceed as follows.

1. The starting point of the study is the puzzling decay and reaction behavior of sub-

atomic particles. For example, the electron radiations and the electron-positron anni-

hilation into photons or quark-antiquark pair clearly shows that there must be interior

structure of electrons, and the constituents of an electron contribute to the making of

photon or the quark in the hadrons formed in the process. In fact, all sub-atomic decays

and reactions show clearly the following conclusion:

There must be interior structure of charged leptons, quarks and mediators. (1)

2. The above conclusion motivates us to propose a model for sub-lepton, sub-quark, and


sub-mediators. It is clear that any such model should obey four basic requirements.

The first is the mass generation mechanism. Namely, the model should lead to con-

sistency of masses for both elementary particles, which we call weaktons to be introduced

below, and composite particles (the quarks, leptons and mediators). Since the media-

tors, the photon γ and the eight gluons gk (k = 1, · · · , 8), are all massless, a natural

requirement is that

the proposed elementary particles—weaktons— are massless. (2)

Namely, these proposed elementary particles must have zero rest mass.

The second requirement for the model is the consistency of quantum numbers for

both elementary and composite particles. The third requirement is the exclusion of

nonrealistic compositions of the elementary particles, and the fourth requirement is the

weakton confinement.

3. Careful examinations of these requirements and subatomic decays/reactions lead us

to propose six elementary particles, which we call weaktons, and their anti-particles:

w∗, w1, w2, νe, νμ, ντ ,

w∗, w1, w2, νe, νμ, ντ ,(3)

where νe, νμ, ντ are the three generation neutrinos, and w∗, w1, w2 are three new particles,

which we call w-weaktons. These are massless, spin-12particles with one unit of weak

charge gw. Both w∗ and w∗ are the only weaktons carrying strong charge gs.

With these weaktons at our disposal, the weakton constituents of charged leptons and

quarks are then given as follows:

e = νew1w2, μ = νμw1w2, τ = ντw1w2,

u = w∗w1w1, c = w∗w2w2, t = w∗w2w2,

d = w∗w1w2, s = w∗w1w2, b = w∗w1w2,

(4)

where c, t and d, s, b are distinguished by the spin arrangements; see (59) and (60).

4. Using the duality given in the unified field theory for four interactions, the mediators

of strong, weak and electromagnetic interactions include the photon γ, the vector bosons

W± and Z, and the gluons gk, together with their dual fields φγ, φ±W , φ0

Z , and φkg . The

constituents of these mediators are given by

γ = cos θww1w1 − sin θww2w2 (�,�),

Z0 = cos θww2w2 + sin θww1w1 (�,�),

W− = w1w2(�,�),

W+ = w1w2(�,�),

gk = w∗w∗(�,�), k = color index,

(5)


and the dual bosons:

φγ = cos θww1w1 − sin θww2w2(↑↓, ↓↑),φ0Z = cos θww2w2 + sin θww1w1(↑↓, ↓↑),

φ−W = w1w2(↑↓, ↓↑),φ+W = w1w2(↑↓, ↓↑),

φkg = w∗w∗(↑↓, ↓↑),

(6)

where θw ∼= 28.76◦ is the Weinberg angle.

Remarkably, both the spin-1 mediators in (5) and the spin-0 dual mediators in (6)

have the same weakton constituents, differing only by their spin arrangements. The

spin arrangements clearly demonstrate that there must be spin-0 particles with the same

weakton constituents as the mediators in (5). Consequently, there must be dual mediators

with spin-0. This observation clearly supports the unified field model presented in [8, 7].

Conversely, the existence of the dual mediators makes the weakton constituents perfectly

fit.

5. Also, a careful examination of weakton constituents predicts the existence of an

additional mediator, which we call the ν-mediator:

φ0ν =

∑l

αlνlνl(↓↑),∑l

α2l = 1, (7)

taking into consideration of neutrino oscillations. When examining decays and reactions

of sub-atomic particles, it is apparent for us to predict the existence of this mediator.

6. One important conclusion of the aforementioned weakton model is that all particles—

both matter particles and mediators—are made up of massless weaktons. A fundamental

question is how the mass of a massive composite particle is generated. In fact, based on

the Einstein formulas:

d

dt�P =

√1− v2

c2�F , m =

√1− v2

c2E

c2, (8)

we observe that a particle with an intrinsic energy E has zero mass m = 0 if it moves in

the speed of light v = c, and possess nonzero mass if it moves with a velocity v < c. Hence

by this mass generation mechanism, for a composite particle, the constituent massless

weaktons can decelerate by the weak force, yielding a massive particle.

In principle, when calculating the mass of the composite particle, one should also

consider the bounding and repelling energies of the weaktons, each of which can be very

large. Fortunately, the constituent weaktons are moving in the “asymptotically-free” shell

region of weak interactions as indicated by the weak interaction potential/force formulas,

so that the bounding and repelling contributions to the mass are mostly canceled out.

Namely, the mass of a composite particle is due mainly to the dynamic behavior of the

constituent weaktons.

7. As we mentioned earlier, one requirement for the weakton model is the consistency

of quantum numbers for both elementary particles and composite particles. In fact, the


weakton model obeys a number of quantum rules, which can be used to exclude unrealistic

combinations of weaktons. The following rules are introduced for this purpose:

a) Weak color neutral rule: each weakton is endowed with a weak color quantum num-

ber, and all weakton composite particles must be weak color neutral.

b) BL = 0, LiLj = 0 (i �= j), where B is the baryon number and L is the lepton

number.

c) L+Qe = 0 if L �= 0 and |B +Qe| ≤ 1 if B �= 0.

d) Angular Momentum Rule: Only the fermions with spin s = 12can rotate around a

center with zero moment of force. The particles with s �= 12will move in a straight

line unless there is a nonzero moment of force present.

The angular momentum is a consequence of the Dirac equations, and it is due to

this rule that there are no spin-3/2 quarks.

8. Remarkably, the weakton model offers a perfect explanation for all sub-atomic de-

cays. In particular, all decays are achieved by 1) exchanging weaktons and consequently

exchanging newly formed quarks, producing new composite particles, and 2) separating

the new composite particles by weak and/or strong forces.

One aspect of this decay mechanism is that we know now the precise constituents of

particles involved in all decays/reactions both before and after the reaction. It is therefore

believed that the new decay mechanism provides clear new insights for both experimental

and theoretical studies.

9. The weakton theory, together with the unified field theory developed in [8, 7], provides

sound explanations and new viewpoints for the twelve fundamental questions given at

the beginning of the Introduction.

We end this Introduction by mentioning that there have been numerous studies on

sub-quark and sub-lepton models; see among others [9, 1, 11, 4]

The paper is organized as follows. A brief introduction to the current understanding

of elementary particles is given in Section 2, focusing on 1) the constituents of subatomic

particles, and 2) decays. Section 3 addresses a few theoretical foundations needed for

introducing the weakton model, which is then introduced in Section 4. All decays are

then perfectly explained using the weakton model in Section 5. An application of the

weakton model to bremsstrahlung is given in Section 6. Section 7 summaries conclusions

of this article, focusing on answers and explanations to the 13 open questions.

2. Current Knowledge of Elementary Particles

The current view on subatomic particles classifies all particles into two basic classes,

bosons and fermions:

bosons = integral spin particles,

fermions = fractional spin particles.


However, based on their properties and laws in Nature, all particles are currently classified

into four types:

leptons, quarks, mediators, hadrons.

Hereafter we recapitulate the definitions and the quantum characterizations of these par-

ticles.

2.1 Leptons

Leptons are fermions which do not participate in strong interaction, and have three

generations with two in each generation:⎛⎜⎝ e

νe

⎞⎟⎠ ,

⎛⎜⎝ μ

νμ

⎞⎟⎠ ,

⎛⎜⎝ τ

ντ

⎞⎟⎠ ,

where e, μ, τ are the electron, the muon, the tau, and νe, νμ, ντ are the e neutrino, the μ

neutrino, the τ neutrino. Together with antiparticles, there are total 12 leptons:

particles: (e−, νe), (μ−, νμ), (τ−, ντ ),

antiparticles: (e+, νe), (μ+, νμ), (τ+, ντ ).

The quantum numbers of leptons include the mass m, the charge Q, the lifetime τ ,

the spin J , the e-lepton number Le, the μ-lepton number Lμ, and the τ -lepton number

Lτ . Table 1 lists typical values of these quantum numbers, where the mass is in MeV/c2,

lifetime is in seconds, and the charge is in the unit of proton charge. Also, we remark

that the left-hand property of neutrinos is represented by J = −12for ν, and J = +1

2for

ν.

2.2 Quarks

Based on the Standard Model, there are three generations of quarks containing 12 parti-

cles, which participate in all interactions:

quarks: (u, d), (c, s), (t, b),

antiquarks: (u, d), (c, s), (t, b).

The celebrated quark model assets that three quarks are bounded together to form

a baryon, and a pair of quark and antiquark are bounded to form a meson. Quarks are

confined in hadrons, and no free quarks have been found in Nature. This phenomena

is called quark confinement, which can be very well explained using the three levels of

strong interaction potentials derived using a unified field theory developed recently in

[8, 7]; see discussions in Section 3.4.

The quantum numbers of quarks include the mass m, the charge Q, the baryon

number B, the spin J , the strange number S, the isospin I and its third component I3,

the supercharge Y , and the parity P . These quantum numbers are listed in Table 2.


Table 1 Leptons

lepton M Q J Le Lμ Lτ τ

e− 0.51 -1 ±1/2 1 0 0 ∞

νe 0 0 −1/2 1 0 0 ∞

e+ 0.51 +1 ±1/2 -1 0 0

νe 0 0 +1/2 -1 0 0 ∞

μ− 105.7 -1 ±1/2 0 1 0 2.2× 10−6

νμ 0 0 −1/2 0 1 0 ∞

μ+ 105.7 +1 ±1/2 0 -1 0

νμ 0 0 +1/2 0 -1 0 ∞

τ− 1777 -1 ±1/2 0 0 1 3× 10−13

ντ 0 0 −1/2 0 0 1 ∞

τ+ 1777 +1 ±1/2 0 0 -1

ντ 0 0 +1/2 0 0 -1 ∞

2.3 Mediators

The standard model shows that associated with each interaction is a class of mediators.

Namely, there are four classes of mediators:

Gravitation: graviton gG,

Electromagnetism: photon γ,

Weak interaction: vector meson W±, Z0,

Strong interaction: gluons gk (1 ≤ k ≤ 8).

The quantum numbers of these mediators include the mass m, the charge Q, the spin

J , and the lifetime τ , listed in Table 3.

With the unified field theory developed in [8, 7], we have obtained a natural duality


Table 2 Quarks

Quarks m Q J B S Y I I3 P

u 3 2/3 ±1/2 1/3 0 1/3 1/2 +1/2 +1

d 7 −1/3 ±1/2 1/3 0 1/3 1/2 −1/2 +1

c 1200 2/3 ±1/2 1/3 0 1/3 0 0 +1

s 120 −1/3 ±1/2 1/3 -1 −2/3 0 0 +1

t 1.7× 105 2/3 ±1/2 1/3 0 1/3 0 0 +1

b 4300 −1/3 ±1/2 1/3 0 1/3 0 0 +1

u 3 −2/3 ±1/2 −1/3 0 −1/3 1/2 −1/2 -1

d 7 1/3 ±1/2 −1/3 0 −1/3 1/2 +1/2 -1

c 1200 −2/3 ±1/2 −1/3 0 −1/3 0 0 -1

s 120 1/3 ±1/2 −1/3 +1 2/3 0 0 -1

t 1.7× 105 −2/3 ±1/2 −1/3 0 −1/3 0 0 -1

b 4300 1/3 ±1/2 −1/3 0 −1/3 0 0 -1

Table 3 Interaction Mediators

Interaction Mediator m Q J τ

Gravitation gG 0 0 2 ∞

Electromagnetic γ 0 0 1 ∞

W+ 8× 104 +1 1 3× 10−25

Weak W− 8× 104 -1 1 3× 10−25

Z0 9× 104 0 1 2.6× 10−25

Strong gk(1 ≤ k ≤ 8) 0 0 1 ∞


between the interacting fields {gμν , Aμ,Waμ , S

kμ} and their dual field {ΦG

μ , φE, φaw, φ

ks}:

{gμν} ←→ ΦGμ ,

Aμ ←→ φE,

{W aμ} ←→ {φa

w},{Sk

μ} ←→ {φks}.

This duality leads to four classes of new dual bosonic mediators:

graviton gG ←→ vector boson ΦG,

photon γ ←→ scalar boson φγ ,

vector bosons W±, Z ←→ scalar bosons φ±W , φ0Z ,

gluons gk (1 ≤ k ≤ 8) ←→ scalar bosons φkg (1 ≤ k ≤ 8)

(9)

These dual mediators are crucial not only for the weak and strong potential/force formulas

given in the next section, but also for the weakton model introduced in this article. In

addition, the dual vector field ΦG gives rise to a unified theory for dark matter and dark

energy [6].

The quantum numbers of these dual mediators are given as follows:

ΦG : m = 0, J = 1, Q = 0, τ =∞,

φe : m = 0, J = 0, Q = 0, τ =∞,

φ±W (Higgs) : m ∼ 105, J = 0, Q = ±1, τ ∼ 10−21s,

φ0Z(Higgs) : m ∼ 1.25× 105, J = 0, Q = 0, τ ∼ 10−21s,

φkg : m =?, J = 0, Q = 0, τ =?.

(10)

2.4 Hadrons

Hadrons are classified into two types: baryons and mesons. Baryons are fermions and

mesons are bosons, which are all made up of quarks:

Baryons = qiqjqk, mesons = qiqj,

where qk = {u, d, c, s, t, b}. The quark constituents of main hadrons are listed as follows:

• Baryons (J = 12) : p, n,Λ,Σ±,Σ0,Ξ0,Ξ−.

p(uud), n(udd), Λ(s(du− ud)/√2),

Σ+(uus), Σ−(dds), Σ0(s(du+ ud)/√2),

Ξ−(uss), Ξ0(dss).

(11)

• Baryons (J = 32) : Δ++,Δ±,Δ0,Σ∗±,Σ∗0,Ξ∗0,Ξ∗−,Ω−.

Δ++(uuu), Δ+(uud), Δ−(ddd), Δ0(udd),

Σ∗+(uus), Σ∗−(dds), Σ∗0(uds),

Ξ∗0(uss), Ξ∗−(dss), Ω−(sss).

(12)


• Mesons (J = 0) : π±, π0, K±, K0, K0, η,

π+(ud), π−(ud), π0((uu− dd)/√2),

K+(us), K−(us), K0((ds), K0(ds),

η((uu+ dd− 2ss)/√6).

(13)

• Mesons (J = 1) : ρ±, ρ0, K∗±, K∗0, K∗0, ω, ψ,Υ,

ρ+(ud), ρ−(ud), ρ0((uu− dd)/√2),

K∗+(us), K∗−(us), K∗0(ds), K∗0(ds),

ω((uu+ dd)/√2), ψ(cc), Υ(bb).

(14)

2.5 Principal decays

Decays are the main dynamic behavior for sub-atomic particles, and reveal the interior

structure of particles. We now list some principal decay forms.

• Lepton decays:

μ− → e− + νe + νμ,

μ+ → e+ + νe + νμ,

τ− → e− + νe + ντ ,

τ− → μ− + νμ + ντ ,

τ− → π− + ντ ,

τ− → ρ− + ντ ,

τ− → K− + ντ .

• Quark decays:

d→ u+ e− + νe,

s→ u+ e− + νe,

s→ d+ g + γ (g the gluons),

c→ d+ s+ u,

• Mediator decays:

2γ → e+ + e−, qq,

W+ → e+ + νe, μ+ + νμ, τ+ + ντ ,

W− → e− + νe, μ− + νμ, τ− + ντ ,

Z0 → e+ + e−, μ+ + μ−, τ+ + τ−, qq.


• Baryon decays:

n→ p+ e− + νe,

Λ→ p+ π−, n+ π0,

Σ+ → p+ π0, n+ π+,

Σ0 → Λ + γ, Σ− → n+ π−,

Ξ0 → Λ + π0, Ξ− → Λ + π−,

Δ++ → p+ π+, Δ+ → p+ π0,

Δ0 → n+ π0, Δ− → n+ π−,

Σ∗± → Σ± + π0, Ξ∗0 → Σ0 + π0,

Ξ∗0 → Ξ0 + π0, Ξ∗− → Ξ− + π0.

• Meson decays:

π+ → μ+ + νμ, π0 → 2γ,

π− → μ− + νμ,

K+ → μ+ + νμ, π+ + π0, π+ + π+ + π−,

K− → μ− + νμ, π− + π0, π− + π+ + π−,

K0 → π+ + e− + νe, π+ + π−, π+ + π− + π0,

η → 2γ, π+ + π− + π0,

ρ± → π0, ρ0 → π+ + π−,

K∗± → K± + π0, K∗0 → K0 + π0,

ω → π0 + γ, π+ + π− + π0,

ψ → e+ + e−, μ+ + μ−,

Υ→ e+ + e−, μ+ + μ−, τ+ + τ−.

3. Theoretical Foundations for the Weakton Model

3.1 Angular momentum rule

It is known that the dynamic behavior of a particle is described by the Dirac equations:

i�∂ψ

∂t= Hψ, ψ = (ψ1, ψ2, ψ3, ψ4)

T (15)

where H is the Hamiltonian

H = −i�c(αk∂k) +mc2α0 + V (x), (16)


V is the potential energy, and α0, αk (1 ≤ k ≤ 3) are the Dirac matrices

α0 =

⎛⎜⎜⎜⎝1 0

1

−10 −1

⎞⎟⎟⎟⎠ , α1 =

⎛⎜⎜⎜⎝0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

⎞⎟⎟⎟⎠ ,

α2 =

⎛⎜⎜⎜⎝0 0 0 −i0 0 i 0

0 −i 0 0

i 0 0 0

⎞⎟⎟⎟⎠ , α3 =

⎛⎜⎜⎜⎝0 0 1 0

0 0 0 −11 0 0 0

0 −1 0 0

⎞⎟⎟⎟⎠ .

By the conservation laws in relativistic quantum mechanics, if a Hermitian operator L

commutes with H in (16):

LH = HL,

then the physical quantity L is conservative.

The main objective of this section is to prove the following angular momentum rule

for quantum systems.

Quantum Rule of Angular Momentum. Only the fermions with spin J = 12and the

bosons with J = 0 can rotate around a center with zero moment of force, and particles

with J �= 0, 12will move on a straight line unless there is a nonzero moment of force

present.

This quantum rule is very useful in describing how elementary particles are binding

together forming subatomic particles. This rule offers a mathematical foundation of the

weakton model and the theory of mediator cloud structure of charged leptons and quarks,

and explains why all stable fermions with mediator clouds possess spin J = 12.

To prove this quantum rule, we first recall the total angular momentum �J of a particle

defined by�J = �L+ s�S,

where �L = �r × �p is the orbital angular momentum, �p = −i�∇, s is the spin, and

�S = (S1, S2, S3), Sk = �

(σk 0

0 σk

).

Here σk (1 ≤ k ≤ 3) are the Pauli matrices.

Conservation laws based on quantum Hamiltonian dynamics (QHD)

Let H be the Hamiltonian energy of a conservative quantum system, which can be

described by the following Hamiltonian equations:

∂Ψ

∂t= HΦ(Ψ,Φ),

∂Φ

∂t= −HΨ(Ψ,Φ),

(17)


where

HΦ =δH

δΦ, HΨ =

δH

δΨ.

Let L be an observable physical quantity with the corresponding Hermitian operator

L for the conjugate fields (Ψ,Φ)T of (17), and L is expressed as

L =

(L11 L12

L21 L22

), LT

12 = L∗21.

Then the physical quantity L of system (17) is given by

L =

∫(Ψ†,Φ†)L

(Ψ

Φ

)dx =

∫ [Ψ†L11Ψ+ Φ†L22Φ + 2Re(Ψ†L12Φ)

]dx. (18)

It is clear that the quantity L of (18) is conserved if for the solution (Ψ,Φ)T of (17)

we havedL

dt= 0,

which is equivalent to∫ [H†

ΦL11Ψ+Ψ†L11HΦ − H†ΨL22Φ− Φ†L22HΨ (19)

+ 2Re(H†ΦL12Φ−Ψ†L12HΨ)]dx = 0.

We remark here that if the QHD is described by a complex valued wave function:

ψ = Ψ+ iΦ,

and its dynamic equation is linear, then (17) can be written as

i�∂ψ

∂t= Hψ, H =

∫ψ†Hψdx. (20)

In this case, the physical quantity L in (18) is in the form

L =

∫ψ†Lψdx, (21)

and the conservation law (19) of L is equivalent to

LH − HL = 0. (22)

The formulas (20)-(22) are the conservation laws of the classical quantum mechanics.

Hence, the conservation laws in (19) are the generalization to the classical quantum

mechanics, which are applicable to all conservative quantum systems, including the Klein-

Gordon systems and nonlinear systems.

Fermions


Consider fermions which obey the Dirac equations as (20) with the Hamiltonian

H = −i�c(αk∂k) +mc2α0 + V (r), (23)

where V is the potential energy of a central field, and α0, αk (1 ≤ k ≤ 3) are the Dirac

matrices

α0 =

(I 0

0 −I

), αk =

(0 σk

σk 0

)for 1 ≤ k ≤ 3, (24)

and σk are the Pauli matrices.

The total angular momentum J of a particle is

J = L+ sS,

where s is the spin, L is the orbital angular momentum

L = (L1, L2, L3) = r × p, p = −i�∇,L1 = −i�(x2∂3 − x3∂2),

L2 = −i�(x3∂1 − x1∂3),

L3 = −i�(x1∂2 − x2∂1),

(25)

and S is the spin operator

S = (S1, S2, S3), Sk = �

(σk 0

0 σk

)for 1 ≤ k ≤ 3. (26)

By (23)-(26), we see that

HL1 − L1H =�2c[(x2∂3 − x3∂2)(α

2∂2 + α3∂3)− (α2∂2 + α3∂3)(x2∂3 − x3∂2)]

=�2c[α2∂3(x2∂2 − ∂2x2)− α3∂2(x3∂3 − ∂3x3)].

Notice that

x2∂2 − ∂2x2 = x3∂3 − ∂3x3 = −1.

Hence we get

HL1 − L1H = �2c(α3∂2 − α2∂3). (27)

Similarly we haveHL2 − L2H = �

2c(α1∂3 − α3∂1),

HL3 − L3H = �2c(α2∂1 − α1∂2).

(28)

On the other hand, we infer from (33) and (26) that

HSj − SjH = −i�2cγ5[∂k(σ

kσj − σjσk)]= −i�2cγ5(2iεkjlσ

l)∂k = 2�2cεkjlαl∂k,

where γ5 is defined by

γ5 = iγ0γ1γ2γ3 =

(0 I

I 0

).


Hence we haveHS1 − S1H = −2�2c(α3∂2 − α2∂3),

HS2 − S2H = −2�2c(α1∂3 − α3∂1),

HS3 − S3H = −2�2c(α2∂1 − α1∂2).

(29)

For J = L+ sS, we derive from (27)-(29) that

HJ − JH = 0 ⇐⇒ spin s =1

2. (30)

When fermions move on a straight line,

H = cα3p3, L = 0.

In this case, by (27)-(28), for straight line motion,

HJ − JH = 0 for any s. (31)

Thus, by the conservation law (22), the assertion of Angular Momentum Rule for fermions

follows from (30) and (31).

Bosons

Now, consider bosons which bey the Klein-Gordon equation in the form (17). It is

known that the spins J of bosons depend on the types of Klein-Gordon fields (Ψ,Φ):

(Ψ

Φ

)=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

a scalar field ⇒ J = 0,

a 4-vector field ⇒ J = 1,

a 2nd-order tensor field ⇒ J = 2,

a real valued field ⇒ neutral bosons,

a complex valued field ⇒ charged bosons.

(32)

For the Klein-Gordon fields (Ψ,Φ)T , the Hamiltonian for a central force field is given

by

H =1

2

∫ [|Φ|2 + c2|∇Ψ|2 + 1

�2(m2c4 + V (r))|Ψ|2

]dx (33)

The Hamiltonian energy operator H of (33) is given by

H =

(HΨ 0

0 HΦ

), HΦ = Φ, HΨ =

[−c2Δ+

1

�2(m2c4 + V )

]Ψ. (34)

The angular momentum operator J is

J =

(L 0

0 L

)+ s�σ, σ = (σ1, σ2, σ3). (35)

where s is the spin of bosons, and L is as in (25).


For scalar bosons, spin s = 0 in (35) and the Hermitian operators in the conservation

law (19) are

L11 = L22 = L, L12 = L21 = 0, HΦ, HΨ as in (34)

Then by ∫Φ†LΨdx = −

∫ΨL†Φdx,∫

H†ΨLΦdx = −

∫Φ†LHΨdx,

we derive the conservation law (19), i.e.∫ [Φ†LΨ+Ψ†LΦ− H†

ΨLΦ− Φ†LHΨ

]dx = 0.

However, it is clear that H and J in (34) and (35) don’t satisfy (19) for spin s �= 0. Hence

the quantum rule of angular momentum for bosons holds true.

For example, the fermionic particles bounded in a ball rotating around the center, as

shown in Figure 1, must possess s = 12.

A B

O

(a)

A

B

O

(b)

Fig. 1 (a) Two particles A,B rotate around the center O; (b) three particle A,B,C rotatearound the center O.

3.2 Mass generation mechanism

For a particle moving with velocity v, its mass and energy E obey the Einstein relation

E = mc2/√

1− v2

c2. (36)

Usually, we regardm as a static mass which is fixed, and energy E is a function of velocity

v.

Now, taking an opposite viewpoint, we regard energy E as fixed, and mass m is a

function of velocity v, i.e. the relation (36) is rewritten as

m =

√1− v2

c2E

c2. (37)

Thus, (37) means that a particle with an intrinsic energy E has zero mass m = 0 if it

moves at the speed of light v = c, and will possess nonzero mass if it moves with a velocity


v < c. All particles including photons can only travel at the speed sufficiently close to

the speed of light. Based on this viewpoint, we can think that if a particle moving at the

speed of light (approximately) is decelerated by an interaction field, obeying

d�P

dt=

√1− v2

c2�F ,

then this massless particle will generate mass at the instant. In particular, by this mass

generation mechanism, several massless particles can yield a massive particle if they are

bounded in a small ball, and rotate at velocities less than the speed of light.

From this mass generation mechanism, we can also understand the neutrino oscillation

phenomena. Experiments show that each of the three neutrinos νe, ντ , νμ can transform

from one to another, although the experiments illustrate that neutrinos propagate at

the speed of light. This oscillation means that they generate masses at the instant of

transformation. This can be viewed as the neutrinos decelerate at the instant when they

undergo the transformation/oscillation, generating instantaneous masses, and after the

transformation, they return to the usual dynamic behavior–moving at the speed of light

with zero masses. In other words, by the mass generation mechanism, we can assert that

neutrinos have no static masses, and their oscillations give rise to instantaneous masses.

3.3 Interaction charges

In the unified field model developed in [8, 7], we derived that both weak and strong

interactions possess charges, as for gravity and electromagnetism:

gravitation: mass charge m

electromagnetism: electric charge e,

weak interaction: weak charge gw,

strong interaction: strong charge gs.

(38)

If Φ is a charge potential corresponding to an interaction, then the interacting force

generated by its charge C is given by

F = −C∇Φ, (39)

where ∇ is the spatial gradient operator.

The charges in (38) possess the physical properties:

1) Electric charges Qe, weak charges Qw, strong charges Qs are conservative. The

energy is a conserved quantity, but the mass M is not a conserved quantity due to

the mass generation mechanism as mentioned earlier.

2) There is no interacting force between two particles without common charges. For

example, if a particle A possesses no strong charge, then there is no strong interacting

force between A and any other particles.

3) Only the electric charge Qe can take both positive and negative values, and other

charges can take only nonnegative values.


4) Only the mass charge is continuous, and the others are discrete, taking discrete

values.

5) We emphasize that the continuity of mass is the main obstruction for quantizing the

gravitational field, and it might be essential that gravity cannot be quantized.

3.4 Strong interaction potentials

Three levels of strong interacting potentials are derived in [7] using the field equations, and

they are called the quark potential Sq, the hadron potential Sh, and the atom/molecule

potential Sa:

Sq = gs

[1

r− Bk2

0

ρ0e−k0rϕ(r)

], (40)

Sh = N0

(ρ0ρ1

)3

gs

[1

r− B1

ρ1k21e−k1rϕ(r)

], (41)

Sa = 3N1

(ρ0ρ1

)3 (ρ1ρ2

)3

gs

[1

r− B1

ρ2k21e−k1rϕ(r)

], (42)

where N0 is the number of quarks in hadrons, N1 is the number of nucleons in an

atom/molecule, gs is the strong charge, B and B1 are constants, ρ0 is the effective quark

radius, ρ1 is the radius of a hadron, ρ2 is the radius of an atom/molecule, and

k0 ∼= 1013cm−1, k1 ∼= 1016cm−1.

It is natural to approximately take

ρ0 ∼= 10−21cm, ρ1 ∼= 10−16cm, ρ2 ∼= 10−8cm. (43)

The function ϕ(r) in (40)-(42) is a power series, approximately given by

ϕ(r) =r

2+ o(r).

Formula (40) and (41) lead to the following conclusions for quarks and hadrons:

1) Based on (38), it follows from (40) that the quark interacting force F has the prop-

erties

F

⎧⎪⎨⎪⎩> 0 for 0 < r < R0

= 0 for r = R0,

< 0 for R0 < r < ρ1,

(44)

where R0 is the quark repelling radius, ρ1 is the radius of a hadron as in (43). Namely,

in the region r < R0 the strong interacting force between quarks is repelling, and in

the annulus R0 < r < ρ1, the quarks are attracting, as shown in Figure 2.

2) In the attracting annulus R0 < r < ρ1 as shown in Figure 2, the binding energy

of quarks is in the Planck level, which explains the quark confinement; see [7] for

details.


R0

attracting annulus

ρ1

repelling ball

Fig. 2 In the ball r < R0 quark strong force is repelling, and in the annulus R0 < r < ρ1 quarkstrong force is attracting.

3) For hadrons, the strong interacting force is determined by (41), which implies that

F =

{> 0 for r < R1,

< 0 for R1 < r < R2,(45)

where R1 is the hadron repelling radius, R2 is the attracting radius, with values

given by

R1 =1

2× 10−13cm, R2 = 4× 10−12cm. (46)

Namely the strong interacting force between hadrons is repelling in the ball r < R1,

and attracting in the annulus R1 < r < R2. In particular the repelling force tends

infinite as r → 0:

F = +∞ as r → 0, (47)

which means that there is a large repelling force acting on two very close hadrons.

These properties will be used to explain the strong interacting decays as well.

3.5 Weak interaction potentials

Two weak interaction potential formulas can also be derived by the unified field equations

in [7]. The weakton potential Φ0w and the weak interacting potentials Φ1

w for any particle

with weak charge, including leptons, quarks and mediators, as well as the weaktons

introduced in the next section, are written as

Φ0w =

(ρ0ρw

)3

gwe−k1r

(1

r− ψ1(r)e

−k0r), (48)

Φ1w = gwe

−k1r(1

r− ψ2(r)e

−k0r), (49)

where gw is the weak charge, ρ0 is the radius of the charged leptons, the quarks and the

mediators, ρw is the weakton radius,

k0 ∼= 1016cm−1, k1 ∼= 2× 1016cm−1,

and ψ1, ψ2 are two power series:

ψ1(r) = α1 + β1(r − ρ0) + o(|r − ρ0|),ψ2(r) = α2 + β2(r − ρ0) + o(|r − ρ0|).


Here α1, β1, α2, β2 are the initial values of a system of second order ordinary differential

equations satisfied by ψ1 and ψ2, and they are determined by the physical conditions or

experiments.

We remark that (49) was derived in [7], and (48) can be derived in the same fashion

as the three level of strong interaction potentials (40)–(42) in [7].

Based on physical facts, phenomenologically we take ρ0, ρw, α1, α2, β1, β2 as

ρw ∼= 10−26cm, ρ0 ∼= 10−21cm, ρ1 ∼= 10−16cm,

α1∼=

2

ρ0, α2 =

1

ρ1, β1 = 0, β2 > 0.

(50)

The potentials (48) and (49) imply following assertions:

1) Weaktons are confined in the interior of charged leptons, quarks and mediators. In

fact, the bound energy of the weaktons has the level

E = gwΦ0w(ρ0)

∼= −1

ρ0

(ρ0ρw

)3

g2w = −1036g2w/cm.

By the Standard Model,

g2w =8√2Gf

(mwc

�

)2

= 10−1�c. (51)

Hence the bound energy is

E = −1035�c/cm = −1021GeV.

This is the Planck level, to sufficiently confine the weaktons in their composite

particles.

2) By (48) and (50), for the weak interacting force F0 between weaktons, we have

F0

{> 0 for 0 < r < 1

2ρ0,

< 0 for 12ρ0 < r < ρ1,

(52)

where ρ0, ρ1 are as in (43).

3) By (49) and (50), for the weak interacting force F1 of a composite particle, we have

F1

{> 0 for 0 < r < ρ1,

< 0 for ρ1 < r < ρ2.(53)

Hence the weak force is repelling if the particles are in the ρ1-ball, and is attracting

if they are in the annulus ρ1 < r < ρ2.

4) F0 and F1 tend to infinite as r → 0:

F0, F1 → +∞ as r → 0.

Namely, the weak interacting force between two very close particles is large and

repelling.

We shall see that these properties of the weak interacting force are crucial for the

weakton model presented in the next few sections.


4. Weakton Model of Elementary Particles

4.1 Decay means the interior structure

From Section 2.5, it is clear that all charged leptons, quarks and mediators can undergo

decay as follows:

• Charged lepton decay:

e− → e− + γ,

μ− → e− + νe + νμ,

τ− → μ− + νμ + ντ .

(54)

• Quark decay:

d → u+ e− + νe,

s → d+ g + γ,

c → d+ s+ u.

(55)

• Mediator decay:

2γ → e+ + e−,

W± → l± + νl± ,

Z0 → l+ + l−.

(56)

All leptons, quarks and mediators are currently regarded as elementary particles.

However, the decays in (54)-(56) show that these particles must have interior structure,

and consequently they should be considered as composite particles rather than elementary

particles:

Decay Means Interior Structure.

4.2 Weaktons and their quantum numbers

The above observation on the interior structure of quarks, charged leptons and mediators

leads us to propose a set of elementary particles, which we call weaktons. These are

massless, spin-12particles with one unit of weak charge gw.

The introduction of weaktons is based on the following theories and observational

facts:

(a) the interior structure of charged leptons, quarks and mediators demonstrated by the

decays of these particles as shown in (54)-(56),

(b) the new quantum numbers of weak charge gw and strong charge gs introduced in

(38),

(c) the mass generating mechanism presented in Section 3.2, and

(d) the weakton confinement theory given by the weak interacting potentials (48).

The weaktons consist of 6 elementary particles and their antiparticles, total 12 parti-

cles:w∗, w1, w2, νe, νμ, ντ ,

w∗, w1, w2, νe, νμ, ντ ,(57)


Table 4 Weakton quantum numbers

Weakton Qe gw gs Qc B Le Lμ Lτ J m

w∗ +2/3 1 1 0 1/3 0 0 0 ±1/2 0

w1 −1/3 1 0 1 0 0 0 0 ±1/2 0

w2 −2/3 1 0 -1 0 0 0 0 ±1/2 0

νe 0 1 0 0 0 1 0 0 −1/2 0

νμ 0 1 0 0 0 0 1 0 −1/2 0

ντ 0 1 0 0 0 0 0 1 −1/2 0

where νe, νμ, ντ are the three generation neutrinos, and w∗, w1, w2 are three new elemen-

tary particles, which we call w-weaktons.

These weaktons are endowed with the quantum numbers: electric charge Qe, weak

charge gw, strong charge gs, weak color charge Qc, baryon number B, lepton numbers

Le, Lμ, Lτ , spin J , and mass m. The quantum numbers of weaktons are listed in Table 4.

A few remarks are now in order.

Remark 4..1. The quantum numbersQe, Qc, B, Le, Lμ, Lτ have opposite signs and gw, gs,m

have the same values for the weaktons and antiweaktons. The neutrinos νe, νμ, ντ possess

left-hand helicity with spin J = −12, and the antineutrinos possess right-hand helicity

with spin J = 12.

Remark 4..2. The weak color charge Qc is a new quantum number introduced for the

weaktons only, which will be used to rule out some unrealistic combinations of weaktons.

Remark 4..3. Since each composite particle contains at most one w∗ particle, there is

no strong interaction between the constituent weaktons of a composite particle. There-

fore, for the weaktons (57), there is no need to introduce the classical strong interaction

quantum numbers as strange number S, isospin (I, I3) and parity P .

Remark 4..4. It is known that the quark model is based on the SU(3) irreducible

representations:

Meson = 3⊗ 3 = 8⊕ 1,

Baryon = 3⊗ 3⊗ 3 = 10⊕ 8⊕ 8⊕ 1.

The weakton model is based on the aforementioned theories and observational facts (a)–

(d), different from the quark model.


4.3 Weakton constituents

In this section we give the weakton compositions of charged leptons, quarks and mediators

as follows.

Charged leptons and quarks. The weakton constituents of charged leptons and

quarks are given by

e = νew1w2, μ = νμw1w2, τ = ντw1w2,

u = w∗w1w1, c = w∗w2w2, t = w∗w2w2,

d = w∗w1w2, s = w∗w1w2, b = w∗w1w2,

(58)

where c, t and d, s, b are distinguished by the spin arrangements. We suppose that

u = w∗w1w1(�↓,�↑, ↑↓↑, ↓↑↓),c = w∗w2w2(�↓,�↑),t = w∗w2w2(↑↓↑, ↓↑↓),

(59)

and

d = w∗w1w2(�↓, ↓↓↑),s = w∗w1w2(↑�, ↓�), (60)

b = w∗w1w2(↑↓↑, ↓↑↓).

Mediators. The duality between mediators given in (9) plays an important role in

the weakton model. In fact, the mediators in the classical interaction theory have spin

J = 1 (graviton has spin J = 2), and are apparently not complete. The unified field

theory in [8, 7] leads to complement mediators with spin J = 0 (graviton dual particle

is J = 1). Thus, the spin arrangements of weaktons in the mediators become perfectly

reasonable.

For convenience, we only write the dual relation for the mediators of electromagnetism,

weak interaction, and strong interaction in the following:

J = 1 J = 0

photon γ ↔ electro-dual boson φγ,

vector bosons W±, Z ↔ weak-dual bosons φ±W , φ0Z ,

gluons gk (1 ≤ k ≤ 8) ↔ strong-dual bosons φkg .

(61)

In view of this duality, we propose the constituents of the mediators as follows:

γ = cos θww1w1 − sin θww2w2 (�,�),

Z0 = cos θww2w2 + sin θww1w1 (�,�),

W− = w1w2(�,�), (62)

W+ = w1w2(�,�),

gk = w∗w∗(�,�), k = color index,


and the dual bosons:

φγ = cos θww1w1 − sin θww2w2(↑↓, ↓↑),φ0Z = cos θww2w2 + sin θww1w1(↑↓, ↓↑),

φ−W = w1w2(↑↓, ↓↑), (63)

φ+W = w1w2(↑↓, ↓↑),

φkg = w∗w∗(↑↓, ↓↑),

where θw ∼= 28.76◦ is the Weinberg angle. Here φ0Z corresponds to the Higgs particle in

the standard model, found in LHC. As all the dual mediators in our theory have the

same constituents as the classical mediators, distinguished by spin arrangements, each

mediator and its dual should possess masses in the same level with slight difference, as

evidence by the masses of Z0 and φ0Z .

Remark 4..5. The reason why we take γ, Z0 and their dualities φγ, φ0Z as the linear

combinations in (62) and (63) is that by the Weinberg-Salam electroweak theory, the

U(1)× SU(2) gauge potentials are

Zμ = cos θwW3μ + sin θwBμ,

Aμ = − sin θwW3μ + cos θwBμ

sin2 θw = 0.23.

Here Aμ, Zμ represent γ and Z0.

The ν-mediator. Now the neutrino pairs

νeνe, νμνμ, ντ ντ (↓↑) (64)

have not been discovered, and it should be a mediator. Due to the neutrino oscillations,

the three pairs in (64) should be indistinguishable. Hence, they will be regarded as a

particle, i.e. their linear combination

φ0ν =

∑l

αlνlνl(↓↑),∑l

α2l = 1, (65)

is an additional mediator, and we call it the ν-mediator. We believe that φ0ν is an

independent new mediator.

4.4 Weakton confinement and mass generation

Since the weaktons are assumed to be massless, we have to explain the mass generation

mechanism for the massive composite particles, including the charged leptons e, τ, μ, the

quarks u, d, s, c, t, b, and the vector bosons W±, Z0, φ±W , φ0Z .

The weakton confinement derived in Section 3.5 and the mass generation mechanism

in Section 3.3 can help us to understand why no free w∗, w1, w2 are found and to explain

the mass generation of the composite particles.


First, by the infinite bound energy (Planck level), the weaktons can form triplets

confined in the interiors of charged leptons and quarks as (58), and doublets confined in

mediators as (62)-(63) and (65). They cannot be opened unless the exchange of weaktons

between the composite particles. Single neutrinos νe, νμ and ντ can be detected, because

in the weakton exchange process there appear pairs of different types of neutrinos such as

νe and νμ, and between which the governing weak force is given by (49), and is repelling

as shown in (53).

Second, for the mass problem, we know that the mediators

γ, φγ, gk, φk

g , φ0ν , (66)

have no masses. To explain this, we note that the particles in (66) consist of pairs

w1w1, w2w2, w∗w∗, νlνl. (67)

The weakton pairs in (67) are bound in a circle with radius R0 as shown in Figure 3.

Since the interacting force on each weakton pair is in the direction of their connecting

line, they rotate around the center O without resistance. As �F = 0, by the relativistic

motion law:d

dt�P =

√1− v2

c2�F , (68)

the massless weaktons rotate at the speed of light. 2 Hence, the composite particles

formed by the weakton pairs in (67) have no rest mass.

ωωO

Fig. 3

Third, for the massive particles

e, μ, τ, u, d, s, c, t, b, (69)

by (58), they are made up of weakton triplets with different electric charges. Hence the

weakton triplets are not arranged in an equilateral triangle as shown in Figure 1 (b),

and in fact are arranged in an irregular triangle as shown in Figure 4. Consequently,

the weakton triplets rotate with nonzero interacting forces F �= 0 from the weak and

electromagnetic interactions. By (68), the weaktons in the triplets move at a speed less

than the speed of light. Thus, by the mass generating mechanism, the weaktons possess

mass present. Hence, the particles in (69) are massive.

2 In fact, a better way to interpret (68) is to take a point of view that no particles are moving at exactly

the speed of light. For example, photons are moving at a speed smaller than, but sufficiently close to,

the speed of light.


ω′

2

ω′

1

ω′

3

Fig. 4

Finally, we need to explain the masses for the massive mediators:

W±, Z0, φ±W , φ0Z . (70)

Actually, in the next weakton exchange theory, we can see that the particles in (67) are

some transition states in the weakton exchange procedure. At the moment of exchange,

the weaktons in (70) are at a speed v (v < c). Hence, the particles in (70) are massive.

Here we remark that the dual mediator φ0Z is the Higgs particle found in LHC.

4.5 Quantum rules for weaktons

By carefully examining the quantum numbers of weaktons, the composite particles in

(58), (62), (63) and (65) are well-defined.

In Section 4.4, we solved the free weakton problem and the mass problem. In this

section, we propose a few rules to solve some remainder problems.

1). Weak color neutral rule. All composite particles by weaktons must be weak

color neutral.

Based on this rule, there are many combinations of weaktons are ruled out. For

example, it is clear that there are no particles corresponding to the following www and

ww combinations, as they all violate the weak color neutral rule:

νew2w2, w∗w1w1, w∗w1w2, etc., νew1, w∗w1, w∗w2, etc.

2). BL = 0, LiLj = 0 (i �= j).

The following combinations of weaktons

w∗ν, νiνj, νiνk (i �= k), νk = νe, νμ, ντ . (71)

are not observed in Nature, and to rule out these combinations, we postulate the following

rule:

BL = 0, LiLj = 0 (i �= j), Li = Le, Lμ, Lτ , (72)

where B,L are the baryon number and the lepton number.

3). L+Qe = 0 if L �= 0 and |B +Qe| ≤ 1 if B �= 0.


The following combinations of weaktons

νw1w1, νw2w2, νw1w2, w∗w∗ etc (73)

cannot be found in Nature. It means the lepton number L, baryon number B, and electric

charge Qe obey

L+Qe = 0 if L �= 0 and |B +Qe| ≤ 1 if B �= 0. (74)

Thus (73) are ruled out by (74).

4). Spin selection.

In reality, there are no weakton composites with spin J = 32as

w∗w1w1(↑↑↑, ↓↓↓), w∗w2w2(↑↑↑, ↓↓↓), w∗w1w2(↑↑↑, ↓↓↓) (75)

and as

νw1w2(↑↑↑, ↓↓↓). (76)

The cases (75) are excluded by the Angular Momentum Rule in Section 3.1. The

reasons for this exclusion are two-fold. First, the composite particles in (75) carries one

strong charge, and consequently, will be confined in a small ball by the strong interaction

potential (40), as shown in Figure 1 (b). Second, due to the uncertainty principle, the

bounding particles will rotate, at high speed with almost zero moment of force, which

must be excluded for composite particles with J �= 12based on the angular momentum

rule.

The exclusion for (76) is based on the observation that by the left-hand helicity of

neutrinos with spin J = −12, one of w1 and w2 must be in the state with J = +1

2to

combine with ν, i.e. in the manner as

νww(↓�, ↓↑↓).

In summary, under the above rules 1)-4), only the weakton constitutions in (58), (62),

(63) and (65) are allowed.

5). Eight quantum states of gluons.

It is known that the gluons have eight quantum states

gk : g1, · · · , g8.

In (62), gk have the form

w∗w∗(�,�).

According to QCD, quarks have three colors

red (r), green (g), blue (b),


and anti-colors r, g, b. They obey the following rules

bb = rr = gg = w(white),

br = g, rb = g,

bg = r, gb = r,

rg = b, gr = b,

rr = r, bb = b, gg = g,

rb = g, rg = b, gb = r.

(77)

Based on (58), w∗ is endowed with three colors

w∗b , w∗r , w∗g .

Thus, by (77) we give the eight gluons as

g1 = (w∗w∗)w, g2 = w∗b w∗r , g3 = w∗b w

∗g , g4 = w∗rw

∗g , (78)

g5 = (w∗w∗)w, g6 = w∗rw∗b , g7 = w∗gw

∗b , g8 = w∗gw

∗r , (79)

where (w∗w∗)w is a linear combination of w∗b w∗b , w

∗rw

∗r , w

∗gw

∗g . Namely, the gluons in (79)

are the antiparticles of those in (78).

In summary, all of the most basic problems in the weakton model have a reasonable

explanation.

5. Mechanism of Sub-atomic Decays

5.1 Weakton exchanges

We conclude that all particle decays are caused by exchanging weaktons. The exchanges

occur between composite particles as mediators, charged leptons, and quarks.

5.1.1 Weakton exchange in mediators

First we consider one of the most important decay processes in particle physics, the

electron-positron pair creation and annihilation:

2γ → e+ + e−,e+ + e− → 2γ.

(80)

In fact, the reaction formulas in (80) are not complete, and the correct formulas should

be as follows

2γ + φ0ν � e+ + e−. (81)

Note that the weakton component of γ is as

γ = cos θww1w1 − sin θww2w2, (82)


which means that the probability of the photon γ at the state w1w1 is cos2 θw, and its

probability at the state −w2w2 is sin2 θw. Namely, for photons, the densities of the

w1w1(�) and −w2w2(�) particle states are cos2 θw and sin2 θw. Hence, the formula (81)

can be written as

w1w1(�) + w2w2(�) + νeνe(↓↑) � νew1w2(↓↑↓) + νew1w2(�↓). (83)

It is then clear to see from (83) that the weakton constituents w1, w1, w2, w2, νe, νecan regroup due to the weak interaction, and we call this process weakton exchange.

The mechanism of this exchanging process can be explained using the weak interacting

potentials (48) and (49).

The potential formula (48) means that each composite particle has an exchange radius

R, which satisfies

r0 < R < ρ1, (84)

where r0 is the radius of this particle and ρ1 is the radius as in (52). As two composite

particles A and B are in a distance less than their common exchange radius, there is a

probability for the weaktons in A and B to recombine and form new particles. Then,

after the new particles have been formed, in the exchange radius R, the weak interacting

forces between them are governed by (53) which are repelling, and then drive them apart.

For example, to see how the weaktons in (83) undergo the exchange process in Fig-

ure 5. When the randomly moving photons and ν-mediators, i.e. w1w1, w2w2 and νeνe,

come into their exchange balls, they recombine to form an electron νew1w2 and a positron

νew1w2, and then the weak repelling force pushes them apart, leading to the decay pro-

cess (81). We remark here that in this range the weak repelling force is stronger than the

Coulomb force. In fact, by (51), g2w = 10−1�c and the electric charge square e2 = 1/137�c.

Hence, the weak repelling force between e− and e+ in Figure 5 is (3gw)2/r2, stronger than

e2/r2.

⊗

�

⊕

�

•

�

w1w1

w2w2

νeνe

⊗ ⊕

•

�

� �

⊗ ⊕

•

�

� �

e−

e+

Fig. 5 ⊗,!,⊕,", •,# represent w1, w1, w2, w2, νe, νe.


5.1.2 Weakton exchanges between leptons and mediators

The μ-decay reaction formula is given by

μ− → e− + νe + νμ. (85)

The complete formula for (85) is

μ− + φν → e− + νe + νμ,

which is expressed in the weakton components as

νμw1w2 + νeνe −→ νew1w2 + νe + νμ. (86)

By the rule LeLμ = 0, the μ neutrino νμ and the e antineutrino νe can not be combined

to form a particle. Hence, νe and νμ appear as independent particles, leading to the

exchange of νμ and νe as shown in (86).

5.1.3 Weakton exchanges between quarks and mediators

The d-quark decay in (55) is written as

d→ u+ e− + νe. (87)

The correct formula for (87) is

d+ γ + φν → u+ e− + νe,

which, in the weakton components, is given by

w∗w1w2 + w1w1 + νeνe → w∗w1w1 + νew1w2 + νe, (88)

In (88), the weakton pair w2 and w1 is exchanged, and νe is captured by the new doublet

w1w2 to form an electron νew1w2.

5.2 Conservation laws

The weakton exchanges must obey some conservation laws, which are listed in the fol-

lowing.

5.2.1 Conservation of weakton numbers

The total weaktons given in (57) are elementary particles, which cannot undergo any

decay. Also, the w-weaktons cannot be converted between each other. Although the

neutrino oscillation converts one type of neutrino to another, at the moment of a particle

decay, the neutrino number is conserved, i.e. the lepton numbers Le, Lμ, Lτ are conserved.

Therefore, for any particle reaction:

A1 + · · ·+ An = B1 + · · ·+Bm, (89)

the number of each weakton type is invariant. Namely, for any type of weakton w, its

number is conserved in (89):

NAw = NB

w ,

where NAw and NB

w are the numbers of the w weaktons in two sides of (89).


5.2.2 Spin conservation

The spin of each weakton is invariant. The conservation of weakton numbers implies that

the spin is also conserved:

JA1 + · · ·+ JAn = JB1 + · · ·+ JBm ,

where JA is the spin of particle A.

In classical particle theories, the spin is not considered as a conserved quantity. The

reason for the non-conservation of spin is due to the incompleteness of the reaction for-

mulas given in Section 2.5. Hence spin conservation can also be considered as an evidence

for the incompleteness of those decay formulas. The incomplete decay interaction formu-

las can be made complete by supplementing some massless mediators, so that the spin

becomes a conserved quantum number.

5.2.3 Other conservative quantum numbers

From the invariance of weakton numbers, we derive immediately the following conserved

quantum numbers:

electric charge Qe, weak charge Qw, strong charge Qs,

baryon number B, lepton numbers Le, Lμ, Lτ .

5.3 Decay types

In particle physics, the reactions as in Section 2.5 are classified into two types: the weak

interacting type and the strong interacting type. However there is no clear definition to

distinguish them. Usual methods are by experiments to determine reacting intensity, i.e.

the transition probability Γ. In general, the classification is derived based on

Weak type: i) presence of leptons in the reactions,

ii) change of strange numbers,

Strong type: otherwise.

With the weakton model, all decays are carried out by exchanging weaktons. Hence

decay types can be fully classified into three types: the weak type, the strong type, and

the mixed type, based on the type of forces acting on the final particles after the weakton

exchange process.

For example, the reactions

νμ + e− → μ− + νe,

n → p+ e− + νe,

π0 → 2γ,

(90)

are weak decays,

Δ++ → p+ + π+ (91)


is a strong decay, and

Λ→ p+ + π−(i.e. Λ + g + 2γ + φγ → p+ + π− + γ) (92)

is a mixed decay.

In view of (90)-(92), the final particles contain at most one hadron in a weak decay,

contain no leptons and no mediators in a strong decay, and contain at least two hadrons

and a lepton or a mediator in a mixed decay. Namely, we derive the criteria based on the

final particle content:

Weak Decay: at most one hadron,

Strong Decay: no leptons and no mediators,

Mixed Decay: otherwise.

5.4 Weak decays

Decays and scatterings are caused by weakton exchanges. The massless mediators

γ, φγ, g, φg (g the gluons), φν (93)

spread over the space in various energy levels, and most of them are at low energy states.

It is these random mediators in (93) entering the exchange radius of matter particles that

generate decays. In the following we shall discuss a few typical weak decays.

5.4.1 νμe− → νeμ

− scattering

First we consider the scattering

νμ + e− → μ− + νe,

which is rewritten in the weakton components as

νμ + νew1w2 → νμw1w2 + νe. (94)

Replacing the Feynman diagram, we describe the scattering (94) using Figure 6. It is

clear that the scattering (94) is achieved by exchanging weaktons νμ and νe.

5.4.2 β-decay

Consider the classical β-decay process

n→ p+ e− + νe. (95)

With the quark constituents of n and p

n = udd, p = uud,


νe

w1 w2

νμ νe

w1w2

νμ

w1 w2

νe

νμ

Fig. 6 νμe− → μ−νe scattering.

the β-decay (95) is equivalent to the following d-quark decay:

d→ u+ e− + νe, (96)

whose complete form should be given by

w∗w1w2(d) + νeνe(φν) + w1w1(γ)→ w∗w1w1(u) + w1w2(W−) + νeνe(φν) (97)

→ w∗w1w1(u) + νew1w2(e−) + νe.

In the β decay (97), w2 and w1 in d quark and photon γ have been exchanged to form u

quark and charged vector boson W−, then W− captures a νe from φν to yield an electron

e− and a νe.

5.4.3 Quark pair creations

Considerg + φγ + γ −→ u+ u,

φg + 2φγ −→ d+ d.

They are rewritten in the weakton constituent forms as

w∗w∗ � (g) + w1w1 ↓↑ (φγ) + w1w1 � (γ) (98)

−→ w∗w1w1 ↑↓↑ (u) + w∗w1w1 ↑� (u),

w∗w∗ ↑↓ (φg) + w1w1 ↑↓ (φγ) + w2w2 ↓↑ (φγ) (99)

−→ w∗w1w2 �↓ (d) + w∗w1w2 �↑ (d).

In (98), w∗ and w∗ in a gluon are captured by a γ-dual mediator φγ and a photon γ to

create a pair u and u. In (99), w1 and w2 in two φγ are exchanged to form φ±W (charged

Higgs), then φ+W and φ−W capture w∗ and w∗ respectively to create a pair d and d.

5.4.4 Lepton decays

The lepton decays

μ− + φν → e− + νe + νμ,

τ− + φν → μ− + νμ + ντ .


are rewritten in the weakton constituents as

νμw1w2 + νeνe → νew1w2 + νe + νμ,

ντw1w2 + νμνμ → νμw1w2 + νμ + ντ .(100)

Here the neutrino exchanges form leptons in the lower energy states and a pair of neutrino

and antineutrino with different lepton numbers. By the rule LiLj = 0 (i �= j) in Sec-

tion 4.5, the generated neutrino and antineutrino cannot be combined together, and are

separated by the weak repelling force in (53). The decay diagram is shown by Figure 7.

νi-

Fig. 7 νi �= νj , νi = νe, νμ, ντ .

5.5 Strong and mixed decays

5.5.1 Strong decays.

Consider the following types of decays:

Δ++ −→ p+ π+.

The complete decay process should be

Δ++ + φg + 2φγ −→ p+ π+. (101)

It is clear that the final particles are the proton and charged π meson π+. Hence (101)

is a strong type of decays. Recalling the weakton constituents, (101) is rewritten as

3w∗w1w1(Δ++) + w∗w∗(φg) + w1w1 + w2w2(φγ) (102)

−→ (2w∗w1w1)(w∗w1w2)(p) + (w∗w1w1)(w

∗w1w2)(π+).

The reaction process in (102) consists of two steps:

weakton exchanges: φg + 2φγ −→ d+ d, (103)

quark exchanges: uuu+ dd −→ uud+ ud. (104)

The exchange mechanism of (103) was discussed in (100), which is a weak interaction,

and the quark exchange (104) is a strong interaction.


Let us discuss the D0 decay, which is considered as the weak interacting type in the

classical theory. But in our classification it belongs to strong type of interactions. The

D0 decay is written as

D0 −→ K− + π+.

The complete formula is

D0 + g + 2γ −→ K− + π+. (105)

The weakton constituents of this decay is given by

(w∗w2w2)(w∗w1w1)(cu) + w∗w∗(g) + 2w1w1(γ) (106)

−→ (w∗w1w2)(w∗w1w1)(su) + (w∗w1w1)(w

∗w1w2)(ud).

This reaction is due to the c-quark decay

c+ g + 2γ → s+ u+ d,

which is given in the weakton constituent form as

w∗w2w2(c) + w∗w∗(g) + 2w1w1(γ) (107)

−→ w∗w1w2(s) + w∗w1w1(u) + w∗w1w2(d).

The reaction (5.28) consists of two exchange processes:

w∗w2w2(c) + w1w1(γ)→ w∗w1w2(s) + w1w2(W−), (108)

w1w2(W−) + w1w1(γ) + w∗w∗(g)→ w∗w1w1(u) + w∗w1w2(d). (109)

It is clear that both exchanges here belong to weak interactions. However, the final

particles of the D0 decay are K− and π+, which are separated by the strong hadron

repelling force.

5.5.2 Mixed decays

We only consider the Λ decay:

Λ→ p+ π−. (110)

The correct form of this decay should be

Λ + g + 2γ + φγ → p+ π− + φγ. (111)

There are three exchange procedures in (111):

g + γ + φγ → u+ u, (112)

s+ γ → d+ φγ, (uds+ γ → udd+ φγ) (113)

udd(n) + uu→ uud(p) + ud(π−). (114)

The procedure (112) was described by (99), the quark exchange process (114) is clear,

and (113) is the conversion from s quark to d quark, described by

w∗w1w2 ↑↓↓ (s) + w1w1 ↑↑ (γ) −→ w∗w1w2 ↑↑↓ (d) + w1w1 ↓↑ (φγ). (115)

Namely, (115) is an exchange of two w1 with reverse spins.


6. Electron Radiations

6.1 Electron structure

The weakton constituents of an electron are νew1w2, which rotate as shown in Figure 4.

Noting that

electric charge: Qνe = 0, Qw1

e = −1

3, Qw2

e = −2

3,

weak charge: Qνw = 1, Qw1

w = 1, Qw2w = 1,

we see that the distribution of weaktons νe, w1 and w2 in an electron is in an irregular

triangle due to the asymptotic forces on the weaktons by the electramagnetic and weak

interactions, as shown in Figure 8.

ρ

Fig. 8 Electron structure.

In addition, by the weak force formula (53), there is an attracting shell region of weak

force:

ρ1 < r < ρ2, ρ1 = 10−16 cm (116)

with small weak force. Outside this region, the weak force is repelling:

Fw > 0 for r < ρ1 and r > ρ2. (117)

Since the mediators γ, φγ, g, φg and φν contain two weak charges 2gw, they are attached

to the electron in the attracting shell region (116), forming a cloud of mediators. The

irregular triangle distribution of the weaktons νe, w1 and w2 generate a small moment

of force on the mediators in the shell region, and there exist weak forces between them.

Therefore the bosons will rotate at a speed lower than the speed of light, and generate a

small mass attached to the naked electron νew1w2.

6.2 Mechanism of Bremsstrahlung

It is known that an electron emits photons as its velocity changes. This is called

bremsstrahlung, and the reasons why bremsstrahlung can occur is unknown in classical


theories. We present here a mechanism of this phenomena based on the above mentioned

structure of electrons.

In fact, as an electron is in an electromagnetic field, which exerts a Coulomb force on

its naked electron νew1w2, but not on the attached neutral mediators. Thus, the naked

electron changes its velocity, which draws the mediator cloud to move as well, causing

a perturbation to moment of force on the mediators. As the attracting weak force in

the shell region (116) is small, under the perturbation, the centrifugal force makes some

mediators in the cloud, such as photons, flying away from the attracting shell region, and

further accelerated by the weak repelling force (117) to the speed of light, as shown in

Figure 9.

e

(a)

Fe

(b)

γ

γ

Fig. 9 (a) The naked electron is accelerated in an electromagnetic field; (b) the mediators(photons) fly away from the attracting shell region under a perturbation of moment of force.

6.3 An alternate explanation of solar neutrino problem

The solar neutrino problem refers to a discrepancy between measurements of the numbers

of neutrinos flowing through the Earth and theoretical models of the solar interior. The

current resolution is the neutrino oscillation theory.

The weakton model, however, provide an entirely different explanation of the problem.

Basically, when the solar electron neutrinos collide with anti electron neutrinos, which

are abundant due to the β-decay of neutrons, they can form ν mediators and flying away,

causing the loss of electron neutrinos. This is consistent with the experimental results on

the agreement between the speed of light and the speed of neutrinos.

7. Conclusions

The main motivation of this article is that the sub-atomic decays amounts to saying

that quarks and charged leptons must possess interior structure. With this motivation, a

weakton model of elementary particles is proposed based on 1) sub-atomic particle decays,

and 2) formulas for the weak and strong interaction potentials/forces. In this weakton

model, the elementary particles consist of six spin-12massless particles, which we call

weaktons, and their antiparticles. The weakton model leads to 1) composite constituents

for quarks, charged leptons and medaitors, 2) a new mass generation mechanism, and 3)

a perfect explanation of all sub-atomic decays and reactions.


With this weakton model and the unified field theory [8, 7], we now present our expla-

nations and viewpoints to the twelve fundamental questions stated in the Introduction.

Q1: Our current view on four interactions is that each interaction has its own charge,

the mass charge m, the electric charge e, the weak charge gw and the strong charge gs,

which are introduced in Section 3.3. Each weakton carries one unit of weak charge, hence

the name weakton, and only w∗ carries a unit of strong charge gs. A particular interaction

can only occur between two particles if they both carry charges of the corresponding

interaction.

The dynamic laws for four interactions are the unified field model, which can be easily

decoupled to study individual interactions. Our theory shows that each interaction has

both attractive and repulsive regions, leading the stability of matter in our universe.

Q2: With the weakton model, it is clear that leptons do not participate strong in-

teractions, as they do not carry any strong charge–the weakton constituents of charged

leptons (58) do not include w∗.

Q3: The weakton model postulates that all matter particles (leptons, quarks) and

mediators are made up of massless weaktons. The basic mass generation mechanism is

presented in Section 3.2. Namely, for a composite particle, the constituent massless weak-

tons can decelerate by the weak force, yielding a massive particle, based on the Einstein

mass-energy relation. Also, the constituent weaktons are moving in an “asymptotically-

free” shell region of weak interactions as indicated by the weak interaction potential/force

formulas, so that the bounding and repelling contributions to the mass are mostly can-

celed out. Hence the mass of a composite particle is due mainly to the dynamic behavior

of the constituent weaktons.

Q4 & Q5: In Sections 5.1-5.5, the weakton model offers a perfect explanation for

all sub-atomic decays and all generation/annihilation precesses of matter-antimatter. In

particular, all decays are achieved by 1) exchanging weaktons and consequently exchang-

ing newly formed quarks, producing new composite particles, and 2) separating the new

composite particles by weak and/or strong repelling forces. Also, we know now the precise

constituents of particles involved in all decays both before and after the reaction.

Q6: Again, the sub-atomic decays and reactions offer a clear evidence for the existence

of interior structure for quarks and leptons, as well as for mediators. The consistency of

the weakton model with all reactions and decays, together with conservations of quan-

tum numbers, demonstrates that both quarks and charged leptons are not elementary

particles.

Q7 (Baryon Asymmetry): Conventional thinking was that the Big Bang should

have produced equal amounts of matter and antimatter, which will annihilate each other,

resulting a sea of photons in the universe, a contradiction to reality. The weakton model

offers a complete different view on the formation of matter in our universe. The weakton

model says that what the Big Bang produced was a sea of massless elementary weaktons

and anti-weaktons, forming all the matter, including mediators such as photon, in the


universe. Hence with the weakton model, the baryon asymmetry problem is no longer a

right question to ask.

Q8–Q11: The decoupled unified field model leads to three levels of strong interaction

potentials and two levels of weak interaction potentials as recalled in (40)–(42), (48)

and (49). These formulas give a natural explanation of both the short-range nature

and confinements for both strong and weak interactions. The different levels of each

interaction demonstrate that in the same spatial region, the interaction can be attracting

between weaktons, and be repelling for newly formed hadrons and leptons. This special

feature of weak and strong interactions plays a crucial rule for decays.

Q12 (Bremsstrahlung): The weak interaction force formulas show that the attract-

ing shell region near a naked electron can contain a cloud of neutral mediators as photon.

As the naked electron changes its velocity due to the presence of an electromagnetic field,

which has no effect on the neutral mediator cloud. The change of velocity of electron gen-

erates a perturbation to moment of force on the mediators causing some of the mediators

flying out from the attracting shell region. This is the mechanism of bremsstrahlung; see

Sections 6.1 and 6.2.

Q13 (Solar neutrino problem): When the solar electron neutrinos collide with anti

electron neutrinos, which are abundant due to the β-decay of neutrons, they can form ν

mediators and flying away, causing the loss of electron neutrinos. This is consistent with

the experimental results on the agreement between the speed of light and the speed of

neutrinos.

References

[1] E. J. Eichten, K. D. Lane, and M. E. Peskin, New tests for quark and leptonsubstructure, Phys. Rev. Lett., 50 (1983), pp. 811–814.

[2] D. Griffiths, Introduction to elementary particles, Wiley-Vch, 2008.

[3] F. Halzen and A. D. Martin, Quarks and leptons: an introductory course inmodern particle physics, John Wiley and Sons, New York, NY, 1984.

[4] H. Harari, A schematic model of quarks and leptons, Physics Letters B, 86 (1979),pp. 83–86.

[5] G. Kane,Modern elementary particle physics, vol. 2, Addison-Wesley Reading, 1987.

[6] T. Ma and S. Wang, Gravitational field equations and theory of dark matterand dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34:2 (2014),pp. 335–366; see also arXiv:1206.5078v2.

[7] , Unified field theory and principle of representation invariance, AppliedMathematics and Optimization, 69:3 (2014), pp. 359–392; see also version 1 ofarXiv:1212.4893.

[8] , Unified field equations coupling four forces and principle of interactiondynamics, Discrete and Continuous Dynamical Systems, Ser. A, 35:3 (2015),pp. 1103–1138; see also arXiv:1210.0448.


[9] J. C. Pati and A. Salam, Lepton number as the fourth ”color”, Phys. Rev. D, 10(1974), pp. 275–289.

[10] C. Quigg, Gauge theories of the strong, weak, and electromagnetic interactions, 2ndedition, Princeton Unversity Press, 2013.

[11] M. A. Shupe, A composite model of leptons and quarks, Physics Letters B, 86 (1979),pp. 87–92.

[12] N.-S. Zhang, Particle Physics, vol. I & II, in Chinese, Chinese Academic Press,1994.


Physics of Currents and PotentialsII. Classical Singlet-Triplet Electroweak Theory with

Non-point Particles

Valerii Temnenko∗

Tavrian National University, 95004, Simferopol, Crimea, Ukraine

Received 16 March 2014, Accepted 19 July 2014, Published 10 January 2015

Abstract: The formulation of classical singlet-triplet (electroweak) theory, in which point

particles are replaced by continual current fields, has been presented. The expression for the

Lagrangian of the theory and complementary algebraic and differential constraints, imposed on

the currents of the theory, have been suggested. Classification of stationary and wave states

for the singlet-triplet theory has been given. Stationary states correspond to massive particles

with the current zone of a finite volume. The theory contains a number of wave states, both

one-sector (singlet or triplet waves) and compound two-sector ones (singlet-triplet waves). Wave

states differ in number of currents: zero-current waves (free singlet or free triplet waves), one-

current, two-current, three-current and four-current ones. Wave states also differ in character

of four-dimensional wave vector (the waves with time-like and space-like wave vector). Some

forms of waves may have negative density of energy. Some wave states can be treated as

classical models of a neutrino. Neutrino states are classified in accordance with the character of

the current which forms the state: singlet (maxwellian) neutrino, Yang-Mills triplet neutrino,

Maxwell-Yang-Mills singlet-triplet neutrino.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Lagrangian; Singlet-Triplet Theory; Classification of the States in the Singlet-Triplet

Theory; Yang-Mills Triplet Waves; Classical Models of Neutrino

PACS (2010): 03.50.-z; 03.50.Kk; 11.10.-z; 12.15.-y

1. Introduction

Let us extend the approach stated in the article [1] over the unified theory of electroweak

interaction. Within the framework of this extension we should postulate the existence

of two sectors of physics of currents and potentials: singlet (or maxwellian) sector and

triplet (or Yang-Mills) sector. Each sector contains a dyad ”current/potential”. Singlet

∗ [email protected]


sector contains singlet (maxwellian) current Jν and singlet (maxwellian) potential Wν

which is dynamically conjugated with it. Triplet sector contains Yang-Mills triplet of

currents Jν =

{a

J ν

}and dynamically conjugated with it Yang-Mills triplet of potentials

Wν =

{a

W ν

}. Over-letter indices a, b, c number the components of Yang-Mills triplets:

a = 1, 2, 3. Greek indices μ, ν, . . . number Lorentz components of 4-vectors. They run

over values 0, 1, 2, 3.

The adjectives ”singlet” and ”maxwellian”, which are synonyms and often appear in

the text, will be sometimes expressed by symbols S- or M-, for example, ”S-current”

or ”M-current” instead of the term ”singlet current”. Generally speaking, the terms

”triplet” and ”Yang-Mills” are not synonyms: octuplet (chromodynamical) sector of

physics deals with Yang-Mills octuplet of currents and dynamically conjugated with it

Yang-Mills octuplet of potentials. However, within the framework of the present article,

which is devoted to classical electroweak theory, we shall treat the terms ”triplet” and

”Yang-Mills” as synonyms, and if necessary, we shall denote them with symbols T- or

YM-, for example, ”T-current” or ”YM-current” instead of the term ”triplet current”.

YM-geometry is Euclidian: there is no necessity to introduce contravariant and covariant

indices. YM-vectors are denoted with bold type in indexless notation. In some cases we

have to use bold type for indexless notation of the three space components of Lorentz

4-vectors. We hope that the reader will be able to tell these objects from YM-vectors.

Twice repeating YM-indices imply summation, such as, for example, scalar YM-product

of YM-vectors Jν and Wν :

Jμ ·Wν ≡a

Jμa

W ν ≡3∑

a=1

a

Jμa

W ν .

In the three-dimensional space of YM-vectors it is possible to enter a vector product

through Levi-Civita 3-symbolabcε in a conventional way: if Cμν = Aμ ×Bν ,

a

Cμν =abcε

b

Aμc

B ν ,

while objecta

Cμν is Yang-Mills 3-vector and Lorentz tensor of the second rank in Minkov-

ski four-dimensional space-time.

Singlet M-potential Wν does not coincide with the electromagnetic potential Aν con-

sidered in the article [1]. Singlet M-current Jν also does not coincide with the electro-

magnetic current which we will denote as Jem

ν . However, as we will show later, in the

”mixed” Maxwell-Yang-Mills’ singlet-triplet world there are ”pure” maxwellian (singlet)

states in which electromagnetic potential Aν , proportional to singlet potential Wν , and

electromagnetic current, proportional to singlet current Jν , can be introduced to describe

a state, thereby we will revert to the classical electromagnetic theory of non-point parti-

cles, formulated in the article [1]. With description of an arbitrary mixed singlet-triplet

state, such extraction of electromagnetic potential in the form of some linear combination

of singlet and triplet potential is also possible, but it is irrational: it overloads simple and


natural algebra of YM-vectors.

Dynamics of the two field dyads ”currents/potentials” of the singlet-triplet world is de-

rived from the principle of the least action at minimization of action functional S:

S =

∫L′dΩ. (1)

The total space-time integration is implied in this expression; L′ is the effective La-

grangian consisting of the sum of the base Lagrangian L and the addition to the La-

grangian Lad, which is caused by the necessity of accounting differential and algebraic

constraints, which are a priori imposed on the four currents (Jν ,Jν).

The base LagrangianL consists of the sum of two sector Lagrangians - the singlet La-

grangian LS and the triplet Lagrangian LT :

L = LS + LT . (2)

The relation (2) means that singlet and triplet sectors of physics do not intermingle with

each other on the level of the base Lagrangian.

Each of the two sector terms of the base Lagrangian (2), in its turn, consists of three terms,

which correspondently describe the current part of the Lagrangian Lcur, the Lagrangian

of current and potential Lint interaction, and the free field Lagrangian Lf :

LS = LS,cur + LS,int + LS,f ,

LT = LT,cur + LT,int + LT,f .(3)

Integrals in (1), containing the Lagrangians of free fields, are taken from all over the

4-space. Integrals in (1), containing the current Lagrangian Lcur and the interaction

Lagrangian Lint are taken over current 4-zones ΩJ , in which at least one of the four

currents is nonzero. It is supposed that in the current zones each of the four currents is

space-like:

JνJν ≤ 0, (4)

1

J ν1

J ν ≤ 0;2

J ν2

J ν ≤ 0;3

J ν3

J ν ≤ 0. (5)

The condition of form (4) was substantiated for electromagnetic current in the previous

article [1]. Current inequalities (4) and (5) are taken here by analogy with the inequality

for Jem

ν as the postulates of physics without which the construction of the classical theory

of non-point particles is impossible.

At the three-dimensional boundaries σJ of the four-dimensional current zones ΩJ at least

one of the currents becomes isotropic, and, correspondingly, the sign of equality is reached

at least in one of the relations (4). Such boundary, termed in the article [1] a ”pomerium”,

can be common for some currents or can disintegrate into different boundaries for different

currents. Beyond such boundary, on which isotropization of one of the four currents took

place, this current is identically zero. At the boundary itself, this isotropic current is

orthogonal to 4-vector of normal to the boundary nμ.

According to the reasons given in the article [1], we have to accept that pseudo-Euclidian


modules of space-like currents are limited by absolute value. Thereby, we must postulate

the existence of such fundamental constant jS, that

−JνJν ≤ j2S (6)

and of such fundamental constant jT , that

−1

J ν1

J ν ≤ j2T ;

−2

J ν2

J ν ≤ j2T ;

−3

J ν3

J ν ≤ j2T .

(7)

It is unlikely that God would have been so wasteful to have created two different fun-

damental constants of the same dimension, and the equality jS = jT does not seem

impossible.

The reasons, given below in p.3.4, make it possible to come to conclusion that jT ≤ jS2.

Adoption of inequalities (6) and (7) is necessary for obtaining finite mass of particles

within the framework of the classical theory under consideration.

The equality sign in relations (6) and (7) is reached at some three-dimensional boundaries

of the four-dimensional current zones. Some cavitated no-current tubes are limited by

these boundaries inside the current zone. Let us call such cavitated no-current tube with

a term ”latebra”. This word was translated from the language of the ancient Romans

as a ”sheltered place”, ”hiding place” and it also had the meaning ”inner cavity”. The

boundary of a cavitated tube will be termed ”latens” (hidden, invisible).

According to the reasons provided in the article [1], it can be expected that diametrical di-

mensions of latebra are of the order of Planckian length rp. The longitudinal dimension

of latebra is determined by the appearing in the theory unknown fundamental constant

of length dimension r0: r0 > rp.

Latebra and latens are inaccessible in any laboratory experiment – as one of the few

survived Heraclitus’ fragments reads, ”nature likes hiding”. Late Roman commentator

Macrobius, while interpreting this idea one thousand years after Heraclitus, said that

”nature feels disgusted to be exposed open and naked”. However, to give sense to the

classical theory of currents and potentials discussed here, we must consider currents and

potentials as measurable physical quantities even in case if the only device, which had

already been known to Isaac Newton, Sensorium Dei, is applicable for their measuring.

According to the reasons given in the article [1], when constructing the theory of non-

point particles, we should take into account Riemannian space-time curvature, generated

by large quantities of energy-momentum tensor components in the region occupied by

currents and inside the cavitated no-current latebrae .

As in the article [1], we will suppose that this account of curvature can be provided

according to Einsteinian recipe:

• obtain the field theory equations in Minkowski coordinates, ignoring the space-time

curvature;

2 See the form below (292), which determines the relation between the constants jS and jT .


• rewrite the obtained equations in arbitrary curvilinear coordinates with arbitrary

metric tensor which possesses Minkowski signature (+– – –) at each point of the

four-dimensional space-time continuum;

• subordinate the metric tensor to Einstein equations, which read that the Ricci ten-

sor is proportional to energy-momentum tensor; the proportionality coefficient is

determined by gravitation constant.

When solving the wave states problems, where the boundaries of pomerium and latens are

missing, the space-time curvature can be ignored, supposing that energy density in the

wave is insufficient for forming substantial deviations from Minkowski plane geometry.

The singlet-triplet theory, constructed here, assumes the existence of the states where

the conditions of isotropy of currents (4) or (5) for one or a few currents are satisfied

in some four-dimensional zone ΩN rather than on the three-dimensional hyper-surface of

pomerium σJ. We shall name these states ”neutrino states”; the corresponding isotropic

currents will be named ”neutrino currents”. Neutrino states should be treated as non-

stationary, transitional states generated by some instability of ”neutrinoless” states. How-

ever, the theory also admits the existence of the model wave neutrino states, in which

neutrino zone ΩN coincides with the whole four-dimensional space. While considering

such states, we have to sacrifice the condition of currents and fields disappearance ”at

infinity”. Generally speaking, these conditions must be always supposed to be satisfied

at minimization of the action functional (1).

Summing up the ”Introduction”, we would like to remind the reader that the constructed

theory, as in the article [1], includes the unknown fundamental dimension constant r0.

This constant is further assumed as a length unit. Velocity of light c is also assumed as

a unit. Constants r0 and c do not explicitly appear in the theory formulas.

2. The Base Lagrangian of the Singlet-triplet Theory

2.1 The Interaction Lagrangian. The Weinberg Angle

Let us suppose, as in classical electrodynamics, that the singlet interaction Lagrangian

LS,int is proportional to scalar product of singlet current Jν and singlet potential Wν :

LS,int ∼ JνWν , (8)

and, similarly, the triplet interaction Lagrangian LT,int is proportional to ”twice scalar”

product of the triplet current Jν and the triplet potential Wν :

LT,int ∼ Jν ·Wν . (9)

The expression (9) implies summation by unwritten explicitly YM-indices and summation

by Lorentz indices (Euclidean scalar product in three-dimensional Euclidean space of

YM-vectors and pseudo-Euclidean scalar product of 4-vectors in four-dimensional pseudo-

Euclidean Minkowski space-time).

Proportionality coefficients in formulas (8) and (9) are negative [2]. We do not have any


a priori grounds to suggest that the expressions (8) and (9) must enter into the general

interaction Lagrangian with identical weight. We shall assume that singlet and triplet

contributions to Lint have different weight factors pS and pT :

Lint = −1

2pSJνWν −

1

2pTJν ·Wν . (10)

(pS > 0; pT > 0).

In the units of measurement used here, currents and potentials have an electric charge

dimension. The Lagrangian and the action functional have a dimension of an electric

charge square. The singlet interaction constant of pS and triplet interaction constant

pT are dimensionless, although their numerical values are determined by selecting an

electric charge unit of measurement (as it is possible to introduce an arbitrary coefficient

of proportionality between the left and the right parts of the relation (1), which determine

the action functional). It is convenient to normalize the sum of squares of constants pSandpT per unit:

p2S + p2T = 1. (11)

The relation (11), implying a definite selection of an electric charge unit of measurement,

by normalizing the general contribution of the interaction Lagrangian (10) to the total

Lagrangian, keeps us from a temptation to consider the model ”one-sector” worlds: the

singlet world with the triplet sector (pT → ∞) ”switched off” or the triplet world with

the singlet sector (pS → ∞) ”switched off”. The relation (11) allows us to use one con-

stant instead of two fundamental constants pS and pT . If we assume that pT = sin θw(0 < θw <

π

2), according to (11) we should assume that pS = cos θw.

The fundamental constant θw is usually called Weinberg angle. It would be more correct

(but more ponderous) to call this constant ”the angle of the singlet-triplet mixing” and,

correspondingly, to denote it with θST .

The same electric charge unit that was used in (11) for Weinberg parameters normaliz-

ing, is used to measure currents and potentials. Accordingly, further the currents and

potentials are considered dimensionless.

2.2 The Current Lagrangian

Based on the results of the preceding article [1], we shall assume that the contribution

of each current to the base Lagrangian in neutrinoless state is proportional to pseudo-

Euclidean square of current. ”Assigning” corresponding weight factors, we shall write

the current part of the Lagrangian as follows:

Lcur = −1

8p2SJνJν −

1

8p2TJν · Jν . (12)

The weight factor in the singlet part of the current Lagrangian (12) is set on the basis

of compliance with the electromagnetic theory of non-point particles, constructed in the

preceding article [1]. Weight factor in the triplet part of the current Lagrangian (12) is


set on the basis of current ”singlet-triplet equality”, by means of substitution of pS → pT ,

Jν → Jν in the singlet part of the current Lagrangian.

In the state which is neutrino by some current, the pseudo-Euclidean square of this

current in the neutrino 4-zone ΩN is equal to zero, and the contribution of this current

to the current Lagrangian is missing (12). However, the pseudo-Euclidean square of a

neutrino current appears in the additional Lagrangian Lad with an arbitrary Lagrange’s

multiplier.

2.3 The Field Lagrangian

The field Lagrangian in (2) and (3) is constructed in the form of the expression which is

quadratic by tensors of the singlet and triplet field Wμν and Wμν with identical weight

factors:

Lf = − 1

16πWμνW

μν − 1

16πWμν ·Wμν . (13)

The singlet field tensor Wμν is constructed in the same way as it is constructed in classical

electrodynamics [2]

Wμν = ∂μWν − ∂νWμ. (14)

Yang-Mills triplet field tensor was constructed in the classical article of these two authors

[3]3. In the symbols we have accepted, it looks as follows:

Wμν = ∂μWν − ∂νWμ + pTWμ ×Wν , (15)

or, with explicitly specified YM-indices:

a

Wμν= ∂μa

Wν −∂νa

Wμ +pTabcε

b

Wμ

c

Wν . (16)

The choice of expressions (15), (16) for YM-field tensor was in due time motivated by

considerations of SU(2) – gauge invariance (in the era of Yang-Mills they spoke of an

”isotopic gauge invariance” [3]). However, the explicit construction of YM-potential and

YM-field tensor , in this way, requires appealing to the notion ”wave function phase” of

point particles and it is inadmissible in the classical theory of non-point charges, which is

developed here. Within the framework of this classical theory it is reasonable to treat the

potential and current as primary, irredundant and non-interpreted concepts, and to treat

YM-field tensor (15) (in the spirit of ”pedagogical” motivations typical for the course of

Landau and Lifshitz [2]), as the simplest antisymmetric Lorentz second-rank tensor (and,

at the same time, YM-vector), which can be constructed from YM-vector.

Fixation of the multiplicative constant in (15) before the nonlinear term suggests a certain

choice of measurement unit for the electric charge. Change of the unit charge in (15)

requires re-scaling of this multiplier and violates unit coefficient normalization (11). The

3 Wolfgang Pauli, as pointed by N. Straumann [4], developed the main aspects and formulas known now

as ”Yang-Mills theory” already in summer 1953, but he didn’t publish his results.


fact that this unit of charge measurement, connecting (11) and (15), coincides with the

charge value of the electron, can be treated as a casual gift4.

2.4 The Total Base Lagrangian of the Singlet-triplet Theory

By substituting the written out expressions for separate terms of the base Lagrangian

(10), (12), (13) into (2) and (3), we obtain the total base Lagrangian of a neutrinoless

state:

L = − 1

8p2SJνJν−

1

8p2TJνJν−

1

2pSJνWν−

1

2pTJν ·Wν−

1

16π(WμνW

μν +Wμν ·Wμν) . (17)

In this Lagrangian the singlet field tensor Wμν is calculated by the formula (14) through

the components of the singlet potential Wμ. The triplet field tensor Wμν is calculated

through the components of the triplet potential Wμ by formula (15). Weight factors

pS and pT (”Weinberg parameters”) are expressed by the formulas mentioned in p. 2.1

through the fundamental constant θw which is exterior for this theory. Currents Jν and

Jν , potentials Wν and Wν are primary, non-interpreted variables of the theory, which

are inexpressible by other more elementary physical concepts. If, in accordance with the

specific initial conditions, the formation of the neutrino 4-zone ΩN , occupied by isotropic

current, is possible in the singlet-triplet problem, within this zone the contribution of

a corresponding neutrino (isotropic) current falls out of the two first terms in the base

Lagrangian (17), but this contribution is restored (with an arbitrary Lagrange’s multi-

plier) in the additional Lagrangian Lad, since the condition of current isotropy must be

considered as holonomic constraint imposed on the solution of the singlet-triplet problem.

3. The Algebraic and Differential Constraints

for Currents. The Additional Lagrangian

3.1 Differential Condition for Singlet Current Conservation

As in the classical electrodynamics [1], the condition for S-current conservation is adopted

in the singlet sector of the theory:

∂μJμ = 0. (18)

The condition (18) requires appearing of the term in the form of1

2p2Sχ∂νJ

ν in the addi-

tional Lagrangian Lad, where χ is the arbitrary Lagrange’s multiplier depending on the

space-time coordinates xμ, and the multiplier1

2p2Sis introduced for the convenience of

the additional and base Lagrangian combination. By extracting from this term the total

4 This ”gift” faintly signifies that the singlet-triplet theory is possibly not a final and unimprovable

version of physics of electroweak interactions.


4-divergence1

2p2S∂ν(χJ

ν), which disappears after the Lagrangian integration over the en-

tire 4-space, we can denote the corresponding contribution to the additional Lagrangian

as follows

Lad(χ) = −1

2p2SJν∂νχ.

As it is known, the differential constraint (18) is natural for the Lagrangian (17). This

constraint is the result of Maxwell field equations, appearing with minimization of the

action functional with the Lagrangian (17) under any Lagrange’s multiplier χ. So, it

would be reasonable to assume from the outset that χ = 0 and not to include the term

Lad(χ) into the additional Lagrangian.

3.2 Differential Condition for the Triplet Current

Generally speaking, the triplet YM-current, by contrast to the singlet M-current, is not

conserved. Differential constraint, imposed on YM-current [3], can be described in the

form of the relation which we are free to adopt as a postulate:

∂μJμ + pTWμ × Jμ = 0. (19)

Multiplying the equation (19) by the arbitrary Lagrange’s multiplier χ (Lorentz scalar

and YM-vector), after the extraction of total 4-divergence, we obtain a corresponding

contribution to the additional Lagrangian coupled with the Lagrange’s multiplier χ:

Lad(χ) =1

2p2T(−Jμ∂μχ+ pTχ · (Wμ × Jμ)) .

In this expression the multiplicative constant1

2p2Tis introduced for the convenience of

combination of the base Lagrangian and the Lagrangian Lad(χ).

By varying the total Lagrangian L with addition of Lad(χ) it is not difficult to estab-

lish that the differential constraint (19) for YM-current is the result of Yang-Mills field

equations with any option of the Lagrange’s multiplier χ which satisfies the condition

∂μχ+ pTWμ × χ = 0.

Supposing that χ = 0, we are satisfying this condition. Such option of χ may be not

convenient enough to solve some problems, but its use is acceptable in any problem.

Consequently, we may avoid including term Lad(χ) into the additional Lagrangian.

Let us call the choice of the Lagrange’s multipliers χ = 0 and χ = 0 the Lorentz gauge.

The possibility of Lorentz gauge choice is connected with the idea of the gauge symmetry,

the essence of which can be expressed in the following way: construct the field tensor

from potentials in such a way that the differential constraints, which are a priori imposed

on the currents and express physical laws of current conservation/interconversion, would

be natural constraints for the base Lagrangian.


When using Lorentz gauge, the differential constraints (18) and (19) should not be in-

cluded into the list of solvable problems under the numerical solution of field problems.

They can be used as a tool for monitoring of accuracy and stability of the computation

process.

3.3 Algebraic condition for the Singlet

and Triplet Current Coupling

Existence of singlet current Jν violates the isotropy of YM-space; it violates the equality

of the three currents of YM-triplet Jν , selecting from three YM-currents the one which

the singlet current Jν is closely coupled with. We shall denote this ”selected” current by3

J ν . Algebraic condition for S-current Jν coupling with the selected third component of

YM-triplet of currents3

J ν has the following form:

3

J ν

(3

Jν −Jν)

= 0. (20)

The algebraic postulate (20) specifies the holonomic constraint, imposed on the currents

of the singlet-triplet theory. It is based on the current structure of Weinberg-Salam elec-

troweak theory (”WS theory”, e.g. [5]). But the current structure of WS theory per se

cannot be a ”legal” basis for the classical theory that we are constructing. The currents

in WS theory are algebraic expressions which are bilinear by the spinor wave functions of

fermions. Point fermions are primary, non-interpretable objects of WS theory. They are

built into the theory ”manually”. In WS theory, just as in the modern relativistic quan-

tum theory in general, the totally quantum object – spinor wave function of a fermion –

is implicitly interpreted as some true classical field under construction of the classical

Lagrangian which afterwards, following the procedure of the secondary quantization (or

Feinman integration by all possible configurations of all fields of the theory), transforms

into a carrier of all the information about the quantum system. Such inconsistency, a

latent ”dragging” of the quantum object into the classical Lagrangian, dating back to the

pioneer works of 1930th (V. Heisenberg, W. Pauli, V. Veiskopf), generates the condition

of modern quantum relativistic physics, which was uncompromisingly described by a se-

vere judge Paul Dirac as ”an ugly and incomplete one” [6].

Here we are constructing the classical theory, in which the wave functions of particles can

not appear in any way; and the massive particles are not built into the theory ”manually”.

They are missing ”at the entrance” of the theory, but they must appear as computable,

constructed objects ”at the exit” of the theory. ”At the entrance” of the theory there are

only currents and potentials, plus the base Lagrangian structure, as well as the holonomic

and differential constraints which are imposed on the currents.

According to these remarks, the algebraic constraint, imposed on the currents of the the-

ory (20), must be considered a postulate of the theory, without any apparent references

to the existing WS theory. The accounting of this constraint leads to appearance in the


additional Lagrangian of a term which we shall denote as Lad(ψ):

Lad(ψ) = −1

2p2Tψ

3

J ν

(3

Jν −Jν). (21)

In formula (21), ψ is the Lagrange’s multiplier, which is the functional of the space-time,

coordinates xμ. This very term in the Lagrangian ”entangles” physics of the singlet

maxwellian sector with physics of triplet Yang-Mills sector in case if current3

J ν is differ-

ent from identical zero. The base Lagrangian (17) splits into singlet and triplet parts.

The terms of the additional Lagrangian Lad(χ) and Lad(χ), generated by the differential

constraints imposed on currents (if these terms are proper to calculate technically by

numerical solution of a definite problem), are also placed according to the principle ”each

one in its own sector”. And only the term (21) in the additional Lagrangian mixes two

sectors of physics.

3.4 Algebraic ”2→ 3” Decomposition of the Coupled Pair of currents Jν

and3

J ν

The algebraic condition (20), coupling S-current Jν with the third YM-component of T-

current Jν , and the conditions for the space-likeness of currents (4), (5) make it possible

to conduct an effective decomposition of a couple of currents Jν ,3

J ν , by representing

them in the form of a linear combination of the triple of mutually orthogonal currents

Nν , lν ,rν , where current Nν is isotropic and currents lν and rν are space-like:

Jν = Nν + lν + 2rν ,3

J ν = lν −Nν ,

N νNν = 0,

lνNν = 0,

rνNν = 0,

lνrν = 0,

lνlν < 0, rνrν < 0.

(22)

It is not difficult to see that the problem (22) has two one-parametric families of solutions.

The condition for the existence of these solutions is the inequalities

JνJν <3

J ν3

Jν< 0. (23)

It is proper to construct an explicit solution to the problem (22) within the ”intrinsic

frame of reference” of current3

J ν , where time component of current3

J 0 is missing and

3-vector of the current space components is oriented along the axisx1:

3

J ν = {0, α, 0, 0}. (24)


With regard to (24) and condition (20), the singlet current Jν can be represented in this

frame of reference in the following form:

Jν = {τ, α, β, 0} ; τ 2 < β2. (25)

In (25) it is established that through the rotation about axis x1 we oriented axes x2 and

x3 in a way that J3 = 0.

Inequality, connecting τ and β, results from the inequality (23).

Thus, rν , which at decomposition (22) enters only into the singlet current Jν , is generally

called the right-handed current . Thus, lν , which enters into both S-current Jν and

the third component of YM-triplet3

J ν , is called the left-handed current . The isotropic

current Nν will be called a neutrino current . The solution to the problem (22) for the

neutrino current can have the following form:

N ν = N{β, 0, τ, ±

√β2 − τ 2

}, (26)

where N is an arbitrary normalization multiplier. It is proper to assume that N = 1.

Fixation of parameter N in a specific frame of reference does not influence Lorentz-

invariance of the solution: for transition to another frame of reference it is enough to

make Lorentz-transformation of 4-vector (26). Under such gauge of Nν , the left-handed

and the right-handed currents have the following form:

lν ={β, α, τ, ±

√β2 − τ 2

},

rν =

{τ

2− β, 0,

β

2− τ, ∓

√β2 − τ 2

}.

(27)

Currents lν and rν depend on the arbitrarily chosen gauge of parameter N , but their sum

jν = lν + rν (”electromagnetic current”) does not depend on the choice of N :

jν =

{τ

2, α,

β

2, 0

}.

Combination of currents qν also does not depend on the choice of normalization constant

N :

qν = rν +N ν .

Electromagnetic current jν and current qν are reduced to the half-sum and, correspond-

ingly, to the half-difference of currents Jν and3

J ν , and, consequently, they are as well-

defined physical quantities as currents Jν and3

J ν . Currents Nν , lν , rν themselves in the

arbitrary singlet-triplet state are not well-defined physical quantities.

The procedure of ”2 → 3”-decomposition of currents, presented by formulas (26) and

(27), does not only depend on the arbitrary gauge of parameter (and the choice of sign

in z-components of vectors N3, l3, r3) – this procedure is also not P-invariant. Indeed,

the inversion of spatial axes means the reversal of sign of spatial components of any

vector, including transformation α → −α, β → −β in the components of currents Jν


and3

J ν (formulas (24) and (25)). But, y-components of vectors N2 and l2, described

by the formulas (26) and (27), do not change sign in the process, but time components

N0 and l0 change the sign. Thus, under the inversion of spatial axes, all three currents

Nν , lν and rν turn into other physical entities. To restore the invariance of the theory

relatively to discrete transformations, it is necessary to make T-transformation (change

of the time coordinate sign) and C-transformation (change of sign of all currents and po-

tentials) along with P-transformation (inversion of spatial coordinates). In so doing, to

follow PCT-invariance of the theory in z-components of the currents (N3, l3, r3) we have

to transit to the second branch of the square root5. PCT-inversion leaves the procedure

of ”2 → 3”-decomposition of currents invariant. This procedure itself is not necessary

for the classical theory. Of course, there are such physical states, in which the terms

”right-hand current”, ”left-handed current”, ”neutrino current” describe the observable

physical quantities. The simplest of these states is a pure singlet state with a nonde-

generate singlet current J (JνJν < 0) and an empty triplet sector (Jν = 0;Wν = 0). In

this case we can assume that lν = 0, Nν = 0, consequently, the singlet current Jν is

reduced to the right-hand current: Jν = 2rν . The extraction of a neutrino current is

effective in special limit conditions when one or both inequality signs in formula (23) are

replaced by equality signs. For example, if currents Jν and3

J ν have identical nonzero

pseudo-Euclidean modules:

JνJν =3

J ν3

Jν< 0, (28)

and, accordingly, in formula (25) τ = β, the system of equations (22) has no solution

which would satisfy the condition of space-likeness of the right-hand current rν . The

condition (28) describes ”left-hand state”:

rν = 0,

lν =

{1

2β, α,

1

2β, 0

},

Nν =

{1

2β, 0,

1

2β, 0

},

and decomposition of currents has a simple form:

Jν = lν +Nν ,3

J ν = lν −N ν .(29)

In the left-hand singlet-triplet state (29) the left-hand current lν and neutrino current

Nν must be regarded as observable physical quantities. In the process, currents1

J ν ,2

J ν

are sure to be present in the triplet sector.

One more special limit condition is a combined isotropization of both currents Jν and3

J ν :

JνJν =3

J ν3

Jν= 0.

5 In language usually used in theoretical physics, this transition to the second branch of square root in the

expression for neutrino current Nν (26) can be interpreted as transition from neutrino to antineutrino.


In this condition, in formulas (23) we should suppose that lν = 0 and rν = 0, and,

consequently, according to (22), we should accept that:

Jν = N ν ,3

J ν = −Nν .(30)

In such mixed singlet-triplet state, the same neutrino current Nν is present both in the

singlet and triplet sector (”Maxwell-Yang-Mills neutrino”).

There are two more special one-sector neutrino states.

The first one is the singlet neutrino state with ”empty” triplet sector:

Jν = Nν ; N νNν = 0

Jν = 0; Wν = 0.

In this state the decomposition formula (22) ”does not work” – there is no triplet partner

for current Jν .

This one-sector singlet neutrino state (”maxwellian neutrino”) is studied in detail in the

article [1] of this series6. Algebraic condition (20), imposed on the currents, will be also

satisfied at the ”empty” singlet sector in case if current3

J ν is a neutrino one:

Jν = 0; Wν = 0.3

J ν = Nν ; NνNν = 0.(31)

In this case the decomposition formulas (22) ”do not work”: current3

J ν does not have a

singlet partner.

Relations (31) describe a one-sector triplet neutrino state (one of the variants of ”Yang-

Mills neutrino”).

3.5 Algebraic Condition for the Inner Normalization

of Yang-Mills Triplet

The current structure of the standard version of electroweak WS-theory with point par-

ticles indicates the existence of one more algebraic constraint, imposed on the three

components of YM-triplet:

21

J ν2

Jν=3

J ν3

Jν . (32)

While we take the condition (32) as a postulate of the triplet theory, it registers some

kind of ”auto-normalization” of the currents inside YM-triplet. This condition creates

its own contribution Lad(η) to the additional Lagrangian Lad:

Lad(η) =η

2p2T

(2

1

J ν2

Jν −3

J ν3

Jν

), (33)

6 The description of this state in [1] contained reservations expressing a doubt as to reality of such

neutrino state: the principal text of the article [1] was already written in the late 1970s, and it took the

author a long period of time to put up with such Maxwellian model of neutrino.


in which η is a Lagrange’s multiplier which depends on the space-time coordinates xν .

This contribution to the Lagrangian is totally placed in YM-sector of the theory.

The very fact of appearing of the algebraic constraints (20) and (32) in the theory, which

are beyond the base Lagrangian of the theory, is rather unpleasant from the point of

view of an estimation of the completeness and correctness of our ideas of fundamentals

of physics. The Lagrange’s multipliers ψ and η, entering into the additional Lagrangian,

actually correspond to some outer ”Higgs-like” fields, which influence the real physical

fields. In the language of Newtonian mechanics they correspond to the ”constraints

force”, imposed on the current fields by someone from outside. The theory could be

rehabilitated by constructing such solutions for the theory equations, for which algebraic

constraints (20) and (32) are natural , i.e. are satisfied under the absence of ”Higgs-like”

fields ψ and η: ψ ≡ 0 and η ≡ 0.

3.6 About Quality Criteria of the Classical Physical Theory

We can assume that a good classical physical theory has to satisfy two of the following

quality criteria:

• the theory contains a short , not requiring any extension, list of primary objects of

the theory which form the base Lagrangian;

• the theory does not contain any external to the Lagrangian constraints imposed on

the primary objects, or these constraints are natural for the Lagrangian of the

theory and do not generate any external ”Higgs-like” constraint forces, at least for

some quite wide range of solutions.

The existing version of theoretical physics – both Weinberg-Salam electroweak theory

and the Standard Model – actually does not satisfy the first criterion. Point fermions

with their wave functions are built into the theory ”manually” as primary objects. There

can be as many fundamental fermions as possible: only experimental data restricts the

length of the list of primary objects of the theory. The existing version of physics also

does not satisfy the second criterion, as it contains external Higgs objects.

The version of classical field theory, developed here, satisfies the first criterion. In this

version the short list of primary objects of the theory is a priori fixed: they are currents

only the singlet and triplet ones (and octuplet, if strong interactions are taken into

account). There is nothing in nature except them and dynamically conjugated with them

potentials with the same Yang-Mills dimension (singlet, triplet and octuplet). Particles

and waves are secondary, derivative, calculated entities; they are the objects ”at the exit”

of the theory – there can be as many of them as possible.

We do not know if the theory, developed here, satisfies the second criterion. We have no

theorem which would contain the proof of the fact that there exists quite a wide range

of solutions to the equations of singlet-triplet theory, such as that algebraic constraints

(20) and (32), imposed on the currents, are natural for the base Lagrangian of the theory


and can be satisfied under zero values of ”Higgs-like” fields ψ and η7. These solutions

describe all singlet-triplet states, which are realized in nature.

Two more criteria have to be added to the two quality criteria of the fundamental physical

theory, formulated above.

• The theory naturally incorporates the accounting of Riemann space-time curvature.

• The theory must contain the solutions with finite energy and should not require

renormalization or regularization.

The theory, developed here, satisfies these criteria, while Weinberg-Salam theory and the

Standard Model do not.

And, finally, a good physical theory has to satisfy the fifth, Einsteinian, quality criterion:

• The theory should not contain arbitrarily specified dimensionless ”empirical” con-

stants. The fine structure constant, the mixing angles, the mass ratio of particles,

etc. must be calculated within the theory8.

There is no theory satisfying this Einsteinian criterion in modern physics. The theory,

developed here, partly satisfies this criterion, apparently, allowing calculating mass ratio

of the particles.

4. Discrete Transformations

in the Singlet-Triplet Theory

4.1 Charge Conjugation in the Singlet-Triplet Theory

The field theory, constructed here, – and its base Lagrangian L (17) and additional

Lagrangian Lad, consisting of the sum of Lad(ψ) (21) and Lad(η) (33), - are invariant

relative to the charge conjugation consisting of two operations:

(1) Sign reversal for all currents and potentials;

(2) Permutation of Yang-Mills indices 1↔ 2.

Formally it can be written with help of two sector operators of the charge conjugation,

operator CS, affecting the singlet vectors and reduced to change of their sign:

Jν → J′ν = CSJν = −Jν ,Wν →W′

ν = CSWν = −Wν ,

and operator CT , affecting YM-triplets of vectors. Operator CT is YM-matrix, changing

vector signs and permutating YM-indices 1↔ 2:

Jν → J′ν = CTJν ,

Wν →W′ν = CTWν ,

7 We also do not know if there exists at least one mathematician who is able to get interested in this

problem.8 ”God had no choice”, – this is probably the answer that must be given by good physical theory to the

famous Albert Einsteins question.


where CT =

⎛⎜⎜⎜⎜⎝0 −1 0

−1 0 0

0 0 −1

⎞⎟⎟⎟⎟⎠.

The structure of singlet-triplet theory itself demonstrates ”inequality” of Lorentz vectors

entering YM-triplet. The third YM-component of current3

Jν is connected with singlet

current Jν by relation (20) and is thereby distinguished in Yang-Mills triple. It is obvious

that this extraction has to be also transferred to the third YM-component of potential3

Wν . ”Inequality” of YM-indices 1 and 2 (”chiral inequality”) is connected with the

availability of the vector YM-product in determining of the triplet field tensor Wμν (15)

and differential constraint (19). In these expressions vector YM-product is summed up

with YM-vector. For this summation operation to make invariant geometric sense in YM-

space, the vector YM-product should not be an axial YM-vector, but a true YM-vector.

Therefore, with the specified third YM-vectors component we must absolutely definitely

choose the numbering scheme of the first and the second components. We should suppose

that we have a possibility to provide a chiral consistency, i.e. to choose the right-handed

(or the left-handed) coordinate system simultaneously and consistently in the coordinate

space and YM-vector space9. To provide chiral consistency at the inversion of a sign of

YM-currents and YM-potentials, we have to permutate YM-indices 1↔ 2.

At the performing of the described transformation of charge conjugation, field tensors

are transformed in the same way as the currents and potentials:

Wμν →W′μν = CSWμν ,

Wμν →W′μν = CTWμν .

The base Lagrangian, quadratic by currents, potentials and field tensors, does not change

with charge conjugation.

To conclude paragraph 4.1, we will dare to make two anticipatory remarks. Probably,

these remarks will help the reader to estimate the defiant beauty and complexity of

Yang-Mills physics.

(1) In octuplet sector of physics, which is responsible for strong interactions and is the

subject for discussion in the subsequent article of this series, there is no octuplet

matrix of charge conjugation similar to triplet matrix CT by its characteristics. In

other words, in contrast to the singlet and triplet sector, the octuplet sector of

physics is charge-asymmetrical. No permutation of vector numbers in Yang-Mills

octuplet provides invariance of the octuplet Lagrangian relative to the inversion of

currents.

No doubt, this fact is not psychologically easy to put up with. Perhaps, it is as

hard to do as it was for the physicists of 1956 to put up with the parity violation.

However, the charge asymmetry, latent in the octuplet algebra itself, obviously makes

9 ”Only a theory defines what can be observable”, – Albert Einstein once said to Werner Heinsenberg.


it possible to understand the reason of the evident and defiant charge asymmetry of

the Universe: in the octuplet sector God had a choice .

(2) The requirement to observe chiral definiteness of the solutions of Yang-Mills equa-

tions actually imposes some unilateral (releasing) constraint on variables of the

triplet and octuplet sectors of physics. Such constraint is not something unusual

for the problems of classical mechanics, for example, the constraint between two

material points, established by means of a nonstretchable line. Problems of mechan-

ics with such unilateral constraint lead to solutions, discontinuous by the velocity

vector: at the moments of ”constraint activation” some velocity components change

a sign for the opposite. If you do not like solutions with velocity discontinuity, you

are free to complicate the constraint model for example, to consider line stretcha-

bility.

However, in the problems of mathematical physics and the field theory, unilateral con-

straints, apparently, did not occur before. Besides, unlike a mechanical problem with

nonstretchable line, we do not have any more accurate and adequate model in reserve:

there is nothing beyond Yang-Mills physics at all, this is all physics.

In particular, considering the problems of Yang-Mills free no-current waves, we easily find

out that Yang-Mills nonlinear wave equations possess chiral symmetry, despite the chiral

asymmetry of the field tensor and the Lagrangian. Accordingly, the solutions of these

equations possess chiral symmetry. Here we encounter not a very pleasant dilemma:

a) to take beautiful, continuous and smooth chiral-symmetric solutions which contain the

most difficult and unpredictable chaotic oscillations and to shut our eyes to the require-

ment of chiral determinacy of the solution;

or

b) to take into account the requirement of chiral determinacy and ”to impose” chiral de-

terminacy upon the solution by means of rough ”chiralization” procedure; the solutions

become discontinuous by derivatives, lose their chaotic character and gain a rather boring

periodicity10.

Choosing the first alternative leads to good Yang-Mills mathematical physics, but to

the knowingly incorrect Yang-Mills physics which violates the requirement of chiral de-

terminacy of solutions.

The choice of the second alternative correctly reproduces physical requirements of Yang-

Mills chiral definiteness, but generates rather ugly version of Yang-Mills mathematics of

solutions with derivatives discontinuities.

The very necessity for discussion of this ”mathematics-physical” dilemma seems strange

after half-a-century existence of Yang-Mills equations. Apparently, it is connected with

the fact that during these fifty years there has been made no serious attempt to turn Yang-

Mills classical field theory into the similar minutely developed section of mathematical

physics and applied mathematics, which is classical electrodynamics, for example.

10 In the subsequent articles of this series the examples of numerical solutions to the problems of Yang-

Mills triplet and octuplet waves both chaotic chiral-symmetric solutions and periodic solutions generated

by the procedure of chiralization will be provided.


4.2 Combined Coordinate Inversion

in the Singlet-Triplet Theory

When making the operation of inversion of coordinate system:

xν → x′ν = −xν , (34)

it should be taken into account that occurring transformation of the three-dimensional

right-handed coordinate system to the left-handed one (or vice versa) requires the same

transformation in YM-space, i.e. permutation of YM-indices 1↔ 2. However, during the

performance of the transformation (34) and YM-permutation, different terms in the ex-

pression for YM-tensor of field Wμν (15) behave differently. In the coordinate inversion,

Lorentz vectors of currents and potentials reverse signs, consequently, the expressions in

the form of ∂νJν or ∂νWμ do not reverse a sign. However, vector products in the form

of Wμ × Jμ or Wμ ×Wν do reverse a sign due to permutation of YM-indices 1↔ 2.

One more operation has to be provided for restoration of the consistency of transfor-

mational qualities of different terms in the same formula – sign inversion of currents

and potentials (without repeated permutation of the indices 1 ↔ 2). As a result, both

the base Lagrangian and additional Lagrangian of the singlet-triplet theory prove to be

invariant relative to this three-way combined coordinate inversion:

(1) Inversion of signs xν , generating the inversion of signs Jν , Wν , Jν and Wν .

(2) Permutation of YM-indices 1↔ 2.

(3) Repeated inversion of the signs of currents and potentials.

Finally, when performing such combined inversion, currents and potentials do not change,

field tensors reverse a sign (with regard to permutation of YM-indices 1 ↔ 2); the

Lagrangian does not change.

Coordinate inversion (34) per se, not accompanied by charge inversion, is not a correctly

determined procedure in the triplet sector of physics11.

5. About the Name of the Theory

The standard name of this branch of physics, going back to Abdul Salam, is ”electroweak

theory”. This name is absolutely justified from the historical point of view: unification

within the framework of one theory of electromagnetic and weak interaction was a great

achievement at the time of WS theory creation. However, from the point of view of logical

structure of this theory, we believe that it is possible to leave the terms ”electromagnetic

interaction” and ”weak interaction” in the past and to speak about ”singlet-triplet the-

ory” (ST theory) or, at least, about ”Maxwell and Yang-Mills theory” (MYM theory).

11The triviality of this statement within the framework of Yang-Mills structure of physics deserves

real amazement. This statement is equivalent to so-called ”Law of parity non-conservation in weak

interactions”. The formulation of this ”law” in due time required some courage from T. D. Lee and C.

N. Yang. Probably, Yang, partly under the influence of Wolfgang Pauli’ unfair criticism, who was always

confident of his own innocence, did not treat Yang-Mills equations, named after him, with due trust.


In fact, besides the idea of C.N. Yang and R.L. Mills about the existence of triplet of

YM- currents and triplet of YM potentials [3] conjugated with it, the idea of Sheldon Lee

Glashow about the necessity of joint interpretation of Maxwell singlet and Yang-Mills

triplet [7] was also needed, so, from the point of view of personification of the theory it

would be, perhaps, better to speak about ”Maxwell-YangMills-Glashow theory”. It was

S. Glashow’s work, in which angle , later called ”Weinberg angle”, appeared for the first

time. Glashow’s work, performed six years after the publication of C.N. Yang and R.L.

Mills’ article [3], did not contain any mentioning of the article [3] at all, so to a certain

extent it is possible to say that anyway, the electroweak theory would have probably

been constructed without C.N. Yang and R.L. Mills’ contribution. However, Yang-Mills’

approach goes beyond the framework of the triplet sector of physics without any slightest

changes in its essence and extends to the octuplet strong interaction. The only technical

change necessary here is the replacement of Levi-Civita three-index three-dimensional

symbol (which is a structural symbol of the group SU(2)) in the definition of vector prod-

uct of YM-vectors by a three-index eight-dimensional structural symbolabc

f of the group

SU(3). (Over-letter YM indices in the octuplet sector of physics run values from one to

eight). Therefore, personification of the name of the theory in this case seems doubtful,

so the best way to name the theory is by its basic algebraic structure: the singlet-triplet

theory (ST -theory) or, with regard to the octuplet sector of physics, a ”singlet-triplet-

octuplet theory” (STO-theory).

The commonly acknowledged version of electroweak theory existing today, Weinberg-

Salam theory, has also added Higgs sector to the singlet and triplet sectors of physics.

Within the framework of the classical field approach, developed in this article, and the

previous article of this series [1], the introduction to the theory of Higgs’ fields seems

needless12: the theory does not require Higgs’ spontaneous symmetry violation for gen-

eration of particles’ own weight; masses of particles are directly present in the ST theory

as its computable parameters ”at the exit” of the theory.

6. The Effective Lagrangian

of the Singlet-Triplet Theory

By coupling the base Lagrangian of the ST - theory (17) with the additional Lagrangian,

which accounts the constraints, imposed on the variables of ST - theory, and accepting

Lorentz gauge of the theory (i.e. omitting terms Lad(χ) and Lad(χ)), we obtain the effective

12 If Steven Weinberg asked me why Higgs’ field is missing from this theory, just like emperor Napoleon

asked Pierre Simon de Laplace about the reason of the Creator’s absence in Laplace’s ”Mecanique

celeste”, I would respond with Laplace’s words: ”Sire, je n’ai pas en besoin de cette hypothese” (”I had

no need of that hypothesis, Your Majesty”) [8]. But Higgs’ fields were inseparable part of theoretical

physics of the recent decades. It is hard to imagine that physicists would easily refuse to use Higgs

language. ”Laplace’s Escapade has never been what belongs to the masses”, – Pierre Chaunu [9] once

said.


Lagrangian of the ST - theory:

L′ = L′S + L′T + L′ST . (35)

In this formula (35), L′S is a part of the effective Lagrangian which contains only the

variables belonging to the singlet sector; L′T contains only the quantities belonging to the

triplet sector and L′ST is a part of the Lagrangian responsible for mixing of singlet and

triplet sectors:

L′S = − 1

8p2SJνJν −

1

2pSJνWν −

1

16πWμνW

μν ;

L′T = − 1

8p2TJνJν −

1

2pTJνWν −−

1

16πWμνW

μν − η

2p2T

(2

1

J ν2

Jν −3

J ν3

Jν

);

L′ST = − ψ

2p2T

3

J ν

(3

Jν −Jν).

(36)

The effective Lagrangian (35), (36) (we will call it ”the four-current Lagrangian”) is

appropriate for description of many physical states which differ in number of currents

involved into the current state, from zero-current states (a free singlet field or a free

triplet field) to total four-current states. The four-current Lagrangian has to be changed

in case of isotropization of at least one of the four currents in some 4-zone with non-

zero 4-volume. In this case the contribution of this current in the current part of the

Lagrangian (36) is missing, but the term of the form λN νNν appears in the additional

Lagrangian, where Nν is isotropic (neutrino) current, and λ is the Lagrange’s multiplier.

Let us note that retention of η and ψ in the Lagrangian is equivalent to introduction into

the theory of some external ”Higgs-like” fields that provide realization of the constraints

laid onto currents. For constructing a closed physical theory, it would be reasonable to

accept that the corresponding current constraints are natural for the singlet-triplet theory

Lagrangian, i.e. constraint equations can be satisfied at zero values of the Lagrange’s

multipliers η and ψ. We have no official proof of the validity of this assumption.

The condition of current isotropy for neutrino states is unnatural, and the term of the

form λNνNν cannot be omitted from the Lagrangian in the process of field equations

obtaining. It means that the neutrino state cannot cover all the four-dimensional space-

time while occupying only some finite zone, which is transitional between other states.

7. Classification of States

of the Singlet-Triplet Theory

All variety of the states, described by the Lagrangian of the ST -theory (36), can be clas-

sified as follows. First of all it is possible to allocate the ”pure” or ”one-sector” states

which completely fit one of the two sectors of the theory: singlet states (S-states or M -

states) and triplet states (T -states or YM -states). The states which are not pure will be

named ”mixed” or ”two-sector” states (ST -states). By means of rescaling of variables it

is possible to except Weinberg angle from the description of pure states. Weinberg angle


Fig. 1 Pure one-sector states in the ST -theory


Fig. 2 Mixed two-sector states in the ST -theory


cannot be excepted from the description of mixed states. Further the states of the ST -

theory can be classified by the number of nonzero currents: zero-current states, i.e. free

fields are maxwellian free field in pure S-state and Yang-Mills free field in pure T -state;

one-current states, two-current, three-current and four-current states. Singlet states can

be only zero-current or one-current. All states, except zero-current ones, can be classified

by number of pomeriums, i.e. outer boundaries of the current zones. ”Wave” states have

no pomerium at all (the current zone is unbounded). Photon, and also massive photon

and maxwellian neutrino, described in article [1] of this series, relate to the wave states

in the singlet sector. In the triplet sector there are wave states which are similar to them.

The states which have at least one pomerium will be named ”discrete states”. The most

important of the discrete states is a ”stationary state”, i.e. a state with one pomerium

for which there is such an intrinsic frame of reference where currents and fields do not

depend on time. Stationary states describe current structure of massive single particles.

Stationary states exist only as pure one-sector states, mixed stationary states do not

exist.

It is possible to distinguish a special case of ”neutrino states” among current wave states

– i.e. the states in which pseudo-Euclidean square of 4-current vector is everywhere equal

to zero.

There are mixed neutrino states simultaneously fitting two sectors of the theory, Maxwel-

lian and Yang-Mills’ (”Maxwell-Yang-Mills neutrino”, or mymino in abbreviated form).

There is a pure singlet neutrino state (makswellian neutrino, or mino in abbreviated

form). There are pure neutrino states which completely fit Yang-Mills sector (”Yang-

Mills neutrino”, or ymino in abbreviated form). We do not know whether this theo-

retical classification of neutrino states – mino, ymino, mymino – corresponds to the

empirical fact of distinction of the lepton neutrinos (electronic neutrino, muonic neutrino,

τ -neutrino)13.

The described classification of states, possible in the ST -theory, is reflected in fig. 1 (pure

states) and fig. 2 (mixed states). These pictures illustrate a variety of particular prob-

lems, which can be formulated and solved within the framework of the ST -theory, and

thereby, the inner wealth of the ST -theory as a section of mathematical and computer

physics.

By describing formulations of the problems of the ST -theory further in the text of this

article, we restrict ourselves to only two types of the problems: the wave problems con-

taining no pomerium at all and stationary problems with one pomerium.

Wave problems are quite correct as problems of classical mathematical physics, however

the search for correspondence between a variety of their solutions and real quantum ob-

jects of the micro-world may appear a difficult task14.

13 For complete description of neutrino states it can be noted that a priori nothing forbids to allow

the possibility of izotropization of any of eight currents of the octuplet sector of physics which is not

considered in this article, and thereby to enter into consideration one more set of neutrinos different from

mino, ymino and mymino.14Within the framework of the wave problems, for example, it is not difficult to consider the theory of

standing Yang-Mills waves – triplet and octuplet (gluon) ones. But how, by means of what equipment,


Stationary classical single-particle problems allow to calculate, in principle, masses, char-

ges and spins of particles and to study the inner structure of massive particles. The

particles of one physical class consisting of one current, differ in topology of the outer

boundaries of pomerium, and also in number and topology of the inner boundaries of

latens. Particles of different physical classes differ in currents they consist of. Within the

framework of this article we are not going to consider the formulations of the problems

which describe the states with more than one pomerium. Within classical physics they

are obviously non-stationary states. These states are the key ones for modern treatment

of physics of micro-world. They are the states of particles interaction, including the states

corresponding to spontaneous decay of unstable particles. The possibility of correct de-

scription of such states within the framework of the classical ST -theory, constructed here,

seems rather doubtful. However, without studying such classical problems, it is difficult

to count on successful construction of the proper quantum theory of non-point particles.

The following restrictions on the variables, entering into the Lagrangian of the singlet-

triplet theory (36), are used for extraction of the particular states reflected in figures 1

and 2.

A. Wave states

A1. One-sector wave states

• Zero-current singlet wave (free maxwellian wave, photon):

Jν ≡ 0; Wν ≡ 0; Jν ≡ 0.

• One-current singlet wave (heavy photon):

Jν ≡ 0; Wν ≡ 0; JνJν < 0.

• Neutrino singlet wave (maxwellian neutrino, myno):

Jν ≡ 0; Wν ≡ 0; Jν �= 0; JνJν = 0.

• Zero-current triplet wave (Yang-Mills free wave):

Jν ≡ 0; Wν ≡ 0; Jν ≡ 0.

• One-current triplet wave (Yang-Mills heavy one-current wave):

Jν ≡ 0; Wν ≡ 0;1

J ν1

Jν< 0;2

J ν ≡ 0;3

J ν ≡ 0.15

• Neutrino one-current triplet wave (Yang-Mills neutrino, ymino):

Jν ≡ 0; Wν ≡ 0;1

J ν �= 0;1

J ν1

Jν= 0;2

J ν = 0;3

J ν = 0.16

• Two-current triplet wave (Yang-Mills heavy two-current wave):

Jν ≡ 0; Wν ≡ 0;1

J ν2

Jν= 0;1

J ν1

Jν< 0;2

J ν2

Jν< 0;3

J ν ≡ 0.

• Neutrino two-current triplet wave:

Jν = 0; Wν = 0;1

J ν �= 0;1

J ν1

Jν< 0;2

J ν �= 0;2

J ν2

Jν= 0;1

J ν2

Jν= 0;3

J ν ≡ 0.

• Three-current triplet wave (Yang-Mills heavy three-current wave):

would it be possible to conduct Yang-Mills analogue of maxwellian experiment on observation of standing

singlet waves which was conducted by O. Wiener for the first time already in 1890th? For observing

standing gluon waves it is necessary to arrange reflecting surfaces in a baryon. And, even if this arrange-

ment is possible at baryon collisions, the characteristic dimensions of atomic nuclei are enormously high

in comparison with the fundamental length r0, appearing in the theory of currents and potentials.15Also two analogous wave states with arbitrary permutation of YM-indices.16Also two analogous neutrino states with arbitrary permutation of YM-indices.


Jν ≡ 0; Wν ≡ 0;1

J ν1

Jν< 0;2

J ν2

Jν< 0;2

J ν3

Jν< 0; 21

J ν2

Jν=3

J ν3

Jν .

• Neutrino three-current triplet wave:

Jν ≡ 0; Wν ≡ 0;1

J ν �= 0;1

J ν1

Jν= 0;2

J ν �= 0;2

J ν2

Jν= 0;1

J ν2

Jν= 0;3

J ν �= 0;3

J ν3

Jν= 0.

A2. Two-sector wave states

• One-current two-sector neutrino wave (Maxwell-Yang-Mills neutrino, mymino):

Jν = N ν ;3

J ν = −N ν ; NνNν = 0;1

J ν ≡ 0;2

J ν ≡ 0.

• Two-current two-sector one-neutrino wave:

Jν = N ν ;3

J ν = −N ν ; NνNν = 0;1

J ν1

Jν< 0;2

J ν ≡ 0.17

• Two-current two-sector two-neutrino wave:

Jν = N ν ;3

J ν = −N ν ; NνNν = 0;1

J ν �= 0;1

J ν1

Jν= 0;2

J ν ≡ 0.18

• Left-handed four-current wave:

Jν = lν +N ν ;3

J ν = lν −Nν ; Nν �= 0; NνNν = 0; lνNν = 0; lνlν < 0;

21

J ν2

Jν= lνlν ;1

J ν1

Jν< 0;2

J ν2

Jν< 0.

• Full four-current wave:

JνJν < 0;1

J ν1

Jν< 0;2

J ν2

Jν< 0;3

J ν3

Jν< 0; 21

J ν2

Jν=2

J ν3

Jν< 0;3

J ν

(3

Jν −Jν)

= 0.

B. Discrete states

B1. One-sector discrete states

• Singlet stationary state (lepton):

JνJν ≤ 0; Jν ≡ 0; Wν ≡ 0; there is only one pomerium; there is intrinsic reference

system of a state in which∂

∂t≡ 0.

• Singlet non-stationary state (electromagnetic interaction of leptons):

JνJν ≤ 0; Jν ≡ 0; Wν ≡ 0; there is more than one pomerium.

• Stationary one-current triplet state (hypothetical particle wark):

Jν ≡ 0; Wν ≡ 0;2

J ν ≡ 0;3

J ν ≡ 0;1

J ν1

Jν≤ 0; there is only one YM-pomerium; there

is intrinsic reference system of state in which∂

∂t≡ 0.19

• Non-stationary one-current triplet state (YM-interaction of warks):

Jν ≡ 0; Wν ≡ 0;2

J ν ≡ 0;3

J ν ≡ 0;1

J ν1

Jν≤ 0; there are more than one YM-

pomeriums20.

• Stationary two-current triplet state(hypothetical particle wion):

Jν ≡ 0; Wν ≡ 0;3

J ν ≡ 0;1

J ν1

Jν≤ 0;2

J ν2

Jν≤ 0;1

J ν2

Jν= 0; there is one YM-pomerium,

common for currents1

J ν and2

J ν ; there is intrinsic state reference system in which∂

∂t≡ 0.

• Non-stationary two-current triplet state (YM-interaction of warks or/and wions):

Jν ≡ 0; Wν ≡ 0;1

J ν1

Jν≤ 0;2

J ν2

Jν≤ 0;3

J ν ≡ 0;1

J ν2

Jν= 0; there is more than one

17Also analogous one-neutrino wave with permutation of YM-indices 1↔ 2.18Also analogous two-neutrino wave with permutation of YM-indices 1↔ 2.19Also analogous stationary state with permutation of YM-indices 1↔ 2.20Also analogous non-stationary state with permutation of YM-indices 1↔ 2.


YM-pomerium.

• Stationary three-current triplet state:

Jν ≡ 0; Wν ≡ 0;1

J ν1

Jν≤ 0;2

J ν2

Jν≤ 0;3

J ν3

Jν≤ 0; 21

J ν2

Jν=3

J ν3

Jν ; there is one YM-

pomerium which is common for the three YM-currents; there is intrinsic reference

system of state in which∂

∂t≡ 0.

• Non-stationary three-current triplet state:

Jν ≡ 0; Wν ≡ 0;1

J ν1

Jν≤ 0;2

J ν2

Jν≤ 0;3

J ν3

Jν≤ 0;1

J ν2

Jν=3

J ν3

Jν ; there is more than

one pomerium.

B2. Two-sector discrete states

• Two-current one-neutrino singlet-triplet state containing at least one YM-pomerium

(wark/mymino interaction):

Jν = N ν ;3

J ν = −N ν ; NνNν = 0;1

J ν1

Jν≤ 0;2

J ν = 0.21

• Three-current one-neutrino singlet-triplet state containing at least one YM-pome-

rium, common for currents1

J ν and2

J ν (wion/mymino interaction):

Jν = N ν ;3

J ν = −N ν ; NνNν = 0;1

J ν1

Jν≤ 0;2

J ν2

Jν≤ 0;1

J ν2

Jν= 0.

• Left-handed four-current singlet-triplet state containing at least one pomerium:

Jν = lν +N ν ;3

J ν = lν −Nν ; lνNν = 0; N νNν = 0; lνlν ≤ 0;1

J ν1

Jν≤ 0;2

J ν2

Jν≤ 0; 21

J ν2

Jν= lνlν .

• Total four-current singlet-triplet state containing at least one pomerium:

JνJν ≤ 0;1

J ν1

Jν≤ 0;3

J ν3

Jν≤ 0; 21

J ν2

Jν=3

J ν3

Jν ;3

J ν

(3

Jν −Jν)

= 0.

The provided classification of the states of singlet-triplet theory is rather formal. It

describes possible states on the basis of a set of non-zero currents. Each state, allocated

in the classification, needs some existence theorem which would claim that the algebraic

conditions, imposed on currents, are compatible22.

Wave states can be researched quite minutely, at least, for the plane waves. Plane waves

can be described by a system of ordinary differential equations. The solutions to the

problems of plane waves within the framework of the singlet-triplet theory can be ob-

tained analytically or numerically.

Discrete states in the singlet-triplet theory are described by an equation system in partial

derivatives which include a priori unknown boundaries of current zones (pomerium and

latens). The lack of existence theorems causes some difficulties for the development of

numerical methods of solving problems of discrete states.

The given classification does not include transitional neutrino states, i.e. such states, in

which 4-zone, occupied with the neutrino current, is finite and non-stationary.

The given classification for some discrete states is not unambiguously definite. For exam-

ple, the state, described as a ”singlet non-stationary state”, describes both the state of

21Also analogous one-neutrino state with permutation of YM-indices 1↔ 2.22But stationary states also need some spectral theorem claiming non-uniqueness of stationary state

with the given set of algebraic conditions for currents.


scattering of a lepton on a lepton, for which the classical theory, constructed here, can

be quite correct, and the state of annihilation of two leptons, for which, probably, the

classical theory can prove to be as unacceptable as Newtonian mechanics for description

of atoms.

However, we believe that classical singlet-triplet theory is worth careful and detailed

development as a section of mathematical physics for clarification of prognostic opportu-

nities of the theory and limits of its applicability.

8. Pure one-sector states: Maxwellian Singlet States

8.1 Getting Rid of Weinberg Angle:

the Electromagnetic Lagrangian

In pure S-state all triplet currents Jν and all triplet potentials Wν are equal to zero.

Only L′S will remain in the effective ST -Lagrangian (36). Supposing that in L′S:

jem

ν =1

2Jν ;

Aν = pSWν ;

Aμν = ∂μAν − ∂νAμ = pSWμν ,

(37)

we can rewrite the singlet Lagrangian L′S in the following form:

L′S =1

p2SLem,

where

Lem

= −1

2jem

ν jνem

− jem

νAν −1

16πAμνA

μν . (38)

Therefore, the singlet Lagrangian L′S differs from the standard Lagrangian of classical

electrodynamics of non-point particles [1] only in insignificant common multiplier.

The electromagnetic Lagrangian (38) comprises the description of both the wave prob-

lems without current boundaries (an ordinary no-current electromagnetic wave – photon;

”heavy” current electromagnetic wave – ”heavy photon”; neutrino wave – maxwellian

neutrino [1]), and the discrete problems containing outer and inner boundaries of current

zones (pomerium and latens). The discrete problems, as a subset, comprise stationary

singlet problems which solution describes inner structure of massive leptons [1].

”Deductibility” of the electromagnetic Lagrangian (38) from the effective ST -Lagrangian

(36) in one-sector maxwellian problem is an absolutely trivial mathematical fact. But this

fact comprises the profound physical meaning: intrinsic characteristics of massive

leptons are formed in the singlet sector of the theory and do not depend

on the triplet interactions . In particular, the formulas of heavy lepton mass ratios,

provided in the article [1], remain reasonable in the ST -theory. The triplet sector of

the theory influences the stability of stationary states of the singlet sector, but does not


influence the stationary states themselves.

Now we do not know whether the correct formulation of the problem on stability of the

singlet stationary discrete state within the framework of the classical theory, considered

here, is possible. Perhaps, the effective ST -Lagrangian (36) comprises the description

of such instability connected with spontaneous ”swelling” of the bubble of the neutrino

triplet currents and this bubble’s ”splashing” beyond the limits of pomerium of a discrete

singlet object.

8.2 Estimate of the Fundamental Constant r0

The classical theory of non-point particles, constructed here, contains as ”entrance” pa-

rameters of the theory the fundamental length r0, the velocity of light c, the fundamental

constant j0 which determines the top limit of the current density, and the dimensionless

constant k which forms Einstein equations and is proportional to the constant of gravity

G [1].

If we had a numerical solution of the stationary one-sector singlet problem, we would

have a possibility to express the intrinsic characteristics of massive leptons – mass m,

charge e and angular momentum M – through these ”entrance” parameters. But now

we have neither such numerical solution, nor even the existence theorem which would

guarantee the existence of at least one solution; nor the spectral theorem which would

guarantee the existence of some finite or countable set of the intrinsic solutions, differing

in topological characteristics. In such situation everything we can afford is dimensional

estimates.

The mass of a lepton can be estimated by the order of magnitude by means of the fol-

lowing formula:

mc2 ≈ r20c2

j20 r2P r0,

and the electrical charge e – by means of the formula:

e ≈ 1

cj0 r

30.

These formulas are based on the following ideas. The main contribution to the total

energy of the massive lepton is made by a small part of the lepton volume in small neigh-

borhood of the ”cavitated” tube latebra, on the boundary of which the current density

achieves a peak value j0. We believe that the diametrical dimension of cavitated tubes is

of the order of Planckian length rP , and the typical diametrical dimension of the current

high density zone, bordering upon latens, is approximately the same. The longitudinal

dimension of this zone is of the order of the general outer dimension of the lepton, and

this dimension, in its turn, is close to the fundamental constant r0. The first dimensional

factor(r20

/c2

)in the formula for the lepton mass is the proportionality coefficient which

forms the expression of the tensor components of energy-momentum the current field (see

[1]).


The expression for charge dimension e, linear by current density, means that the notice-

able contribution to the total value of an electric charge is made not only by a narrow

zone in the neighborhood of ”cavitated” tube latebra, but by the whole lepton volume

as well. The assumptions, on which basis the above mass and charge estimates are con-

structed, are open to criticism. But let us see the consequences these estimates lead to.

From the given estimates for m and e we find out that

e2

mc2≈ r30

r2P,

bute2

mc2= rT ,

where rT is a classical Thomson lepton radius.

So, on the grounds of these lepton mass and charge estimates, we get a possibility to

estimate the unknown fundamental constant r0:

r0 ≈(rT r

2P

)1/3. (39)

If we use Thomson electron radius rT ≈ 2.8 · 10−13 cm as a quantity rT , (39) gives the

following estimate for r0:

r0 ≈ 0.9 · 10−26cm. (40)

The given estimate (40) is probably not too cogent. Formula (39) contains not only the

fundamental Planckian length rP , but also Thomson radius of a specific lepton; the di-

mension r0 itself has to be fundamental, i.e. to have no ”binding” to any specific particle.

However, we have no other estimate of the fundamental constant r0.

Quantity r0 (40) is enormously small from the point of view of the possibility ever ”to

consider” the electron structure in any feasible physical experiment. However, this quan-

tity is quite large in comparison with Planckian length rP determining the diametrical

dimensions of latebrae. Physics of leptons contains two dimensionless constants: the

constant of fine structure α and ratio rP/r0. Undoubtedly, these two constants have to

be somehow connected with each other.

By estimating, within the framework of the same ”naive estimate”, the own angular mo-

mentum of lepton M by the formulas provided in the article [1], it is possible to find

that

M ≈ r20c2· 1c· j20 · r40.

Determining ratioe2

Mcby means of this estimate, we get the estimate which is not very

substantial:e2

Mc= O(1), (41)

at the same time, as the correct estimate could be M ≈ � ande2

Mc≈ α, where �

is Planckian constant and α is a fine structure constant. We do not know now whether


estimate (41) is the evidence of a rough and not removable imperfection of the constructed

theory, or whether this estimate can be improved under the accurate numerical solution

of a stationary discrete singlet problem and accurate calculation of quantities e and M.

9. Pure One-sector States: Yang-Mills Triplet States

9.1 Getting Rid of Weinberg Angle: the Weak Lagrangian

The set of pure YM-states, represented in fig.1 have an ”empty” singlet sector: Jν = 0;

Wν = 0. The missing of the singlet current, in accordance with the algebraic condition

(20), means that3

J ν3

Jν= 0. (42)

Condition (42) means that current3

J ν , if it is not identically zero, can be only a neu-

trino one. Algebraic condition of the inner normalization of currents of YM-triplet (32)

requires, in this case, the orthogonality of the first and second components of YM-triplet.

1

J ν2

Jν= 0. (43)

Weinberg angle can be eliminated from the triplet part of the effective ST -Lagrangian

by means of scaling which comprises the triplet interaction coefficient pT :

pTWν = Aν ;

Aμν = ∂μAν − ∂νAμ +Aμ ×Aν = pTWμν ,(44)

Let us also re-scale the triplet currents, assuming that:

1

j ν =1

2

1

J ν ;2

j ν =1

2

2

J ν ; Nν =1

2

3

J ν , (45)

and, in accordance with (42) and (43):

1

j ν2

jν= 0, (46)

NνNν = 0. (47)

After substitution of the relations (44) and (45), with regard to (46) and (47), the triplet

Lagrangian L′T (36) can be formulated in the following form:

L′T =1

p2TLw, (48)

Lw =− 1

2

(1

j ν1

jν +2

j ν2

jν

)−

(1

j ν1

Aν +2

j ν2

Aν +Nν3

Aν

)− 1

16πAμν ·Aμν − η

1

j ν2

jν −1

2λN νNν .

(49)


The Lagrangian Lw does not include Weinberg angle θw.

Let us name the Lagrangian Lw ”weak Lagrangian”, the triplet of currents

(1

j ν ,2

j ν , N ν

),

which satisfies the conditions (46) and (47), – ”weak currents triplet”, triplet Aν – ”weak

potentials triplet”, and tensor Aμν – ”weak field tensor”. Coefficients η and λ under

corresponding terms in the weak Lagrangian are Lagrange’s multipliers.

9.2 Zero-current Triplet States: Yang-Mills Free Triplet Waves

9.2.1 Yang-Mills Wave Equation

All the currents of YM-triplet are missing in the wave triplet state:

1

j ν =2

j ν = Nν = 0.

The Lagrangian of such zero-current state is reduced to Yang-Mills free field Lagrangian

Lw,f :

Lw,f = − 1

16πAμν ·Aμν . (50)

Variation of action functional with the Lagrangian (50) by weak potentials Aμ gives the

equations of the free weak field:

∂μAμν +Aμ ×Aμν = 0. (51)

By eliminating field tensor Aμν from Yang-Mills field equations (51) by means of (44),

we can present the equations (51) in the following form:

−�Aν +Aμ × (2∂μAν − ∂νAμ +Aμ ×Aν) = ∂ν�+Aν × �. (52)

In equations (52) � is 4-divergence of the weak potential Aμ:

� = ∂μAμ,

and � is D’Alembert operator:

� = −∂μ∂μ.

One arbitrary gauge condition can be imposed on potentials Aμ. It is proper to assume

that 4-divergence � is identically zero. This equation is compatible with the field equations

(52). Finally, the free weak field of the zero-current triplet state is described by the

following system of the equations:

�Aν = Aμ × (2∂μAν − ∂νAν +Aμ ×Aν) ; (53)

∂μAμ = 0. (54)


9.2.2 Yang-Mills Plane Zero-current Wave

The simplest solution to field equations (53), (54) can be tried in the form of a plane

wave with wave vector kν , supposing that 12 unknown functions of Aμ depend only on a

single argument – a wave phase φ:

φ = kμxμ.

For the plane wave, the field equations (53), (54) transform into a system of ordinary

differential equations:

kμkμAν ′′ + 2kμA

μ ×Aν ′ − kνAμ ×Aμ′ +Aμ × (Aμ ×Aν) = 0. (55)

kμAμ′ = 0. (56)

In these equations the prime next to the potentials stands for the derivative with respect

to wave phase φ.

The plane wave equations (55) have apparent scale invariance with respect to wave vector

kμ.

Let kμ be a time-like vector:

kμkμ = ε2 > 0.23

Let us make scale transformation of potentials Aμ, coordinates xμ and wave vector kμ,

supposing thatkν = εkν ; kν kν = 1.

Aν = εAν; xν =

1

εxν .

At this transformation the wave phase remains invariable:

φ = kνxν = kν xν .

The specified scale transformation turns a wave vector into a unit vector without changing

the form of the wave equations (55), (56). Let us rewrite these equations with new

variables, omitting the symbol ∼ over the corresponding quantities in the form:

Aν ′′ + 2kμAμ ×Aν ′ − kνAμ ×Aμ

′ +Aμ × (Aμ ×Aν) = 0; (57)

23The fact that both W. Pauli and C.N. Yang with R.L. Mills in 1953-1954 passed by the classical Yang-

Mills wave theory with the time-like wave vector, can be considered an inexplicable historical paradox.

With persistence, that is worth regrets, there was considered only the version of the theory with isotropic

wave vector kμkμ = 0, generating the idea of zero mass of Yang-Mills field quantum (”one will always

obtain vector mesons with rest mass zero”, wrote Wolfgang Pauli in 1953 December [4]) under general

quantum treatment.

It is obvious that the construction of the solution (55) does not require any wave vector isotropy at

all. This sight aberration of the triplet theory authors is probably caused by the tendency of the

physicists of that time to treat nonlinear equations by means of perturbation methods, which is absolutely

unacceptable for Yang-Mills essentially nonlinear equations.

Finally, this stable aberration led to the introduction of Higgs mechanism for explanation of particles

masses. The possibility of the correct treatment of Yang-Mills waves with time-like wave vector, described

in this article, shows the existence of a rest mass of such waves, but within the framework of the classical

theory, this mass, proportional to the pseudo-Euclidean module of the wave 4-vector, remains arbitrary.


kμAμ′ = 0; (58)

kμkμ = 1. (59)

From the condition (58) follows that convolution kμAμ does not depend on the wave

phase:

kμAμ = q = const;

which allows to rewrite the plane YM-wave equation in the form:

Aν ′′ + 2q×Aν − kνAμ ×Aμ′ +Aμ × (Aμ ×Aν) = 0. (60)

9.2.3 Energy Integral for Yang-Mills Plane Zero-current Wave

By multiplying the wave equation (60) scalarly by the derivative of potential A′ν , and

integrating by the wave phase φ, it is possible to obtain the first integral of the wave

equation (60) which is appropriate to be called ”energy integral”:

K+ U = E = const;

K = −1

2Aν ′ ·Aν

′;

U =1

4

((Aν ·Aν)

2 − (Aμ ·Aν) (Aμ ·Aν)

).

(61)

It is easy to show that physical quantity U, which can be treated as a potential wave

energy density, is positive-definite. Physical quantityK is the first term in the energy

conservation law (61) which can be interpreted as the density of kinetic wave energy; it

is also positive-defined. Therefore, wave energy is positive.

In wave equation (60) and energy integral (61) it is easy to see that Yang-Mills plane

wave has a self-similarity relative to the energy quantity E.

The scale transformation which uses E as a parameter:

Aν = E1/4∗A ν ,

φ = −E1/4∗φ,

q = E1/4 ∗q,

(62)

normalizes the wave energy to unit value without altering the form of the wave equation

(60) and energy integral (61).

We will afford once again to rewrite almost tautologically the problem formulation of

Yang-Mills plane wave in variables∗A ν ,

∗φ,

∗q introduced by the scale transformation (62)

omitting symbol * over the letters:

Aν ′′ + 2q×Aν ′ − kνAμ ×Aμ′ +Aμ (A

μ ·Aν)−Aν (Aμ ·Aμ) = 0, (63)

−1

2Aν ′ ·Aν

′ +1

4

((Aν ·Aν)

2 − (Aμ ·Aν) (Aμ ·Aν)

)= 1, (64)

kμAμ = q, (65)


kμkμ = 1. (66)

Let us pay attention to the fact that the invariance of Yang-Mills plane wave theory, rel-

ative to scale transformations generated by pseudo-Euclidean module of the wave vector

and energy (or wave amplitude), means that Yang-Mills wave theory has neither geo-

metrical optics approximation, nor asymptotics of small amplitudes (low energies). Both

wave vector and wave energy can always be re-scaled to unit values.

9.2.4 Nonholonomic Constraint Equations for Yang-MillsPlane Zero-current Wave

By multiplying the wave equation (63) scalarly by wave vector kν , taking into account

(65) and (66), we will get the identity which has to be satisfied with the solution (63):

Aμ ×Qμ = 0. (67)

where

Qμ = A′μ +Aμ × q. (68)

The equation (67) with vector Qμ (68) can be treated as a non-holonomic constraint

imposed on Yang-Mills dynamic system (63). It is sufficient to require the initial data to

meet the condition (67).

9.2.5 Yang-Mills Plane Zero-current Wave in the Intrinsic Frame of Reference

It is proper to make further analysis of Yang-Mills plane wave in the intrinsic frame of

reference of the wave, in which the wave vector kμ has the form:

kμ = {1; 0; 0; 0} . (69)

In this frame of reference the wave phase coincides with the intrinsic time of the frame,

and the phase derivatives coincide with the intrinsic time derivatives. We shall denote

these derivatives with a dot as a dot upper index after Lorentz indices. In this frame

none of the potential components depend on spatial coordinates: the wave ”flashes” and

”dies out” simultaneously over the whole infinite three-dimensional space.

Our deep-rooted Lorentz intuition, undoubtedly, protests against this picture, however

there is no violation of Lorentz-invariance: Yang-Mills plane wave with a fixed wave

vector kν is the artificial mathematical entity which fills up the whole three-dimensional

space. Such object requires the infinite stock of energy for its formation. The real wave,

covering the finite spatial domain and bearing the finite total energy, is a wave packet,

i.e. superposition of waves with different wave vectors. Such wave can be presented in

the form of Fourier integral on four-dimensional space of the wave vectors. The more

compact the space-time localization of the wave packet is, the wider is the spread of

waves within the space of wave vectors. The notion ”intrinsic frame of reference” makes

sense only for a plane wave with a fixed wave vector. For a wave packet this notion has

no sense. However, the mathematics of Yang-Mills wave packets, which requires Fourier

transformation of Yang-Mills wave equations (63), seems to us almost ”uninterpretable” in


the continual situation. The problem of Yang-Mills stationary waves, requiring a discrete

Fourier transformation of the wave equation (63), is quite ”interpreted”. However, we do

not know whether the engineering-physical problem of creation of ”resonators” of Yang-

Mills waves, which are analogous to the radio engineering resonators of electromagnetic

waves, is solvable in principle.

Let us enter ordinary ”1+3” notations for the components of the weak potential Aν :

a

A ν =

{a

T;a

U

}, (70)

wherea

T is a time component of YM-potential vector a,a

U is the three spatial components

of YM-potential vector a. Vectorsa

U (for each a) are ordinary three-dimensional vectors

in Euclidean three-dimensional coordinate space. The separation of the components of 4-

vector into time and spatial part is not Lorentz - invariant. Let us assume that separation

(70) is carried out in YM-wave intrinsic frame of reference. Transition to any other frame

of reference is carried out by means of Lorentz- transformation of 4-vectora

A ν .

Substituting a ”separation formula” (70) into the condition (65) and taking into account

the mode of the wave vector in the wave intrinsic frame of reference (69), we find that all

the time components of YM-potentials are constant:

a

T=aq= const.

Further we will assume that allaq are zero:

aq= 0. (71)

A non-zero value of constantsaq would mean the presence of some outer stationary field,

which is hardly compatible with the problem of studying Yang-Mills free wave .

By substituting (70) and (71) into the wave equation (63) and nonholonomic constraint

(67), (68), we can represent the formulation of the problem of Yang-Mills plane wave in

the following form:a

U ••+b

U ×(

a

U ×b

U

)= 0; (72)

abcε

b

U ·c

U • = 0. (73)

In these formulasa

U is the triple of ordinary three-dimensional vectors, the symbol ”×”means the ordinary three-dimensional vector product; a, b, c are Yang-Mills indices taking

the values 1, 2, 3;abcε is a three-dimensional Levi-Civita symbol; summation is made by

YM-indices. Opening the triple vector product in (72), Yang-Mills equation (72) can be

rewritten as follows:a

U ••+ab

Sb

U −bb

Sa

U= 0. (74)

Yang-Mills matrixab

S appears in the relation (74):

ab

S=a

A νb

Aν=a

Tb

T −a

U ·b

U, (75)


or, regarding the condition (71) in the wave intrinsic system:

ab

S= −a

U ·b

U . (76)

A more compact notation (74) is formed if Yang-Mills ”inertia tensor”ab

I is introduced:

ab

I=ab

S −ab

δcc

S, (77)

whereab

δ is Kronecker symbol in the space of YM-indices. In the intrinsic wave system,

Yang-Mills inertia tensorab

I with (76) looks as follows:

ab

I= U2ab

δ −a

U ·b

U, (78)

where U2 =∑a

a

U 2. By using tensorab

I , introduced here, Yang-Mills equations (74) can

be rewritten in the form which remotely resembles the form of Euler equations in the

dynamics of a perfectly rigid body:

a

U ••+ab

Ib

U= 0. (79)

Yang-Mills equations (79) (which would probably be appropriate to be called Yang-Mills-

Euler ones) describe a three-vector dynamical system with nine degrees of freedom, where

three nonholonomic constraints are imposed on this system’s behavior (73). Let us name

this dynamical system ”tripletic Yang-Mills’ oscillator” (tymos). During quantization of

triplet Yang-Mills wave24 there must appear a particle with a nonzero rest mass. By

analogy with a photon – the singlet field quantum – this particle can be named ymon .

There are no reasons for identification of ymon with W- and Z-particles of Weiberg-

Salam theory.

In notations (70) and (75) energy integral for tymos (79) has the following form:

1

2

a

U •·a

U • +1

4

((ab

S

)2

−ab

Sab

S

)= 1, (80)

or, using formula (76) for YM-matrixab

S:

1

2

a

U •·a

U • +1

2

((1

U ×2

U

)2

+

(2

U ×3

U

)2

+

(3

U ×1

U

)2)

= 1. (81)

24 In the article on the classical theory of Yang-Mills fields it is irrelevant to discuss these rules of

quantization: it is sufficient to assume that they exist.


9.2.6 The Detailed Notation of the Tymos Dynamic Equations

Let us present here the detailed notation of Yang-Mills-Euler equations (79) with explic-

itly written Yang-Mills’ and vector indices in Cartesian coordinates:

1

U1••+

11

I1

U1 +12

I2

U1 +13

I3

U1= 0,1

U2••+

11

I1

U2 +12

I2

U2 +13

I3

U2= 0,1

U3••+

11

I1

U3 +12

I2

U3 +13

I3

U3= 0,2

U1••+

21

I1

U1 +22

I2

U1 +23

I3

U1= 0,2

U2••+

21

I1

U2 +22

I2

U2 +23

I3

U2= 0,2

U3••+

21

I1

U3 +22

I2

U3 +23

I3

U3= 0,3

U1••+

31

I1

U1 +32

I2

U1 +33

I3

U1= 0,3

U2••+

31

I1

U2 +32

I2

U2 +33

I3

U2= 0,3

U3••+

31

I1

U3 +32

I2

U3 +33

I3

U3= 0.

(82)

In the system of equations (82) the subscripts number Cartesian coordinates of the vectora

U; the over-letter indices are Yang-Mills indices.

The explicit notation of the components of YM-inertia tensor tymos , according to (76),

has the following form:

11

I =2

U 2+3

U 2 =2

U12+

2

U22+

2

U32+

3

U12+

3

U22+

3

U32,

22

I =3

U 2+1

U 2 =3

U12+

3

U22+

3

U32+

1

U12+

1

U22+

1

U32,

33

I =1

U 2+2

U 2 =1

U12+

1

U22+

1

U32+

2

U12+

2

U22+

2

U32,

12

I =21

I = −1

U ·2

U= −(

1

U1

2

U1 +1

U2

2

U2 +1

U3

2

U3

),

13

I =31

I = −1

U ·3

U= −(

1

U1

3

U1 +1

U2

3

U2 +1

U3

3

U3

),

23

I =32

I = −2

U ·3

U= −(

2

U1

3

U1 +2

U2

3

U2 +2

U3

3

U3

).

(83)


The explicit notation of energy integral (81) has the following form:

K+ U = 1,

K =1

2

((1

U1•)2

+

(1

U2•)2

+

(1

U3•)2

+

(2

U1•)2

+

+

(2

U2•)2

+

(2

U3•)2

+

(3

U1•)2

+

(3

U2•)2

+

(3

U3•)2

),

U =1

2

((1

U1

2

U2 −1

U2

2

U1

)2

+

(2

U1

3

U2 −2

U2

3

U1

)2

+

(1

U1

3

U2 −1

U2

3

U1

)2

+

+

(1

U1

2

U3 −1

U3

2

U1

)2

+

(2

U1

3

U3 −2

U3

3

U1

)2

+

(1

U1

3

U3 −1

U3

3

U1

)2

+

+

(1

U2

2

U3 −1

U3

2

U2

)2

+

(2

U2

3

U3 −2

U3

3

U2

)2

+

(1

U2

3

U3 −1

U3

3

U2

)2).

(84)

The explicit form of three nonholonomic constraints (73) notation looks as follows:

2

U ·3

U •−3

U ·2

U • =(

2

U1

3

U1•+

2

U2

3

U2•+

2

U3

3

U3•)−

(3

U1

2

U1•+

3

U2

2

U2•+

3

U3

2

U3•)

= 0,

3

U ·1

U •−1

U ·3

U • =(

3

U1

1

U1•+

3

U2

1

U2•+

3

U3

1

U3•)−

(1

U1

3

U1•+

1

U2

3

U2•+

1

U3

3

U3•)

= 0,

1

U ·2

U •−2

U ·1

U • =(

1

U1

2

U1•+

1

U2

2

U2•+

1

U3

2

U3•)−

(2

U1

1

U1•+

2

U2

1

U2•+

2

U3

1

U3•)

= 0.

(85)

The nonholonomic constraints (85) are the restrictions imposed on derivativesa

U • at theinitial time. If they are satisfied at the initial time, they are always satisfied under the

equations of motion (82).

It is easy to specify two trivial methods of meeting the constraint conditions (85) at the

initial time.

1) Zero initial conditions for all potentials area

U:a

U (0) at the arbitrary initial conditions

for derivativesa

U •(0)25. Let us name this method ZIPO (”zero-initial-potentials”).

2) Zero initial conditions for all derivatives of potentials area

U •(0) = 0 at the arbitrary

initial conditions for the potentials26. Let us name this method ZIDE (”zero-initial-

derivatives”).

In general case it is possible to cope with a problem of conditions (85) as follows:

(1) To set arbitrarily the trial initial values for all nine component of the three vectors

of YM-potential triplet ∗a

Ui (0),(a = 1, 3; i = 1, 3

).

(2) To set arbitrarily the trial initial values of six ”non-diagonal” components of the

derivatives from time potentials ∗a

Ui• (a �= i).

(3) To solve the system of equations (85) as a system of the three linear equations

relative to the three ”missing” initial values of the derivatives from time potentials

25Excepting the requirement of unit normalization by energy tymos.26The proviso, restricting this arbitrary rule, is the same as for ZIPO.


∗ a

Ui•(0) (a = i).

It is easy to demonstrate that the condition for this linear system solvability is the

following condition:

Δ =1

U2

2

U3

3

U1 −2

U1

3

U2

1

U2 �= 0,

imposed on the ”trial” initial values of potentials ∗a

Ui.

(4) To calculate the ”trial” energy value ∗E through formulas (84) for kinetic energy K

and potential energy U. This value will be arbitrary, and will not be equal to unit.

(5) To set the ”correct” initial conditionsa

Ui (0) anda

Ui•(0) by means of scale transfor-

mation of the ”trial” initial conditions:

a

Ui (0) =∗ a

Ui (0) · (∗E)−14 ,

a

Ui•(0) =∗

a

Ui•(0) · (∗E)−

12 .

With these ”correct” initial conditions the equations of nonholonomic constraints (85)

will be satisfied and tymos energy will be normalized to a unit.

The positive definiteness of diagonal elements of matrixab

I (83) and the positive definite-

ness of potential energy U (84) allow to claim that the solution to a nonlinear system

of the ordinary differential equations of the 18th order (82), describing tymos dynamics,

have oscillating character. Numerical solutions of the system (82) which will be presented

in one of the subsequent articles of this series show that these ostsillations have a diffi-

cult, almost unpredictable character. Dependences of potential componentsa

Ui on time

are subjected to a ”butterfly effect”. All reliable data on tymos dynamics is concentrated

in statistical characteristics of these dependences. The conservative system (82) does not

have attractors. The solution trajectory gradually fills some area in the nine-dimensional

configurational spacea

Ui. The density of this filling is an important characteristic of ty-

mos dynamics. It is convenient to use energy integral (84) and nonholonomic constraints

(85) as a prover of accuracy and stability of calculations in numerical solution of the

differential equation system (82).

9.2.7 Chirality problem

Yang-Mills-Euler equations (82) are chiral-symmetric, i.e. they do not distinguish the

right-hand and the left-hand. Their solutions are also chiral-symmetric. But the original

variables of the free triplet field Lagrangian (50) have no chiral symmetry. This problem

has been previously mentioned, in p.4.1.

The very fact of the presence of Yang-Mills’ vector product in the triplet theory means

that we have no right to renumber the vectors in YM-triple arbitrarily: this triple must

have a fixed orientation. In particular, for the problem on the plane wave, considered here,

it means that orientation of the triple of three-dimensional vectorsa

U must coincide with

the chosen orientation of the three-dimensional spatial coordinate system (”the principle

of chiral consistency”). It is convenient to watch the triple orientationa

U by the sign of


the mixed scalar-vector product of three vectorsa

U. Let us call this product a ”chiral

determinant” CD:

CD

(1

U,2

U,3

U

)=

(1

U ×2

U

)·

3

U=1

U ·(

2

U ×3

U

). (86)

The chirality determinacy condition, or to be short, ”chirality condition” looks as follows:

CD

(1

U,2

U,3

U

)≥ 0. (87)

The value of CD (86) is not Lorentz-invariant, since the procedure of separation of the

components of 4-potential vectora

Aμ into time and space parts is not Lorentz-invariant.

But CD sign (and it is only this sign that is important in chiral condition (87)) is Lorentz-

invariant.

We can control the sign of CD, calculating CD in any Lorentz system, for example, in

the intrinsic frame of the triplet wave.

The initial conditions for the system (82) must satisfy the condition (87). In addition, the

condition (87) should be satisfied at any time. However, the continuous smooth solutions

of Yang-Mills-Euler equations (82) do not obey to the chiral condition (87): the value of

CD oscillates in time, changing the sign. It is possible to satisfy the condition (87) only

by sacrificing the continuity or smoothness of the solutions.

The method, based on the loss of continuity of the solution, consists of permutation of

vectors1

U and2

U when CD = 0. Component3

U should not be affected by this permutation

on the following reasons. The third YM-component of YM-current is coupled with the

singlet current. In other words, the third component of the YM-triplet vector is initially

distinguished and ”unequal” with the components number 1 and number 2. Components

1 and 2 are equal in the condition of inner normalization of the triplet of currents (32) and

differ only in the condition of chiral determinacy (87). When this condition is violated,

these vectors should be just rearranged to satisfy the condition again. Let us name this

rearrangement ”the hard chiralization of solution”.

Dwelling on the reasonableness of the ”hard chiralization” in the framework of classical

(non-quantum) physics would be as useless as dwelling on the reasonableness of Pauli

Exclusion Principle in non-relativistic quantum mechanics: for non-relativistic theory,

Pauli principle is a phenomenological principle which is a part of the theory, but has no

justification within the theory. The procedure of ”hard chiralization” within the frame-

work of non-quantum theory should also be taken as a phenomenological rule. ”Hard

chiralization” generates discontinuous solutions, which are as chaotic as continuous and

smooth chiralo-symmetric solutions of the equations (82). Actually, these are the same

chiral -symmetric solutions, but they have been re-interpreted by the sporadic permuta-

tions 1↔ 2 in YM-indices.

However, this ”hard chiralization”, a shocking but a valid method to provide chiral deter-

minacy of the solutions, acceptable in the triplet sector of physics, is inadmissible for the

octuplet sector. In order to ensure the chiral definiteness in the group of eight Yang-Mills


vectors of the octuplet wave (such wave is a classical model of a quantum object, gluon),

it is necessary to provide a vector permutation in several triples of vectors simultaneously

(the structure of these triples is no more determined by Levi-Civita symbol, but by the

structure constants of the group SU(3),specifying the analogue of vector product in the

eight-dimensional YM-space), which is found to be impossible.

Therefore, let us turn to studying another method for providing chiral determinacy (87),

which is based on the use of continuous, but not smooth, solutions.

This method, which we will call ”the soft chiralization”, is generated by the possibility

of mechanical interpretation of the condition (87) as a one-sided, unilateral constraint,

imposed on the equations solution (82).

Such constraint, set by inequality, appears, for example, in the problem of the plane

motion of mathematical pendulum on nonstretchable line. If the line is stretched, the

point pendulum swings, moving along the circular arc of the radius equal to the line’s

length. But if the line is not stretched, the point pendulum, as a free material point,

moves along the arc of parabola in the homogeneous gravitational field. At the moments

when the line stretches, the radial velocity component of the pendulum reverses the sign

abruptly, the azimuthal velocity component remains the same. If there is no dissipation

in the line, such swings of the pendulum along the parabolic trajectories continue for

unlimitedly long time.

Based on this mechanical analogy, we will formulate the following method for ”chiralizat-

sion” of the solution: at the moments when CD

(1

U,2

U,3

U

)= 0, the signs of all velocity

vectorsa

U • are reversed.

This method of ”soft chiralization” generates continuous, but not smooth, chiral-determi-

ned solutions: the condition of the chiral determinacy is always satisfied; and at the mo-

ment of ”nulling” of the chiral CD determinant (86), the velocitiesa

U • discontinue.This method of ”soft chiralization” is literally transferred into the octuplet sector of

physics: at ”nulling” of at least one of the set of chiral determinants of the octuplet

sector, the signs of rates of change for all eight Yang-Mills octuplet vectors are reversed.

As it was previously noted in p. 4.1., in the problem of mathematical pendulum motion,

it is possible to get rid of the solutions with velocity discontinuity by introducing a more

complex model of constraint, for example, taking into account the resilient extensibility

of the fiber on which the pendulum is hung. There is no such possibility in the problems

of Yang-Mills physics: we have no more accurate physics beyond Yang-Mills physics.

We have to put up with the roughness of the chiral-determined YM-solutions. These

discontinuities in derivatives mean a sudden, instantaneous occurrence and vanishing of

YM-currents in a zero-current problem. The dependence of these currents on time is

described by δ-delta function.

In general, CD disappears when the three YM-vectorsa

U are in the same plane. This is

a ”reflection plane” – hitting against it, all three vectorsa

U rebound resiliently, providing

chiral definiteness of the solutions for the equations (82). Two of such fixed ”planes of re-

flection” appear at the arbitrary initial conditions, and the points representing the termini


of the three YM-vectorsa

U, make periodic motions between the two planes. Numerical

solutions of the equations (82), accounting the conditions of the chiral determinacy (87),

will be presented later in one of the articles of this series.

9.2.8 Yang-Mills Orthogonal Zero-current Triplet Oscillator

The problem of Yang-Mills plane wave takes especially simple and attractive form if we

assume thata

Ui= 0, with i �= a. (88)

The condition (88) means that all three Yang-Mills vectorsa

U are orthogonal to one

another and each of them has a stationary direction, which, according to (88), we will

identify with the direction of the corresponding coordinate axis of the three-dimensional

Euclidean space in the intrinsic wave system.

When the condition of orthogonality (88) is fulfilled, the equations of non-holonomic

constraints (85) are identically satisfied and YM-inertia tensor (83) becomes diagonal.

Taking into account the condition (88), we no longer need to use two kinds of indices –

the over-letter YM-indices, labeling the vectors of YM-triplet, and the subscripts labeling

the spatial Cartesian components of each YM-triplet vector.

In the problems of tymos dynamics, considering the orthogonality condition (88), there

remain only three non-trivial dynamic variables which have the same Yang-Mills and

spatial indices. In the process it is convenient to use one- index notation system, assuming

that

ui ≡a

Ui

(i = a = 1, 3

). (89)

In this recording Yang-Mills-Euler equations (82) take the following ”canonical” form:

u1 + u1

(u22 + u2

3

)= 0,

u2 + u2

(u23 + u2

1

)= 0,

u3 + u3

(u21 + u2

2

)= 0.

(90)

In such one-index notation, due to the absence of explicit YM-indices, it is proper to put

a dot which labels the intrinsic wave time derivative, over the corresponding letter.

Energy integral (84) for the system (90) also takes an elegant ”canonical” form:

1

2

((u1)

2 + (u2)2 + (u3)

2) + 1

2

(u21 u

22 + u2

2 u23 + u2

3 u21

)= 1. (91)

Let us name the three-dimensional system of motion (90) with energy integral (91) a

”Yang-Mills orthogonal triplet oscillator”, denoting it with ”⊥ −tymos”. With certain

conditionality we can treat the three one-index values (89) as three Cartesian components

of one vector depicting tymos in a three-dimensional configuration space, which allows

us to treat ⊥ −tymos as a ”one-vector tymos” (1-vector-tymos).

Assuming that in (90) one of the components, for example u3, is identically zero, we get

even simpler problem of two-dimensional tymos:

u1 + u1u22 = 0,

u2 + u2u21 = 0,

(92)


with the energy integral:

1

2

((u1)

2 + (u2)2) + 1

2u21 u

22 = 1. (93)

Let us call the system of motion (92) with energy integral (93) a ”doubletic Yang-Mills’

oscillator”, dymos .

The essential nonlinearity of the equations (90) or (92), as well as of the whole Yang-Mills

wave theory should be noted. Such problems in principle do not allow any interpretation

within the framework of the perturbation theory. The linearization of these problems

by neglecting nonlinear terms in (90) or (92) leads to such significant degeneracy of the

problem that it can not be corrected by constructing a chain of successive approximations.

9.2.9 Yang-Mills Plane Zero-current Wave with Space-like Wave Vector

There are no a priori reasons that allow to regard Yang-Mills waves only with time-like

wave vector.

Let us make the transformations, made previously in pp. 9.2.2. – 9.2.6. and p. 9.2.8.

again, now supposing that the wave vector of the plane zero-current wave kμ is space-like:

kμkμ = −ε2 < 0.

After performing the scale transformation (p. 9.2.2) which normalizes the wave vector

per unit, and the subsequent scale transformation (p. 9.2.3) which normalizes the wave

energy per unit27, assuming that there is not any stationary field external to the wave,

we can reduce the problem of Yang-Mills plane zero-current triplet wave with a space-like

wave vector kμ to the following form:

a

A ν ′′−ab

Ib

A ν = 0,

abcε

b

Aμc

A′μ= 0,

kμa

Aμ= 0,

kμ kμ = −1.

(94)


I is determined by formulas (75) and (77).

The energy integral for system of motion (94) differs from formula (61) which is fair for

the wave with time-like wave vector kμ, in ”the wrong” sign in ”kinetic” energy K and in

the absence of energy E sign-definiteness:

1

2

(a

A′ν ·a

A ν ′)+

1

4

((a

Aν ·a

A ν

)2

−(

a

Aμ ·a

Aν

) (b

Aμ·b

A ν

))= signE = ± 1.28 (95)

27 In this transformation, in contrast to formulas (62) which are fair for the wave with a time-like wave

vector kμ and sign-definite energy E, it is not energy E that appears, but its module | E |, since energy

E loses its sign-definiteness for the wave with space-like wave vector kμ. A particular case E = 0 must

be considered separately.28The wave under consideration also can have zero energy.


In relations (94) and (28) the sign ′ on the right side from potentiala

A ν means the wave

phase φ derivative.

Dynamical system (94) with ”the wrong” sign does not have any oscillate solutions, in

contrast to the system of motion (79).

9.2.10 Yang-Mills Plane Zero-current Wave with the Space-like Wave Vector in theIntrinsic Frame of Reference

Let us call the frame of reference, in which a space-like wave vector, normalized per unit,

has the following form, the intrinsic frame of reference:

kμ = {0; 1; 0; 0}.

In this frame of reference, axis x is directed along the wave vector, and the wave phase

derivative coincides with the arbitrary coordinate x derivative. The wave in this frame

of reference is static, its characteristics do not depend on time.

Longitudinal components of 4-potential triplet in this frame of reference are equal to zero.

The notation of the potentials in the form ”1+3” (70) looks as follows:

a

A ν = {a

T; 0;au2;

au3}.

Substitution of this formula into the equations of motion (94) results in the following nine-

component dynamical problem relative to the nine nontrivial components of Yang-Mills

triplet:

−1

T ′′+11

I1

T +12

I2

T +13

I3

T= 0,

− 1u2′′+

11

I1u2 +

12

I2u2 +

13

I3u2= 0,

− 1u3′′+

11

I1u3 +

12

I2u3 +

13

I3u3= 0,

−2

T ′′+21

I1

T +22

I2

T +23

I3

T= 0,

− 2u2′′+

21

I1u2 +

22

I2u2 +

23

I3u2= 0,

− 2u3′′+

21

I1u3 +

22

I2u3 +

23

I3u3= 0,

−3

T ′′+31

I1

T +32

I2

T +33

I3

T= 0,

− 3u2′′+

31

I1u2 +

32

I2u2 +

33

I3u2= 0,

− 3u3′′+

31

I1u3 +

32

I2u3 +

33

I3u3= 0.

(96)


The components of Yang-Mills inertia tensorab

I in (96) have the following form:

11

I =(

2u2

)2

+(

2u3

)2

+(

3u2

)2

+(

3u3

)2

−(

2

T

)2

−(

3

T

)2

;

22

I =(

3u2

)2

+(

3u3

)2

+(

1u2

)2

+(

1u3

)2

−(

3

T

)2

−(

1

T

)2

;

33

I =(

1u2

)2

+(

1u3

)2

+(

2u2

)2

+(

2u3

)2

−(

1

T

)2

−(

2

T

)2

;

12

I =21

I =1

T2

T −(

1u2

2u2 +

1u3

2u3

);

23

I =32

I =2

T3

T −(

2u2

3u2 +

2u3

3u3

);

31

I =13

I =3

T1

T −(

3u2

1u2 +

3u3

1u3

).

(97)

The explicit notation of non-holonomic constraints (67), (68), imposed on the solution of

the system (96), has the following form:(2

T3

T ′−3

T2

T ′)−

(2u2

3u ′2+

2u3

3u ′3

)−

(3u2

2u ′2+

3u3

2u ′3

)= 0,(

3

T1

T ′−1

T3

T ′)−

(3u2

1u ′2+

3u3

1u ′3

)−

(1u2

3u ′2+

1u3

3u ′3

)= 0,(

1

T2

T ′−2

T1

T ′)−

(1u2

2u ′2+

1u3

2u ′3

)−

(2u2

1u ′2+

2u3

1u ′3

)= 0.

(98)

Non-holonomic constraints (98) are actually imposed only on the initial values of the

variables with some arbitrary initial value of the longitudinal coordinate x. If these con-

ditions are fulfilled at one value of x, they will be fulfilled at all values of x on the ground

of the equations (96).

A trivial method of accounting the conditions (98) consists of setting zero initial condi-

tions for all the nine non-trivial components of triplet of potential or for all of the nine

potential derivatives along the longitudinal coordinate x. A less trivial method is similar

to the one described above in p 9.2.6 for a wave with a time-like wave vector. It has

to solve the constraints, linear by derivatives, relative to any three initial derivatives,

considering the other six trial initial values of the derivatives and all the nine trial initial

components of the potentials. For example, so that the three equations (98) were solv-

able relative to the initial values1

T ′,2u ′2,

3u ′3, the corresponding determinant Δ must be

nonzero:

Δ =1u2

2u3

3

T − 1u3

3u2

2

T�= 0.

This condition must be fulfilled for the trial initial values of the potential components.

Then, it is necessary to calculate the trial wave energy E by formula (28) and to rescale

the initial values of the potentials and their derivatives in such a way that the wave energy

was equal to +1 (if E > 0) or 1 (if E < 0). In the process, it is convenient to use an


explicit, but lengthy, expression for energy in the notation ”1+3” (70):

E = K+ U = ±1,

K =1

2

((1

T ′)2

+

(2

T ′)2

+

(3

T ′)2

−

−((

1

u′2

)2

+

(1

u′3

)2

+

(2

u′2

)2

+

(2

u′3

)2

+

(3

u′2

)2

+

(3

u′3

)2))

,

U =1

2

(((1u2

2u3 −

1u3

2u2

)2

+(

1u2

3u3 −

1u3

3u2

)2

+(

2u2

3u3 −

2u3

3u2

)2)−

−((

1

T2u2 −

2

T1u2

)2

+

(1

T2u3 −

2

T1u3

)2

+

(1

T3u2 −

3

T1u2

)2

+

+

(1

T3u3 −

3

T1u3

)2

+

(2

T3u2 −

3

T2u2

)2

+

(2

T3u3 −

3

T2u3

)2))

.

(99)

The introduction of Yang-Mills triple of complex three-dimensional vectorsaτ :

aτ= {

a

T; iau2; i

au3}− (100)

allows to enter the energy expression in a more compact and visible form:

K =1

2

((1

τ ′)2

+

(2

τ ′)2

+

(3

τ ′)2

);

U =1

2

((1τ × 2

τ)2

+(

2τ × 3

τ)2

+(

3τ × 1

τ)2

).

The simplicity of these formulas, in the language of tripleaτ , allows to assume that chiral

determinant CD for this wave can also be entered in the form of (86) with the replacementa

U→ aτ :

CD(

1τ ,

2τ ,

3τ)=

(1τ × 2

τ)· 3τ=

1τ ·

(2τ × 3

τ)≥ 0. (101)

However, we cannot advance any argument in favor of this formula29.

As a matter of fact, the nine-component object under consideration, the Yang-Mills plane

zero-current triplet wave with space-like wave vector, should not be called a ”wave” at all,

since the equations (96) have no continuous, smooth, oscillatory solutions; their solutions

grow progressively along the longitudinal coordinate x. Oscillations (i.e. wave ridges and

decreases) may appear with the use of the above-mentioned (p. 9.2.7.) procedure of ”soft

chiralization” with the use of chiral determinant (101) which is denoted in the terms of

29The only argument that can be advanced consists of the naturalness of the appearance of vectorsaτ and

expression (100) for CD within Yang-Mills mathematical apparatus. The arguments of mathematical

naturalness and simplicity are quite traditional for mathematical physics, for example, while constructing

the Lagrangians. However, it is unknown whether such arguments always lead to a true physical theory:

perhaps, God does not use any mathematics at all.


complex Yang-Mills vectorsaτ .

The most shocking fact in description of this nine-component ”wave” is the possibility of

the existence of an object with negative energy. The world-view of any physicist-theorist

of post-Einstein epoch is hardly able to cope with the existence of such object. We tend

to cautiously assume that such object cannot exist as an independent wave, but it can

appear in a couple with some other object that has an excess of positive energy in the

process of some third object’s decay. Probably, the appearance of objects with negative

energy in the apparatus of classical field theory is not a more shocking fact than the

appearance in modern quantum theory of virtual particles which temporarily violate the

energy conservation law: both the first and the second ”strangeness” may finally describe

the same circle of phenomena.

We shall call this ”strange” nine-component wave with a Latin term ”terriculum” (a

scarecrow, a fright)30.

9.2.11 Yang-Mills Orthogonal Terriculum

Instead of a lengthy nine-component Yang-Mills terriculum, described by the system of

equations (96), we can consider a three-component object, which is simpler for analysis:

an orthogonal terriculum. Let us suppose that vectorsaτ in the triple (100) are orthogonal

to each other and directed along the corresponding Cartesian axes. In other words, only

three of nine components of terriculum potential,1

T,2

U2,3

U3, are nonzero components

(further in this p. 9.2.11, to make the notation simple, the over-letter YM-indices for

designating three of these values, can be omitted). With such choice of tripleaτ , the non-

holonomic constraints (98) are identically satisfied, and the system of motion equations

(96) takes quite a simple form:

T′′ − T(U2

2 +U23

)= 0,

U′′2 +U2

(T2 − U2

3

)= 0,

U′′3 +U3

(T2 − U2

2

)= 0.

(102)

Energy integral (99) for the system (102) takes a more visible form:

K+ U = ±1,

K =1

2

((T′)2 − (U2)

2 − (U3)2),

U =1

2

(U2

2U23 − T2

((U2)

2 + (U3)2)) .

(103)

The condition of chiral definiteness (101) for system (102) takes the following form:

TU2U3 ≤ 0. (104)

30 If terriculum had been discovered by the theorists of 1950s, it would have, undoubtedly, undermined

the physicists’ interest for Yang-Mills fields. Six decades later, when no one doubts the reality of Yang-

Mills fields, the detailed numerical research of terriculum, with the condition of chiralization (101) would

be useful and would hardly be able to undermine the trust to Yang-Mills theory.


Without considering the condition (104), chiral-symmetrical solutions (102) (continuous

and smooth) have, as it is easy to see in the form (102), a simple, but unacceptable

behavior with x → ∞: an unbounded growth of T at more frequent oscillations of U2

and U3. The procedure of ”soft chiralization” similar to the one described in p. 9.2.7,

interrupts the unbounded growth of the variable T and generates oscillatory solutions for

the system (102) with derivative discontinuity.

The procedure of ”soft chiralization” for solution of the system (102) is the following:

for those quantities of the longitudinal coordinate x, under which at least one of the

three decision functions T, U2, U3 vanishes, (and, correspondingly, by (104), the chiral

determinant vanishes) the signs of the derivative from all the three functions are reversed:

T′ → −T′, U′2 → −U′2, U′3 → −U′3.

9.2.12 Yang-Mills Dilemma

The procedure of ”soft chiralization” is able to interrupt the unbounded growth of po-

tential components in the terriculum wave, and, thereby, to make this object more ac-

ceptable for analysis in theoretical physics. However, the chiralization procedure itself is

introduced into theoretical physics for the first time, and, probably, it does not seem rea-

sonably sufficient to the reader (within the framework of classical theory, its sufficiency

is probably impossible). But if the chiralization procedure is not used, relative to the

terriculum waves, we discover two possible modes of behavior:

(1) To reject the possibility of terriculum wave existence itself as a physically unac-

cepted object; to accept that Fourier decomposition of Yang-Mills potentials must

be performed not over the whole four-dimensional space of the wave vectors, but

only within the boundaries of 4-cone kμkμ > 0.

(2) To put up with the fact of terriculum wave existence, supposing that the problem

of terriculum ”wrong behavior” on the space infinity can be removed by instability

of Yang-Mills free zero-current waves in the cone kμkμ < 0, relative to the build-up

of the current states (one-current, two-current, three-current, four-current) and the

subsequent production from them of discrete particles with pomerium.

Let us call the choice of one of these lines a ”Yang-Mills dilemma”.

We can cite a hypothetical ”Dirac’s dilemma” as a historically remote analogue: after the

discovery of solutions with ”the wrong” negative energy in Dirac’s equation, these solu-

tions could be neglected as ”non-physical” (within the framework of the experimentalists’

understanding of physics in 1928), and, by doing so, the positron, still not discovered by

those experimentalists, could be ”missed”; or it could be possible to put up with such

solutions and to look for an opportunity to ”build” them into the common picture of

physical reality.

It is not ruled out that it is the peculiarities of the ”monstrous” behavior of the terricu-

lum that have something to do with the processes in which a ”weak” triplet interaction

”demolishes” heavy leptons and quarks, which per se would be stable in ”maxwellian-

and-chromodynamical world”, the world without triplet interactions.


9.2.13 Yang-Mills Photon: Plane Zero-current Triplet Wavewith Isotropic Wave Vector

Let us have a look at the plane triplet wave with isotropic wave vector kν :

kνkν = 0.

Wave vector isotropy means that the phase velocity of a wave is equal to the velocity

of light. We shall call this wave (to be precise, the quantum object, hypothetically

corresponding to it) ”The Yang-Mills’ photon” (ymiton).

Ymiton is a degenerate object of Yang-Mills mathematical wave theory. The condition of

a wave vector isotropy removes higher derivatives from the wave equation (55), turning

it into the first-order equation:

2kμAμ ×Aν ′ − kνAμ ×Aμ′ +Aμ × (Aμ ×Aν) = 0. (105)

The gauge condition (56), imposed on 4-potential divergence, means that

kμAμ = q = const = 0. (106)

As in other wave problems considered above, we suppose that vector q (106) is vanishing.

Availability of nonzero q would correspond to the wave propagation on the background

of some external stationary field.

With regard to (106), the first term in the wave equation (105) vanishes, and the wave

equation takes the form:abcε kν

b

Aμ

c

Aμ′+ab

Ib

A ν = 0. (107)

whereab

I is Yang-Mills inertia tensor, introduced above.

The twelve equations (107) are not independent from each other. The scalar multipli-

cation (107) by the wave vector kν gives three identities of the form 0 = 0. The scalar

multiplication (107) by the potential Aν generates the algebraic identity, which should

be followed by the solution (107):

U =ab

Sab

S −(

bb

S

)2

= 0. (108)

The expression (108) is a certain ”pale shadow” of the energy conservation law for ymiton.

For the degenerate wave (107) it is impossible to introduce anything resembling the kinetic

energy K. The analogue of potential energy, determined by formula (108), is identically

zero.

The system, in which

kν = {1; 1; 0; 0},will be named the ymiton ”intrinsic frame of reference”.

This is a ”frame of unit frequency”, in which axis x is directed along the wave direction.

In this frame φ = t− x. Using the notation of the form ”1+3” for potentiala

A ν again:

a

A ν = {a

T;a

U},


we can reduce the ymiton wave problem to the following form:

a

T=a

U1; (109)

ab

Ib

U2= 0; (110)

ab

Ib

U3= 0; (111)

ab

Ib

U1=ag . (112)

The equations (109) allow to exclude time componentsa

T of all the three potentials of

YM -triplet from consideration.

YM -vector g, entering into (112), similarly to the components of YM -inertia tensorab

I ,

is expressed by spatial triplet componentsa

Ui

(i = 2, 3

):

1g=

3

U2

2

U2′+

3

U3

2

U3′ −

(2

U2

3

U2′+

2

U3

3

U3′);

2g=

1

U2

3

U2′+

1

U3

3

U3′ −

(3

U2

1

U2′+

3

U3

1

U3′);

3g=

2

U2

1

U2′+

2

U3

1

U3′ −

(1

U2

2

U2′+

1

U3

2

U3′).


I for the ymiton wave problem looks as follows:

11

I =

(2

U2

)2

+

(2

U3

)2

+

(3

U2

)2

+

(3

U3

)2

;

22

I =

(3

U2

)2

+

(3

U3

)2

+

(1

U2

)2

+

(1

U3

)2

;

33

I =

(1

U2

)2

+

(1

U3

)2

+

(2

U2

)2

+

(2

U3

)2

;

12

I =21

I = −(

1

U2

2

U2 +1

U3

2

U3

);

13

I =31

I = −(

1

U2

3

U2 +1

U3

3

U3

);

23

I =32

I = −(

2

U2

3

U2 +2

U3

3

U3

).

The expression for ”potential energy” U (108) is reduced to the form which contains the

sum of three squares:

U = p21 + p2

2 + p23,

where

p1 =1

U2

2

U3 −2

U2

1

U3;

p2 =1

U2

3

U3 −1

U3

3

U2;

p3 =2

U2

3

U3 −2

U3

3

U2 .


The equality U = 0 forms three algebraic conditions, imposed on the spatial components

of the potential:p1 = 0;

p2 = 0;

p3 = 0.

(113)

The relations, provided here, allow formulating the following algorithm of the construction

of ymiton wave parameters:

(1) The equations (110) and (113) give, with regard to the expressions forab

I and pi, the

system of six algebraic equations for determination of ”six-vector”a

Ui, (a = 1,2,3; i = 2,3)

three homogeneous cubic equations (110) and three homogeneous equations of the

second-order (113). Let us assume that there is at least one solution to this algebraic

system. Apparently, this solution will also satisfy the equations (111) which result

from (110) by substitution of the Cartesian subscripts 2↔ 3 in the equations (110)

and in tensorab

I which is invariant relative to such substitution.

Due to homogeneity of the equations (110) and (113), ifa

Ui=aκi is some solution to

these equations, the expression

a

Ui= f(φ)aκi (a = 1,2,3; i = 2,3) .

is also a solution to this system under the arbitrary function f(φ), which depends

on the wave phase, but does not depend on Yang-Mills and Cartesian indices. It

goes without saying that it is worth speaking about a ”wave” only in case if f(φ) is

a bounded oscillatory function.

(2) By substituting the relation fora

Ui into the expressions which determine YM-vector g

and matrixab

I , we solve the equations (112) as a system of three linear heterogeneous

algebraic equations relative to the longitudinal components of potentialsa

U1.

It is easy to see that the components of the inertia tensorab

I are proportional to the

square of arbitrary function f(φ), and the components of vector g are proportional to

the product of f(φ) · f′(φ). Therefore, the dependence of the longitudinal components

of the potential on the wave phase φ has the following form:

a

U1≈f′

f,

which inevitably generates singularities in the longitudinal componentsa

U1 with the

continuous function f(φ) oscillating around zero.

The condition of chiral determinacy for this wave, in forms (86) and (87), imposes

restriction on the behavior of f(φ).

This restriction requires a constant sign of product f · f′, which, actually, requiresdiscontinuous oscillations of function f(φ).

On the whole, the provided description of ”Yang-Mills photon” looks rather confusing.

Unlike tymos wave (kνkν > 0) and terriculum wave (kνkν < 0), isotropic YM-ymiton


wave is an ”algebraic object” which does not require differential wave equations for its

description.

9.2.14 About Free Zero-current Triplet Field Statistics

In the preceding parts of this article (9.2.10-9.2.13) some certain difficulties in the descrip-

tion of Yang-Mills waves in the wave cone kνkν ≤ 0 were mentioned. These difficulties

include unbounded growth of potentials in the terriculum wave – if the procedure of chi-

ralization is not used, and the discontinuous behavior of waves ymiton. These difficulties

may indicate that the problem of describing the free wave state of Yang-Mills triplet is

not a correctly-set problem in theoretical physics. The ”free field” can probably turn into

a ”non-free” field in space-like cone kνkν < 0 of the four-dimensional space of the wave

vectors, generating YM -currents. In the linear singlet sector of physics there is nothing

similar to this phenomenon.

But, at the very least, the description of the free triplet waves in time-like cone kνkν > 0

is quite correct mathematically - no matter whether we use the chaotic chiral-symmetric

solutions of the wave equations, or impose chiral determinacy to a solution by using the

procedure of ”soft chiralization”.

Consequently, we can formulate a classical problem of the statistical characteristics of

the free triplet field in thermodynamic equilibrium - the problem, similar to the one

that Max Planck successfully solved in 1900 for maxwellian singlet by constructing the

frequency energy distribution function. Unlike the one-dimensional Planck distribution,

the corresponding Yang-Mills distribution must be two-dimensional, due to the absence

of constraint between frequency ω and three-dimensional wave vector |k| in the triplet

sector.

It can be supposed that the study of the thermodynamics of the triplet field is an un-

conventional problem. The language of Fourier expansion of the triplet field into plane

waves may not appear to be very convenient language for the nonlinear theory.

”Photon” in the electromagnetic theory retains its identity from one process of inter-

action with matter (emission or scattering) to the other one (absorption or scattering).

Due to the lack of dispersion, wave packets in the electromagnetic theory spread with-

out changing the profile, while retaining its identity. All these identities make Fourier

language useful for the singlet sector of physics. All these identities are not retained in

Yang-Mills nonlinear theory of the triplet waves. Yang-Mills wave packets spread and

interact with each other, changing the profiles and absorbing other packets. Perhaps, the

thermodynamics of Yang-Mills fields requires some other variables for its description.

9.3 One-current Triplet States

9.3.1 Classification and Lagrangians of One-current Triplet States

According to the scheme of the pure states, demonstrated in figure 1, and the structure

of the weak Lagrangian Lw (49), we can describe all the possible pure triplet one-current

states in the following way.


1. There are two similar ”heavy” current states with a space-like current vector. The

Lagrangian of these states, according to (49), can be formulated in the following way:

Lw =1

2jν jν − jν

k

Aν −1

16πAμν ·Aμν , (114)

where jν = 12

k

j ν ; k = 1 or k = 2.

In these states only one of the three YM-currents with YM-number equal to 1 or 2, is

nonzero.

There are three types of states, described by the Lagrangian (114):

• ”Heavy” wave states, in which the current vector is nonzero all over the 4-dimensio-

nal space (there is no one pomerium).

• Stationary states, in which there is one stationary pomerium in a certain intrinsic

system of states the current zone boundaries in which current isotropization jν

(jν jν = 0) takes place; there is no current beyond this boundary; inside the current

zone there are cavitated zones (latebrae) inside which there is no current; on the

boundaries of these cavitated tubes, the pseudo-Euclidean square of 4-current has

the maximum possible value (7).

While solving the problems of stationary one-current states, it is worth taking into

account the curved space-time, generated by a high density of energy-momentum

inside the current zone and around it. As well as quarks, the stationary one-current

states of the triplet sector (in contrast to leptons – the stationary states of the singlet

sector) may turn out to be physically unrealizable due to the ”wrong behavior” of

stationary potential triplet Aν away from the current zone, resulting in divergence

of the total energy of the stationary state.

• Non-stationary states, or the ”interaction states”, in which there is one, or more

than one, non-stationary boundary of the current zones31.

Varying the action functional with the Lagrangian (114) by the triple of potentials Aν

and by current jν , we obtain the following field equations for a ”heavy” one-current triplet

state:

∂μAμν +Aμ ×Aμν = 4π jν

ke, (115)

jν+k

A ν = 0. (116)

In Yang-Mills equations (115)ke is a Yang-Mills unit vector of k YM-direction, i.e. YM-

vector with YM-components {1; 0; 0} for k = 1 and {0; 1; 0} for k = 2.

The second field equation, connecting the current and potential, is true in the current

zone, where j �= 0.

One gauge condition can be imposed on the solution of Yang-Mills equations. As in the

problem on Yang-Mills free fields, it is convenient to apply potential 4-divergence gauge:

∂νAν = 0. (117)

31Currently having no numerical solutions to stationary problems and having no version of the existence

theorem which would guarantee the existence of solutions to stationary problems, we are not going to

deal with formulation of non-stationary problems in this article.


The differential condition for triplet currents (19), which, by means of the symbols applied

here, can be written in the form:

ke ∂νj

ν + jνAν×ke ν = 0, (118)

gives the following relation in YM-projections:

∂νjν = 0, (119)

3

Aν jν = 0,kAν jν = 0, (120)

where � k = 1 with k = 2; � k = 2 with k = 1.

The condition of current conservation jν (119) is compatible with gauge (117) and the

field equation (116).

The orthogonality conditions (120) are the result of differential condition for currents

(118), which, in its turn, is the result of the field equations (115) and (116). Therefore,

the orthogonality conditions (120) should not be included (together with undetermined

Lagrange’s multipliers) into the Lagrangian expansion of one-current problem (114).

The following dilemma can be formulated: either there are solutions to field equations

(115), (116), (117), (119), compatible with the orthogonality conditions (120) at zero

Lagrange’s multipliers, or one-current triplet states should not be treated as real physical

states at all – they exist only on the background of some external non-physical ”Higgs-

like” field which is mathematically reflected in the Lagrange’s multipliers for the extended

Lagrangian.

Getting rid of the weak field tensor Aμν in the equations (115) and considering the gauge

(117), we can formulate the Yang-Mills equations of one-current problem in the following

way:

−�Aν + 2Aμ × ∂μAν −Aμ × ∂νAμ + IAν = 4π jνke, (121)

where YM-inertia tensor I, according to (75) and (77) has the following YM-components

ab

I=a

Aμb

Aμ −ab

δc

Aμ

c

Aμ, (122)

2. Besides the two ”heavy” one-current triplet states, there are three similar neutrino

states, with isotropic current vector Nν (NνNν = 0). The Lagrangian of these neutrino

YM-states, which were named ymino above, looks as follows:

Lw,N = −Nνk

Aν −1

16πAμν ·Aμν −

1

2λN νNν , (123)

(k = 1, 2, 3).

The last term in (123) is the ”penalty” for current isotropy condition, λ is a Lagrange’s

multiplier. The current N ν in (123) corresponds to a nonzero k – YM-component of

YM-current triplet. With k = 1 or k = 2 along with the neutrino states (123), there are

current states with the Lagrangian (114). If k = 3, there is only neutrino state with the

Lagrangian (123).


The field equations, following from the Lagrangian (123), look similar to the equations

(115), (116) and (121):

∂μAμν +Aμ ×Aμν = 4π N ν k

e, (124)

−�Aν + 2Aμ × ∂μAν −Aμ × ∂νAμ + IAν = 4π N ν ke, (125)

λN ν+k

A ν = 0, (126)

∂νAν = 0, (127)

∂νNν = 0, (128)

NνkAν= 0, (129)

NνNν = 0. (130)

In orthogonality conditions (129) � k = 2 and 3 with k = 1; � k = 3 and 1 with k = 2;

� k = 1 and 2 with k = 3.

From field equation (126) and the condition of neutrino current isotropy (130) there

follows the orthogonality condition

Nνk

Aν= 0. (131)

Condition (131), combined with condition (129), means that neutrino current Nν must

be orthogonal to all the three triplet potentials. Besides this, as it follows from (126),

potentialk

A ν is an isotropic vector.

Differentiating the field equation (126) by coordinates xν , and taking into account (127)

and (128), we find that 4-gradient of the Lagrange’s multiplier λ satisfies the orthogonality

condition:

N ν∂νλ = 0. (132)

This condition will be anyway satisfied, if λ = const.

9.3.2 ”Heavy” One-current Plane Triplet Wave

We will try the solution to Yang-Mills wave equations (121) in the form of a plane wave,

supposing that current jν and potential Aν depend on the only argument – the wave

phase φ = kμxμ. The wave equation (121) for a plane wave gets the following form

kμkμAν ′′ + 2kμAμ ×Aν ′ − kνAμ ×Aμ ′ + IAν + 4π

k

A ν ke= 0. (133)

In equation (133) the prime denotes a wave phase derivative. In the last term, field

equation (116) is taken into account. The summation by ”stationary” index k is not

made in this term.

If homogeneous equations (133) are solved, the current is calculated by field equation

(116).

Gauge condition (117) in the plane wave is reduced to the condition:

kνAν ′ = 0,


with integral:

kνAν = q = const,

but, as in the other wave problems, we will assume that q = 0: nonzeroness of q would

correspond to a wave propagation on the background of a certain external stationary

field. Considering zero value of q, the second term in the equation (133) vanishes:

kμkμAν ′′ − kνAμ ×Aμ ′ + IAν + 4π

k

A ν ke= 0. (134)

By scalar multiplying the plane one-current wave equation (134) by wave vector kν , we

find with anisotropic vector kν (kνkν �= 0) that:

Aμ ×Aμ ′ = 0. (135)

As in Yang-Mills free wave problem, relation (135) can be considered a non-holonomic

constraint, imposed on the initial values of potentials and their derivatives. Considering

condition (135), the wave equation (134) takes even a simpler form:

kμkμAν ′′ + IAν + 4π

k

A ν ke= 0. (136)

Let ε be a non-Euclidean module of wave vector kν :

kνkν = ±ε2,

where a plus sign corresponds to the wave with a time-like wave vector, but a minus sign

corresponds to the wave with a space-like wave vector. Making a scale transformation in

(136),

kν → εkν ; φ→ εφ,

normalizing wave vector k for a unit module, we get rid of the module of wave vector ε

in equation (136):

±Aν •• + IAν + 4πk

A ν ke= 0. (137)

A plus sing in (137) corresponds to the wave with time-like wave vector kν , normalized

for unit:

kνkν = 1.

A minus sign in (137) corresponds to the wave with space-like wave vector kν , normalized

for unit:

kνkν = −1.

The dots on the right-side of the potential symbol imply the derivative with respect to ε

- rescaled wave phase. In the wave intrinsic frame of reference, wave vector kν has the

form:

kμ = {1; 0; 0; 0}−

for time-like vector kν and

kμ = {0; 1; 0; 0}−


for space-like vector kν .

In the wave intrinsic frame of reference the phase derivative coincides with the time

derivative (all wave characteristics in the intrinsic frame depend only on time) – for the

time-like wave vector, or with longitudinal coordinate x derivatives – for the wave with a

space-like wave vector (in such wave all the characteristics depend only on longitudinal

coordinate; in the intrinsic frame of reference such a wave is a kind of ripple of current

and potentials stiffed in space).

Multiplying one-current plane wave equation (137) by Aν • and integrating by the wave

phase, we can obtain the energy integral for this wave:

K+ U = E = const,

where

U =1

4

((aa

S

)2

−ab

Sab

S +8πk

A νk

Aν

)−

is a wave potential energy, andab

S=a

Aμb

Aμ .

In the expression for potential energy U, the summation is made by YM-repetitive indices

a, b. But the summation by the ”stationary” YM-index k is missing.

”Kinetic” energy K for the waves with space-like and time-like wave vector differs in sign:

K = Kt = −1

2Aν • ·A•

ν , (time-like wave),

K = Ks =1

2Aν • ·A•

ν , (space-like wave).

For time-like wave, energy E is positive, but for a space-like wave the energy sign E can

be either positive, or negative.

In the expression for potential energy U the first two terms have the fourth order by

potentialsa

Aμ, and the last term, generated by current availability in the wave, is quadratic

by potentials.

Availability of this term destroys the self-similarity of a plane wave relative to the energy

that is inherent to a zero-current wave. For a one-current wave, energy E is a significant

parameter (not just the scale one).

Equations (137) describe some dynamical system. By analogy with dynamical system

(79), which describes a zero-current wave, let us call dynamical system (137) with time-

like vector kν a ”1c-tymos” (one-current-tripletic-yang-mills’ oscillator). The choice of

a plus sign before the first term of the equation corresponds to 1-tymos object (137).

The choice of a minus sign before this term, corresponding to space-like wave-vector kν ,

generates a one-current analogue of the ”terriculum” object. Let us name this object a

1c-terriculum.

9.3.3 Detailed Notation of the 1-tymos Dynamics Equations

Let us use ”1+3” – the notations for the components of weak potentiala

A ν (70). In these

notations, it follows from the condition (117) that in the intrinsic frame of reference of


1-tymos wave all the time components of the potential are vanishing:

a

T= 0, a = 1, 3.

By choosing ”fixed” YM-index k equal to one, for definiteness, we can write the equations

of a plane wave (137) as a system of equations for the three three-dimensional vectorsa

U

which specify the spatial components of YM-potentials

a

U ••+ab

Ib

U +4π1

U ·1a

δ= 0, (138)

where tensorab

I has the same form (78), (83) as for the zero-current wave. If in system

(138) we explicitly write out the YM-indices of all vectorsa

U and all their Cartesian

indices, we will get a system of nine differential equations, in which only the first three

equations with a unit YM-index are different from the analogous system for a zero-current

wave (82):1

U1•• +

(11

I +4π

)1

U1 +12

I2

U1 +13

I3

U1= 0,

1

U2•• +

(11

I +4π

)1

U2 +12

I2

U2 +13

I3

U2= 0,

1

U3•• +

(11

I +4π

)1

U3 +12

I2

U3 +13

I3

U3= 0,

(139)

but the other six equations for YM-indices 2 and 3 coincide with the corresponding

equations (82).

Energy integral for system (138) has the form which is analogous to the energy integral

for zero-current wave (81):

1

2

a

U •·a

U • +1

2

((1

U ×2

U

)+

(2

U ×3

U

)+

(3

U ×1

U

)+ 4π

1

U 2

)= E.

In contrast to a zero-current wave problem, the 1c-wave cannot be ”scaled” by energy E.

The equations of non-holonomic constraints (135) in this problem have the same form

(85) as in the problem on a zero-current wave. Chiral determinacy conditions (87) also

coincide in these problems.

As in the case of a zero-current wave, the one-current wave problem takes especially

simple form for the ”orthogonal” wave, in which

a

Ui= 0, with i �= a.

For such wave the equations of non-holonomic constraints are satisfied identically, and

for the three nontrivial potential componentsa

Ui with i = a, it is possible to formulate

the system of equations, similar to the system (90):

U••1 +U1

(4π +U2

2 +U23

)= 0,

U••2 +U2

(U2

3 +U21

)= 0,

U••3 +U3

(U2

1 +U22

)= 0.

(140)


Energy integral for (140) looks as follows:

1

2

( •U1

2+•U2

2+•U3

2

)+

1

2

(U2

1U22 +U2

2U23 +U2

3U21 + 4πU2

1

)= E.32 (141)

The chiral determinacy condition for the solution (140) has a simple form:

U1U2U3 ≥ 0. (142)

The procedure of ”soft chiralization” for (140) includes the sign reverse for derivatives•Ui(

i = 1, 3)at vanishing of any potential component Ui.

Probably, with higher energy of E the one-current wave behaves like a zero-current wave:

in equations (140) constant terms 4π, compared to the large (average) squares of po-

tentials, can be neglected. At low oscillation amplitude of potentials, the first equation

(140) turns into harmonic oscillator equation, and the second and third equations are

transformed into Mathieu equations. In the situation with small-amplitudes the three-

component system (140) is similar to the behavior described in the article [1] maxwellian

”heavy photon”, but modulated by the energy exchange ”heavy photon” U1 with two

other YM-degrees of freedom U2 and U3 in the mode as it is dictated by the Mathieu

equation. In particular, without taking into account the condition of the chiral determi-

nacy (142), system (140) may have periodic solutions.

9.3.4 Dynamics of a One-current-terriculum

Repeating the calculations, described in p.9.2.10 for zero-current terriculum, as applied

to the problem of a one-current wave with space-like wave vector, we find that all of the

longitudinal components of the potentials are equal to zero:a

U1= 0, and nine non-trivial

componentsa

T,a

Ui with i = 2, 3 satisfy the system of equations similar to the system of

equations (96), which describes zero-current terriculum. In the equations of a one-current

wave only the three first equations, corresponding to YM-index 1, differ from the system

of equations (96). These equations take the form:

−1

T •• +(

11

I +4π

)1

T +12

I2

T +13

I3

T= 0,

−1

U2•• +

(11

I +4π

)1

U2 +12

I2

U2 +13

I3

U2= 0,

−1

U3•• +

(11

I +4π

)1

U3 +12

I2

U3 +13

I3

U3= 0.

(143)

The other six equations of the system (96) are transferred into the problem of a one-

current wave terriculum unchanged. No changes take place while the components of

tensorab

I (97) and the equations of nonholonomic constraints (98) are transferred into

this problem. As in case of the zero-current terriculum problem, the energy integral of a

32 It can be noted that the ”energy” of oscillator (140), determined by the formula (32), differs from the

wave energy density in the intrinsic frame of reference in multiplier 4π.


one-current terriculum is proper to formulate by using three complex potentialsaτ (100).

When using vectorsaτ , the energy integral for (143) takes the following form:

K+ U = E,

K =1

2

((1τ •

)2

+(

2τ •

)2

+(

3τ •

)2),

U =1

2

((1τ × 2

τ)2

+(

2τ × 3

τ)2

+(

3τ × 1

τ)2

+ 4π(

1τ)2

).

Chiral determinant CD for this problem is defined by the same formula (101) as for the

zero-current problem. When CD is vanishing, all derivativesaτ • signs must be reversed. If

we do not appeal to the chiralization procedure, the potential components in one-current

wave terriculum grow indefinitely along the longitudinal coordinates of the wave.

The problem of 1c-terriculum object takes particularly simple form for the ”orthogonal

wave”, in which1τ= {T; 0; 0},2τ= {0; iU2; 0},3τ= {0; 0; iU3}.

For this three-component wave the equations of nonholonomic constraints are satisfied

identically. The wave equations take the form:

− T•• + E(4π +U2

2 +U23

)= 0,

− U••2 +U2

(U2

3 − T2)= 0,

− U••3 +U3

(U2

2 − T2)= 0.

(144)

Energy integral for (144) looks as follows:

K+ U = E,

K =1

2

((T•)2 − (U•2)

2 − (U•3)2) ,

U =1

2

(U2

2U23 + T

(4π − U2

2 − U23

)).

Chiral determinant CD(

1τ ,

2τ ,

3τ)has the form of (104). The ”soft chiralization” proce-

dure for orthogonal one-current terriculum wave is similar to the zero-current orthogonal

terriculum wave, described above.

We do not provide here a description of a one-current triplet wave with an isotropic wave

vector. This description leads to the results which are completely analogous to those given

in part 9.2.13 for zero-current triplet wave with isotropic wave vector. Such a wave is a

degenerate ”algebraic” object which behavior is not controlled by differential equations.

9.3.5 One-neutrino Plane Triplet Wave

The four-dimensional neutrino zone, which state of currents and potentials is described

by the one-neutrino Lagrangian (123) and, correspondingly, by the field equations (125) –


(130), must border the zones in which there are anisotropic currents on some unknown in

advance three-dimensional boundaries. The conditions for solutions splicing on the three-

dimensional boundaries of these different types of 4-zones must allow to determine not

only the currents / potentials, but also the Lagrange’s multipliers λ. By abstracting from

the existence of these boundaries and expanding the area of the one-neutrino Lagrangian

(123) over the whole space-time, we get some hypothetical state that does not exist in a

”pure form” in nature. Let us name this state a ”Yang-Mills neutrino” (ymino). In the

state of ymino the Lagranges multiplier remains arbitrary, obeying only to the condition

of orthogonality (132).

Isotropization of any of the four currents of the singlet-triplet theory in some 4-zone

generates a neutrino state. Consequently, the ST-theory contains a few types of neutrino:

a pure maxwellian neutrino, totally located in the singlet sector; three indistinguishable

YM-neutrinos, located in the triplet sector of the theory (ymino); and a mixed Maxwell-

Yang-Mills neutrino, appearing under the simultaneous isotropization of singlet current

Jν and the third component of YM-current3

J ν . This mixed neutrino state, not fitting

into either singlet or triplet sectors, occupies the both sectors of the theory.

It can be assumed that historically determined binding of a neutrino type to a massive

lepton type (”electron neutrino”, ”muonic neutrino”, ”τ -neutrino”) is incorrect since it

does not find a natural reflection in the apparatus of the ST-theory. The interpretation

of a neutrino as a lepton is also incorrect itself. Within the framework of the theory

constructed here, leptons are pure, stationary singlet states. Neutrino states cannot be

stationary.

It should be noted that there is no a priori forbidding for the possibility of isotropization

of any of the eight currents of the Yang-Mills octuplet which is responsible for the strong

interaction. So, there are no reasons to except the existence of ”quark” or ”gluon”

neutrinos, which are not connected with the triplet sector of physics.

We will try a solution to the field equations of one-neutrino state of ymino (125) – (130)

in the form of a plane wave with wave vector kν . All of the physical variables in a plane

wave depend only on wave phase φ = kνxν . The wave equation (125) for the plane wave

takes the form:

kμkμAν ′′ + 2kμAμ ×Aν ′ − kνAμ ×Aμ ′ + IAν − 4π p

k

A ν ke= 0, (145)

kνAν ′ = 0, (146)

k

A νa

Aν= 0, (a = 1, 2, 3) . (147)

In these equations p = −1

λ, p = const; the prime denotes a wave phase derivative; Yang-

Mills inertia tensor I is determined by formula (122); YM-index ”k” is ”immovable”, the

summation is not made by it. This index denotes the YM-component which contains

the only current of a one-current problem. Further we will fix k = 1. If homogeneous

equations (145) - (147) are solved, the neutrino current is determined from the condition

Nν = pk

Aν .


From the condition (146) it follows that

kνAν = q = const.

As in other wave problems, we will assume that q = 0. If vector q was nonzero, it would

mean that we deal with a neutrino wave on the background of some stationary field.

With q = 0 the second term vanishes from the field equation (145):

kμkμAν ′′ − kνAμ ×Aμ ′ + IAν − 4π p

k

A ν ke= 0. (148)

By scalar multiplying (148) by wave vector kν , with anisotropic wave vector kν we find

that:

Aμ ×Aμ ′ = 0. (149)

Relation (149) is a standard Yang-Mills nonholonomic constraint, which also appears in

the other wave problems studied above. With regard to (149), the equation (148) is

simplified:

kμkμAν ′′ + IAν − 4π p

1

A ν 1e= 0. (150)

From relation (147) follows that potential1

A ν is an isotropic 4-vector:1

A ν1

Aν= 0, or in

the notation of the form ”1+3”:1

T 2−1

U 2 = 0.

Consequently, the three spatial components1

U of 4-vector1

A ν can be formulated as

follows:1

U=1

T e; e2 = 1,

where e is some three-dimensional unit vector. Using wave vector kν in the notation of

the form ”1+3”, kν = {ω; k}, the orthogonality condition

kν1

Aν= 0

can be presented this way:1

T (ω − ke) = 0,

or ω = ke. But | ke |≤| k |, so, ω2 ≤ k2, i.e. the wave vector of the neutrino wave ymino

(as well as the wave vector of the maxwellian neutrino wave, considered in the article [1])

cannot be time-like. Leaving aside the degenerate case of isotropic wave vector, let us

suppose that

kνkν = −k2S,

where kS is a Lorentz-invariant pseudo-Euclidean module of space-like wave vector kν ;

kS �= 0.

By making the scale transformation, which we have also used for the wave problems

above,

φ→ kSφ, kν → kSkν ,


we can normalize a wave vector for unit in the field equations (150):

−Aν ′′ + IAν − 4π p1

A ν 1e= 0, (151)

kνkν = −1, (152)

kνAν = 0, (153)

1

A νAν = 0. (154)

System (151) – (154) together with nonholonomic constraint (149) makes formulation of

the problem of Yang-Mills neutrino – ymino.

By scalar multiplying the field equation (151) by A′ν and integrating by the wave phase,

we find, with regard to (154), the energy integral of the neutrino ymino wave, which

identically coincides with the energy integral of a zero-current wave with a space-like

vector. Like for a zero-current wave with nonzero energy, it is possible to rescale the

variables per unit energy without changing field equations (151):

Aν →| E | 14 Aν ,

φ→| E |− 14 φ,

p→| E | 12 p.

As for a zero-current wave with space-like wave vector, the energy of a neutrino wave does

not have a fixed sign – the energy sign is determined by arbitrarily set initial conditions.

It is hardly doubtful that here the reader’s relativistic intuition is also ready to protest

angrily against the theory in which there appear some objects with negative energy and,

therefore, negative mass. But perhaps the appearance of such objects is just a different

way of description in the classical language of those phenomena which in the language of

quantum are described as the production of virtual particles off-mass-shell surface with a

virtual violation of the energy conservation law for short time intervals: the shorter this

time interval is, the higher, according to the uncertainty relation, the uncertainty in the

energy of state is.

For example, the process of transformation of a muon into an electron is described in the

language of quantum through the transformation of a muon into a virtual W-particle with

the mass exceeding the muon mass on three decimal orders, with concomitant production

of a muon neutrino and fast subsequent transformation of W-particle into an electron and

concomitant electron neutrino.

The classical theory, developed here, presupposes the continuous dependence of all phys-

ical variables of the theory on the space-time arguments.

It requires a different language to describe such processes, since the classical theory does

not allow virtual particles: this is a non-stationary transformation of a particle (i.e. an

object with the current boundary pomerium) into another, more massive particle, ac-

companied by generation of a non-stationary neutrino zone with negative energy – in this

case the energy conservation law is not violated.


Undoubtedly, there is a considerable distance between this speculative picture and a clear

formulation of the initial boundary value problem, describing the muon decay in the clas-

sical theory under consideration: it is even unobvious that this distance generally can

be surmountable in the classical theory framework by combining the total four-current

Lagrangian with neutrino Lagrangians in some previously unknown 4-zones. However,

the opening possibility to use the language of waves with negative energy an alternative

one to the language of virtual particles – seems fruitful in itself.

In the ymino intrinsic frame of reference, in which kν = {0; 1; 0; 0}, anda

A ν = {a

T;a

U ν},from the orthogonality condition (153) follows that all longitudinal potential components

vanish:a

U1= 0,

and the other 9 components of YM-triplet of potentials are controlled by the system of

equations (151), which in notation ”1+3” take the following form:

v′′ + v

(4πp−

11

I

)= 0, (155)

where v =1

T,1

U2,1

U3 is any of the three nonzero components of vector1

A ν , and

11

I =2

U 2+3

U 2−2

T 2−3

T 2,

(in vectors2

U and3

U the longitudinal components2

U1 and3

U1 are missing).

Equations for the two other time components of 4-vector Aμ, according to (151), have

the form:2

T ′′−2

T3

U 2+3

T2

U ·3

U= 0,3

T ′′+2

T2

U ·3

U −3

T2

U 2 = 0,

(156)

and the equations for spatial components of 4-vector Aμ have the form:

2

U ′′+2

U3

T 2−3

U2

T3

T +3

U ×(

3

U ×2

U

)= 0,

3

U ′′+3

U2

T 2−2

U2

T3

T +2

U ×(

2

U ×3

U

)= 0.

(157)

The orthogonality conditions (147) in notations ”1+3” have the form:

1

T 2−1

U 2 = 0,1

T2

T −1

U ·2

U= 0,1

T3

T −1

U ·3

U= 0.

(158)

Nonholonomic constraints (149) in the language of variablesa

T anda

U take the form:

2

T3

T ′−3

T2

T ′ −(

2

U ·3

U ′−3

U ·2

U ′)

= 0. (159)


and two more equations are obtained from (159) by means of cyclic permutation of YM-

indices: 1 → 2 → 3 → 1. For a plane-polarized ymino wave, the drawn wave equations

allow further simplifications. In such wave all spatial vectorsa

U have fixed directions.

By turning axes y and z around axis x, oriented along wave vector k, it is possible to

combine axis y with the direction of vector1

U oscillations.

Then, with regard to (158), we can formulate all the three-dimensional vectorsa

U as

follows:1

U= {0;1

T; 0},2

U= {0;2

T; u},3

U= {0;3

T; v},

(160)

where u and v are z-components of vectors1

U and2

U. The choice of the axes orientation

(160) turns the ymino wave into a ”five-component” mathematical object, characterized

by five functions1

T,2

T,3

T, u, v.

Subject to the form of vectorsa

U (160), the orthogonality conditions (158) become iden-

tities. Field equations (155), (156), (157) subject to (160) are simplified:

1

T ′′+1

T(4πp− u2 − v2

)= 0, (161)⎧⎨⎩

2

T ′′−2

T v2+3

T uv = 0,3

T ′′−3

T v2+2

T uv = 0.

(162)

{u′′ = 0,

v′′ = 0.(163)

Only trivial solutions to equations (163) are physically acceptable:

u = 0; v = 0. (164)

General solution (163) with linear growth of components u and v along longitudinal co-

ordinate x, in accordance with equations (162) and (161), result in exponential growth of

time componentsa

T. For ymino wave we have no possibility to stop the solution growth

and to transfer it into an oscillating regime by means of ”soft chiralization” procedure,

since chiral determinant CD (101) for such a wave is identically zero. So, we have either

to accept (164) as the only solution that has a physical meaning, or to assume that the

ymino wave turns into a wave with anisotropic current due to some instability which is

not reflected in the neutrino Lagrangian (123).

Accepting solution (164), we have to, due to the same reasons, accept a zero solution for

components2

T and3

T, which are controlled by equations (162). Equation (161) for compo-

nent1

T turns into the equation of a harmonic oscillator with frequency ω =√4πp (p > 0).

What is the physical meaning of the obtained solution?


Subject to (160), the two YM-potential components2

A ν and2

A ν vanish, and the compo-

nent2

A ν describes the ”stiffed” in the wave intrinsic frame of reference sinusoidal ripple,

i.e. the ymino wave, nominally located in the triplet sector of physics, is actually a one-

component singlet object, which does not differ from the maxwellian neutrino described

in the article [1]. When calculating the energy of this one-component ymino by formula

(28), applicable to the triplet wave with space-like wave vector, we find that the energy

of ymino is equal to zero.

Indeed, ymino wave with such properties is a strange, confusing object. This object is

generated by the Lagrangian (123) and we have no a priori reasons to throw ymino away

from physics, to recognize it as a non-existent object. Even the total ”five component”

ymino wave (161), (162) with potentials growing along the wave longitudinal coordinate

should probably be considered as a real physical object, as the description of the tran-

sition state associated with the decay of one current state with anisotropic current and

the production of another current state.

9.3.6 One-current Stationary Triplet State

One-current stationary triplet state is described by the one-current Lagrangian (114)

in the zone, occupied by current jν , and by the zero-current Lagrangian of Yang-Mills

free field (50), where the current is missing. At the outer boundary of the current zone

(pomerium), the isotropization of current jν is taking place. At the inner boundary of the

current zone (latens) the pseudo-Euclidean module of the space-like current jν reaches

its maximum value (7). In the cavitated tubes of latebra, which have latens as the outer

boundary, there is only a free triplet field. At the boundaries of pomerium and latens,

3-current j has no component normal to the surface. At these boundaries potentials and

their space-like coordinate derivatives are continuous.

While deriving the one-current triplet state equations, we will use the Lagrangians (50)

and (114), written in Minkowski coordinates. To account the Riemannian space curva-

ture, generated by a high density of energy-momentum in the current zone, the equations

of stationary one-current state, presented below, must be rewritten in arbitrary curvi-

linear coordinates and the metric tensor must be subjugated to the Einstein equations.

Accounting of this curvature can significantly change the quantitative characteristics of

the solutions, but can hardly influence the fact of the existence or non-existence of a

solution.

The equations of one-current stationary state are convenient to write in the intrinsic

frame of reference of this state. In this frame, the pomerium and latens boundaries are

stationary and the time derivatives of all physical values are equal to zero. Fixation of

the frame of reference and separation of time and spatial vector components deprives

field equation (53) and (121) of Lorentz and Yang-Mills elegance, but makes them more

convenient for numerical investigations.

To write the field equations in the intrinsic frame of reference, it is convenient to intro-

duce into this frame a special mathematical object – Yang-Mills bracket.


Let u and v be two YM-vectors depending on spatial coordinates xk (k = 1, 2, 3). We

do not explicitly write out YM-indices over u and v. Geometrical nature of u and v is

inessential: with respect to the transformations of spatial coordinates, u and v may be

either scalars or vectors. Let us call Yang-Mills bracket over u and v YM-vector {u, v}such that its YM-component of number a is as follows:

{u, va

}=abcε

bu ∇ c

v . (165)

In formula (165) ∇ is the gradient operator in spatial coordinates. Yang-Mills bracket,

introduced in (165), is a bilinear differential form of the first order.

Using the Yang-Mills bracket, the equations of stationary one-current state in the intrinsic

frame of reference, derived from the Lorentz-invariant notation of the field equations (53)

and (121), can be written in the following form:

a) In the current zone:

∇a

T −2{U, Ta

} −ab

Ib

T −4π1

T1a

δ= 0,

∇a

U −2{U, Ua

} +{Uk, Uk

a

} −{T, Ta

} −ab

Ib

U −4π1

U1a

δ= 0.

(166)

∇·a

U= 0. (167)

1

T2

T −1

U ·2

U= 0,1

T3

T −1

U ·3

U= 0,

−(1

2jT

)2

≤1

T 2−1

U 2 ≤ 0.

(168)

Yang-Mills inertia tensor has the form

ab

I=a

Tb

T −a

U ·b

U −ab

δ ·∑c

(T2

c −U2c

), (169)

and, subject to orthogonality conditions (168), it has some zero components inside the

current zone:12

I =21

I = 0,13

I =31

I = 0.

(170)

If equations (166) are solved, the current j = {ρ, j} is determined by its coupling with

potential1

A ν (116):

ρ = −1

T;

j = −1

U .

(171)

b) In current-free zone, potentialsa

A ν obey the equations of the free stationary triplet

field, i.e. equations (166) without the last term in each of equations (166), without


orthogonality conditions (168) and without the condition of ”nulling” of some tensorab

I

elements (170):

∇a

T −2{U, Ta

} −ab

Ib

T= 0,

∇a

U −2{U, Ua

} +{Uk, Uk

a

} −{T, Ta

} −ab

Ib

U= 0.

(172)

Vectorsa

U obey the condition (167).

) The continuity conditions fora

T,a

U and their derivatives are fulfilled at the boundaries

between zones a and b. The vector component1

U, normal to the boundary, vanishes at

these boundaries:1

U ·n = 0, (173)

where n is the unit vector of normal to the surface.

Vector1

A ν becomes isotropic on the pomerium:

1

T 2−1

U 2 = 0. (174)

The current jν reaches its peak value jT on latens:

1

T 2−1

U 2 = −(1

2jT

)2

. (175)

Pomerium and latens boundaries are not pre-set and must be found in the process of the

problem solving.

We would like to remind that the system of twelve field equations (166) or (172) must

be written in curvilinear coordinates and supplemented with ten Einstein field equations,

as it has been noted previously for the similar stationary problem of electrodynamics [1].

We do not have any version of the theorem of existence / non-existence of the solutions

to this problem. Perhaps this is one of the most cumbersome problems in mathematical

physics.

It is obvious that the set problem (166) – (175) does not withstand the charge inversiona

T→ −a

T,a

U→ −a

U due to the presence of the Yang-Mills bracket, but it is invari-

ant relative to the charge conjugation method, described in part 4.1, which requires the

permutation of YM-indices 1↔ 2 simultaneously with the charge inversion. This permu-

tation transforms the one-current problem with the current having a YM-number 1 into

a similar problem with the current number 2.

The problem of constructing chiral-definite solutions for the stationary one-current prob-

lem causes certain difficulties. If we formulate the condition for solution chiral-determina-

cy in the same form (86), (87) as it was formulated in part 9.2.7 for the wave zero-current

problem, i.e. if we require sign-definiteness of the chiral determinant, formed by Yang-

Mills triple of spatial components of the 4-potentials, the question will arise: what is to

be done when this determinant turns into zero?


Some fixed surfaces in the intrinsic frame of reference of a stationary one-current prob-

lem will correspond to zero values of the chiral determinant. At the intersection of such

surface we have not only to reverse all the signs of spatial derivatives for all the twelve

potential componentsa

T anda

U, but also to permute YM-indices 1 ↔ 2. Such chiraliza-

tion procedure looks deformed, but the change of sign for gradients, without YM-indices

permutation, does not preserve the form of the equations of a stationary one-current

problem.

By agreeing to such chiralization procedure, we distort the very term ”one-current state”,

making it conventional. If in the zone, occupied by the current , there are surfaces on

which the chiral CD determinant vanishes, this zone splits into sub-zones occupied by

currents1

J ν or2

J ν . Such picture of one-current stationary state can hardly be called

inspiring. However, the choice is limited: we either put up with such picture (and thus,

generally speaking, recognize the non-terminality and imperfection of Yang-Mills’ descrip-

tion of nature), or we ignore the problem of chiral determinacy and construct continuous

solutions to the stationary one-current triplet problem, despite probable violation of the

terms of chiral determinacy (87).

The problem of the stationary one-current triplet state (if it has a solution) provides a

classical description of a particle33. In order not to increase the number of new terms,

we shall call this particle a ”wark”, removing the first sound from the word ”quark”. In

the subsequent article of this series, we shall demonstrate that in the octuplet sector of

physics there are one-current states with triplet field shell which contains only three of

eight potentials of the octuplet sector34. Such stationary states in the octuplet sector of

physics are a classical model of quarks. The mathematical formulations of the problems

of quark and wark structure completely coincide. The physical difference between the

wark and quark particles is that wark is made of currents and potentials of the triplet

sector of physics and quark – of currents and potentials of the octuplet sector of physics.

The fact of free quarks (and hence, the warks) non-existence, which is known from the

experience, means that the corresponding one-current stationary triplet problem has no

solution with finite energy. This divergence may be connected with ”bad” asymptotic

behavior of solutions of the free stationary triplet field equations (172) away from the

wark’s pomerium, with r → ∞ (r is the distance from the center of the particle wark):

components of the potentials do not decrease with increasing, Yang-Mills field tensor

components (15) decrease proportionally to r−1, and the integral of YM-field energy den-

sity diverges at infinity.

Despite this divergence, the problem of one-current stationary triplet state is of interest.

Its solution, which does not allow normalization of energy, can be normalized with respect

to a charge and magnetic moment [1].

33Or, to be precise, the classical description of some Yang-Mills’ particle family, where particles differ

from each other in pomerium topology, as well as in a number and topology of cavitated tubes latebrae.

This family is a kind of Yang-Mills triplet ”parallel” to a singlet lepton family.34While describing such states, a mathematician would say about sub-algebras of Lie algebra, corre-

sponding to the group SU (3).


We cannot say whether there is such a particle in the vast empirical multitude of ”ele-

mentary” particles which would more or less correspond to the model of wark.

9.4 Two-current Triplet States

9.4.1 Classification and the Lagrangians of Two-current Triplet States

In compliance with the general classification of the singlet-triplet states (p.7 of the present

article) and the overall view of the ”weak” triplet three-current Lagrangian (49), we can

consider the following forms of two-current triplet states: two-current neutrinoless state,

two-current one-neutrino state and two-current two-neutrino state.

9.4.1.1 Two-current Neutrinoless State

The third component of YM-current is missing; components1

j ν and2

j ν are space-like and

orthogonal to each other by (46).

Such state can exist in two forms: in the form of a wave that has no boundaries of the

current zone, or in the form of a stationary state that has common for both currents

inner and outer zone boundaries occupied by currents; this stationary state corresponds

to a certain particle existing in the triplet sector of physics.

A more general treatment of the two-current problem allows simultaneous existence of

two non-overlapping or partly overlapping one-current zones with different pomeriums

for current1

j ν and currents2

j ν : this is the problem of nonstationary interaction of the two

particles. This interaction can hardly allow an adequate treatment in the framework of

the classical field theory. The Lagrangian of two-current neutrinoless state has the form

(see (49)):

Lw = −1

2

(1

j ν1

jν +2

j ν2

jν

)−

(1

j ν1

Aν +2

j ν2

Aν

)− 1

16πAμν ·Aμν − η

1

j ν2

jν . (176)

The last term in (176) is a ”penalty” for orthogonality of currents1

j ν and2

j ν (46), η is

the Lagrange’s multiplier.

Probably, the concept of the two-current state per se has scientific sense only in case if

the condition of current orthogonality (46) is a ”natural” condition for the Lagrangian

(176), i.e. in case if the field equations, following from (176), have a solution satisfying

(46) at η = 0. On this basis, while varying (176), we will omit the last term.

The field equations for two-current problem have the form which is similar to the equations

of the one-current problem (121) and (116):

−�Aν + 2Aμ × ∂μAν −Aμ × ∂νAμ + IAν = 4π

(1

j ν1e +

2

j ν2e

); (177)

(Yang-Mills equations);

1

j ν+1

A ν = 0,2

j ν+2

A ν = 0.

(178)


(current equations).

Due to current equations, potentials1

A ν and2

A ν are space-like and orthogonal to each

other.

The current equations (178) are satisfied in the zone occupied by currents. Outside this

zone, potentialsa

A ν obey Yang-Mills homogeneous equation (177) with a zero right-hand

side.

Differential conditions of the first-order must be added to the field equations (177) and

(178): the gauge condition for 4-divergence of potentiala

A ν (117) and the condition for the

triplet current (19), from which two orthogonality conditions follow for the two-current

problem:1

A ν3

Aν= 0,2

A ν3

Aν= 0.

(179)

The third orthogonality condition must be added to the conditions (179):

1

A ν2

Aν= 0, (180)

which follows from the condition of current orthogonality (46) and current equations

(178).

Under relations (179) and (180) in the zone, occupied by the two currents, all three

4-vectors of YM-potential are orthogonal to each other.

9.4.1.2 Two-current One-neutrino State

In this state, one of YM-triplet currents is identically zero, and one of the other two is

isotropic and the second one is space-like35. This state can be considered in two variants:

V ariant”1/2”

3

j ν ≡ 0;2

j ν = Nν ; NνNν = 0;1

j ν = jν ; jνjν < 0; jνNν = 0. (181)

The last relation in (181) expresses the requirement for currents jν and2

j ν orthogonality

in pure triplet states (43).

According to (49), the Lagrangian of this state has the form:

Lw = −1

2

1

j ν1

jν −jν1

Aν −N ν2

Aν −1


λ

2NνNν . (182)

The last term in (182) is a ”penalty” for neutrino (isotropic) character of current2

j ν , λ

is the Lagranges multiplier.

The field equations of this state only slightly differ from the equations of the two-current

problem (177) and (178):

−�Aν + 2Aμ × ∂μAν −Aμ × ∂νAμ + IAν = 4π(jν

1e +N ν 2

e); (183)

35Current3

Jν in pure triplet state cannot be a space-like vector.


jν+1

A ν = 0,

λN ν+2

A ν = 0.

(184)

Gauge condition (117) must be added to these equations.

Similarly to the neutrinoless two-current problem, the condition of pairwise orthogonal-

ity of all the three YM-components of potentials (179), (180) must be added to these

equations:1

A ν2

Aν= 0,2

A ν3

Aν= 0,3

A ν1

Aν= 0. (185)

Besides these conditions, the neutrino problem includes isotropy condition2

A ν :

2

A ν2

Aν= 0. (186)

Relations (185) and (186) specify a set of holonomic constraints for dynamical system

(183), (184). These constraints are generated by the conditions of two-current one-

neutrino problem and differential conditions for YM-triplet currents (19). The same

conditions (19) of the given problem explicitly fix the Lagranges multiplier λ:

λ = 1. (187)

Conditions (185), (186) and (187) appear to be ”rigid” and incompatible with condition

jνjν < 0. Therefore, the two-current state does not exist in variant ”1/2”.

V ariant”1/3”2

j ν ≡ 0;3

j ν = N ν ; NνNν = 0;1

j ν = jν ; jνjν < 0. (188)

In this variant the orthogonality condition of currents jν and Nν vanishes.

The Lagrangian of this state has the form:

Lw = −1

2jνjν − jν

1

Aν −Nν3

Aν −1


λ

2NνNν . (189)

The field equations of this problem almost do not differ from the equations of variant

”1/2”:


(jν

1e +Nν 3

e); (190)

jν+1

A ν = 0,

λN ν+3

A ν = 0.

(191)

Condition for 4-divergence (117) must be added to these equations.

Similarly to variant ”1/2”, in variant ”1/3”, according to (19), λ = 1, and the additional

conditions for the potentials analogous to conditions (185) and (186), take the form:

1

A ν2

Aν= 0,2

A ν3

Aν= 0,3

A ν3

Aν= 0. (192)

In this variant of the two-current one-neutrino state, the number of holonomic constraints

(192), imposed on the solution of the field equations (190) and (191), is one unit less than


in variant ”1/2” (185), (186).

It is possible to construct the solution to field equations (190) and (191), satisfying the

conditions (192), in the form of a plane wave with a space-like wave vector. This solution

grows unrestrictedly along the longitudinal wave coordinate. In this case, the solution

growth cannot be stopped through chiralization procedure since the chiral determinant

(101) for this wave is identically zero. But the current growth is restricted by the condition

(7). Application of this condition allows to construct restricted solutions with derivative

discontinuity.

9.4.1.3 Two-current Two-neutrino State

In this state both currents are isotropic. This state can also be considered in two variants:

variant ”1/2” (isotropic currents1

j ν and2

j ν , orthogonal to each other, are nonzero), and

variant ”1/3” (isotropic currents1

j ν and3

j ν are nonzero, but their orthogonality is not a

necessary condition).

V ariant”1/2”1

j ν =1

Nν ;

2

j ν =2

Nν ; ,

3

j ν ≡ 0.1

Nν

1

Nν= 0;2

Nν

2

N ν= 0;1

Nν

2

Nν= 0.(193)

The Lagrangian of this state has the form:

Lw = −1

Nν

1

Aν −2

Nν

2

Aν −1


λ1

2

1

Nν

1

Nν −λ2

2

2

Nν

2

Nν . (194)

In (194), λ1 and λ2 are the Lagranges multipliers.

The field equations of a two-neutrino problem with the Lagrangian (194) are similar to

the field equations of the two-current neutrinoless problem (177) and (178):


(1

Nν 1e +

2

Nν 2e

); (195)

λ1

1

Nν+

1

A ν = 0,

λ2

2

Nν+

2

A ν = 0.

(196)

Under the current equations (196) and current conditions (193), potentials1

A ν and2

A ν

are isotropic and orthogonal to each other:

1

A ν2

Aν= 0,2

A ν2

Aν= 0,1

A ν2

Aν= 0. (197)

The differential condition (19), which must be satisfied by YM-triplet currents, will pro-

vide three more relations for the physical variables of the two-neutrino problem:

1

A ν3

Aν= 0,2

A ν3

Aν= 0, λ1 = λ2 = λ. (198)

The relations (197) and (198) are five holonomic constraints, which are imposed on the 12

components of triplet potentiala

A ν , satisfying Yang-Mills equations (195). The Lagranges


multiplier λ remains arbitrary in this problem.

It is easy to see that the wave with at least one isotropic current can have only a space-

like wave vector. In the intrinsic system of such wave, we can try solutions to the field

equations (195) as follows:1

A ν =

{1

T, 0,1

T, 0

},

2

A ν =

{2

T, 0,2

T, 0

},

3

A ν =

{3

T, 0,3

T, v

},

(199)

Potentials of the form (199) satisfy the holonomic constraints (197), (198) and the non-

holonomic constraints which appear in the plane wave theory. The substitution of (199)

into (195) gives, subject to (196) and (198):

1

T ′′+1

T(4πp− v2

)= 0,

2

T ′′+2

T(4πp− v2

)= 0.

(200)

3

T ′′ = 0,

v′′ = 0,(201)

where p = −1

λ, and the prime stands for the longitudinal coordinate x derivative.

The general solution (201) gives linear growth of components3

T and v along the longi-

tudinal coordinate. With substitution of this solution into (200), we get the exponential

growth of components1

T and2

T. This growth cannot be stopped by means of chiralization

procedure, since the chiral determinant CD (101) is identically zero for this wave.

So, we have to choose only a trivial solution (201):3

T= 0, v = 0. In this case, potential3

A ν

vanishes, and the two-component couple1

A ν and2

A ν , in accordance with (200), at p > 0

splits into two independent harmonious waves – two non-interacting singlet maxwellian

neutrinos with the same spatial period.

V ariant”1/3”1

j ν =1

Nν ;

2

j ν ≡ 0; ,3

j ν =3

Nν .

1

Nν

1

Nν= 0;3

Nν

3

Nν= 0.(202)

The Lagrangian of this state has the following form:

Lw = −1

Nν

1

Aν −3

Nν

3

Aν −1


λ1

2

1

Nν

1

Nν −λ3

2

3

Nν

3

Nν . (203)

Field equations, generated by the Lagrangian (203), get the following form:


(1

Nν 1e +

3

Nν 3e

); (204)


λ1

1

Nν+

1

A ν = 0,

λ3

3

Nν+

3

A ν = 0.

(205)

Potentials1

A ν and2

A ν are isotropic:

1

A ν1

Aν= 0,3

A ν3

Aν= 0. (206)

Differential relations for the triplet currents (19) generate three other conditions for the

state (202), (205):1

A ν2

Aν= 0,2

A ν3

Aν= 0, λ1 = λ3 = λ. (207)

The Lagranges multiplier λ remains arbitrary in this state.

Trying partial solution (204), (205) in the form of a plane wave with a wave vector , we

can suggest that1

A ν =

{1

T, 0,1

T, 0

},

2

A ν =

{2

T, 0,2

T, v

},

3

A ν =

{3

T, 0,3

T, 0

},

(208)

When choosing a potential in the form (208), the holonomic constraints (206), (207),

as well as nonholonomic ones, will be satisfied, and the field equations (204) with (205)

give the system of equations which is completely analogous to the system (200), (201)

p = −1

λ:

1

T ′′+1

T(4πp− v2

)= 0,

3

T ′′+3

T(4πp− v2

)= 0.

(209)

2

T ′′ = 0,

v′′ = 0.(210)

For the same reasons that have been formed for variant ”1/2”, we have to choose the

trivial solution (210):3

T= 0, v = 0, at which potential2

A ν vanishes, and potentials1

A ν and3

A ν form a couple

of interacting singlet harmonic waves (a couple of maxwellian neutrino) with spatial

frequency√4πp.

9.4.2 Two-current Triplet Neutrinoless State as a Plane Wave

9.4.2.1 Triplet Plane Wave with a Time-like Wave Vector

and Linear Polarization

Let us write the equations of two-current neutrinoless wave in the intrinsic frame of

reference of the wave, supposing that the time-like wave vector is normalized for a unit.


In this frame of reference all the time components of the potentials and currents are

vanishing, and the spatial components of potentialsa

U, subject to (179) and (180) are

orthogonal to each other. Considering the wave with linear polarization, we can direct

the spatial coordinate axes in their intrinsic wave system so that each of the three vectorsa

U would oscillate along the same coordinate axis. It allows us to use one Cartesian index

for numbering all the potential components while omitting YM-indices:

a

Ui= 0 for i �= a;a

Ui≡ ui for i = a. (211)

For the three values of ui introduced by the relation (211) from Yang-Mills equations

(177) and current equations (178), we can obtain the following system of differential

equations:u1 +

(4π + u2

2 + u23

)u1 = 0,

u2 +(4π + u2

3 + u21

)u2 = 0,

u3 +(u21 + u2

2

)u3 = 0.

(212)

The dots in the equations (212) denote differentiation with respect to the intrinsic time

of the wave. While choosing potential components in the form (211), the holonomic

constraints (179) and (180) are satisfied. The nonholonomic constraints, imposed on the

triplet potential, are also satisfied. For the two-current wave, these constraints have the

same form as for the zero-current wave. The chiral determinant CD for the wave (211)

has the form

CD = u1 u2 u3,

and, correspondingly, the condition of chiral determinacy for the system (212) has the

form:

u1 u2 u3 ≥ 0. (213)

The procedure of ”soft” chiralization for this wave is the replacement of of those instants of

time when at least one of the values vanishes. The procedure of ”hard” chiralization would

be a permutation of the Cartesian indices 1↔ 2 in solutions to the system of equations

(212) at these instants of time. The system (212) ”does not notice” such permutation.

The dynamical system (212), which we call ”two-current Yang-Mills oscillator”, has an

obvious energy integral:

1

2

(u21 + u2

2 + u23

)+

1

2

(u21 u

22 + u2

2 u23 + u2

3 u21 + 4π

(u21 + u2

2

))= E. (214)

System (212) is not self-similar by energy. We cannot normalize the wave energy (214)

per unit by scale transformation. Therefore, for a two-current wave, unlike a zero-current

wave, it is possible to talk about the ”wave of high energy” or the ”wave of low energy”.

The wave of high energy behaves like a zero-current wave. For the wave of low energy

in the potential wave energy, the values of the fourth order can be neglected in contrast

to the values of the second order. Yang-Mills two-current oscillator (if we ignore the

condition of chiral determinacy and not provide a ”soft” chiralization) generally has

a chaotic behavior. However, there are also initial conditions, under which periodic


solutions to the equations (212) appear. ”Soft” chiralization ”kills” the chaos, turning

chaotic solutions into periodic, but does it at the expense of appearance of derivative

discontinuities ui.

9.4.2.2 Doublet Plane Wave with Time-like Wave Vector

and Circular Polarization

Along with linear-polarized wave (212) it is possible to consider degenerate doublet wave

in which3

A ν ≡ 0, and vectors1

U and2

U rotate around axis being orthogonal to each other

and to the axis:1

U= X {0, cosψ, sinψ} ;2

U= X {0, −sinψ, cosψ} ;3

U= 0.

(215)

In relations (215) X is the wave amplitude, ψ is the phase angle of the wave. The

amplitude and phase angle depend on time.

Yang-Mills equations (177) and (178) for the doublet wave (215) take the following form:

X + X(4π +X2 − ψ2

)= 0, (216)

X2ψ = M = const. (217)

The integral of the doublet wave M (217) can be identified with the angular momentum

of this wave. Substitution of (217) into the equation of motion (216) transforms the

problem of the doublet wave into quite a simple one-dimensional problem:

X + X

(4π +X2 − M2

X4

)= 0, (218)

One-dimensional oscillator (218) has an obvious energy integral:

K+ U = E = const,

K =1

2X2,

U = 2πX2 +X4

4+

M2

2X2 .

(219)

The problem (218) describes periodic radial oscillations of doublet vectors in the potential

well with potential energy U (219), followed by rotation of vectors around axis x according

to (217). The energy of the doublet wave cannot be less than the minimum value of Emin:

Emin = 2πX2∗ +

X4∗4

+M2

2X2∗,

where X∗, in accordance with (218), is the root of the equation

X6∗ + 4πX4

∗ = M2.


In the state with minimum energy, doublet vectors1

U and2

U, rotate around axis with a

constant angular velocity ψ, conserving a constant length.

This doublet problem is simply appealing for the naive Bohrs wave angular momentum

quantization. But, probably, it would be premature to leave the territory of Yang-Mills

classical waves, not fully opened by us yet, for the sake of this ”Bohr adventure”.

9.4.2.3 Two-current Triplet Neutrinoless State as a Plane Wave

with a Space-like Wave Vector and Linear Polarization

For a two-current neutrinoless wave with space-like wave vector kν = {0; 1; 0; 0}, nor-malized for unit, Yang-Mills field equations (177), subject to the current equations (178),

can be written as follows:

1

τ ′′ − 1τ

(4π− 2

τ 2− 3τ 2

)= 0,

2

τ ′′ − 2τ

(4π− 3

τ 2− 1τ 2

)= 0,

3

τ ′′ − 3τ

(− 1

τ 2− 2τ 2

)= 0,

(220)

whereaτ=

{a

T; ia

U

}are the three-dimensional complex YM-vectors, that have been in-

troduced here before, in which spatial componentsa

U are orthogonal to the wave direction:

a

U=

{0;

a

U2;a

U3

}.

The primes in (220) stand for differentiation along the longitudinal coordinate .

Non-holonomic constraints for the waves (220) have the following form:

abcε

bτ · c

τ ′ = 0,

and holonomic constraints (179) and (180) take the form:

1τ · 2

τ= 0,2τ · 3

τ= 0,3τ · 1

τ= 0,

or:1

T ·2

T −1

U ·2

U= 0,2

T ·3

T −2

U ·3

U= 0,3

T ·1

T −3

U ·1

U= 0.

Obviously, these equations are compatible only in case of satisfying the condition of

solvabilityD ≥ 0,

where D =

(1

U ·2

U

)(2

U ·3

U

)(3

U ·1

U

),

or cosα12 · cosα23 · cosα31 ≥ 0,


where αab is the angle between vectorsa

U andb

U.

IfD �= 0, the equations of holonomic constraints can be solved relative to time components

of potentialsa

T:1

T= ±√D∣∣ 2

U ·3

U∣∣ ;

2

T= ±√D∣∣ 3

U ·1

U∣∣ ;

3

T= ±√D∣∣ 1

U ·2

U∣∣ .

In these relations, the sign of one of valuesa

T can be chosen arbitrarily, the signs of the

other two time components are determined by the equations of holonomic constraints.

These explicitly written relations fora

T can be used instead of differential equations fora

T of the system of nine equations of motion (220), using only six spatial components of

vectorsaτ at the numerical solution of (220).

Nonholonomic constraints must be taken into account only at the formation of the initial

conditions for the system (220) with some arbitrarily chosen value of longitudinal coor-

dinate x.

The condition for chiral determinacy of such wave is formed as a condition, imposed on

complex vectorsaτ (see formulas (100) and (101)).

It is easy to construct a partial solution to this problem, formed by three orthogonal

vectorsaτ , in which vectors

aτ have the following form:

1τ= {0; 0; iv},2τ= {0; iu; 0},3τ= {T; 0; 0}.

For this type of wave, the equations of holonomic and nonholonomic constraints are

identically satisfied, and the dynamic equations (220) take the following form:

T′′ − T(u2 + v2

)= 0,

v′′ − v(4π + u2 − T2

)= 0,

u′′ − u(4π + v2 − T2

)= 0.

(221)

Energy integral for system (220) has the form:

1

2

((T′)2 − (u′)2 − (v′)2

)+

1

2

(u2v2 +

(4π − T2

) (u2 + v2

))= E.

The energy of such wave is not positively determined.

Chiral determinant for such orthogonal wave has the form:

CD = Tu v.

The procedure of soft chiralization of the solutions to system (221) consists in a sign

change of derivatives T′, u′, v′, with those values of the longitudinal coordinate , for

which at least one of functions T, u or v, vanishes.

The same procedure must be applied when one of the two or both currents reach their

limiting value, i.e. the signs of derivatives T′, u′, v′ are changed if |u| = 1

2jT or |v| = 1

2jT.


9.4.2.4 Doublet Plane Wave with Space-like Wave Vector

and Circular Polarization

A doublet wave with circular polarization can be considered as a partial solution to the

equations of motion (220). In this wave, the third YM-component of potential3

A ν is

identically zero (i.e.3τ≡ 0), and doublet

1τ ,

2τ can be written in a form similar to (215):

1τ= iX{0; cosψ; sinψ},2τ= iX{0; −sinψ; cosψ},

where X = X (x) is the doublet wave amplitude and ψ = ψ (x) is the phase angle of

the doublet wave. With phase angle change, vectors1τ and

2τ rotate around the wave

direction, while remaining orthogonal to axis and to each other. Yang-Mills equations

(220) for the doublet wave look as follows:

X′′ − X(4π +X2 + (ψ′)2

)= 0; (222)

X2ψ′ = M = const. (223)

Constant M in the integral of motion (223) can be identified with the angular momentum

of circularly – polarized wave in the intrinsic system of the wave.

By extracting angular velocity ψ′ from (222) with help of (223), we reduce the problem

of the doublet wave to some one-dimensional amplitude problem:

X′′ − X

(4π +X2 +

M2

X4

)= 0; (224)

Equation (224) has an obvious energy integral:

1

2(X′)2 − 2πX2 − X4

4+

M2

2X2 = −E = const. (225)

Equations (224) and (225) describe a wave with amplitude X, infinitely growing along

the longitudinal wave coordinate. However, the growth of the current is limited by the

condition (7):

|X| ≤ 1

2jT. (226)

When in relation (226) the sign of equality is reached at a certain value of the longitudinal

coordinate x = x∗, the current growth ceases, and we have no right to continue solving

the equation (224) uninterruptedly further along the longitudinal coordinate of the wave

with x > x∗.Here we find ourselves faced with a choice of one of two options, A and B:

A) We can assume that, with x > x∗, currents1

j and2

j are missing, and there is only a

free zero-current doublet stationary Yang-Mills field which is available;

B) We can accept that with x = x∗ there is a sign change of derivatives1τ ′ and

2τ ′, i.e.

X′ → −X′ and ψ′ → −ψ′. Wave energy does not change, but the angular momentum


changes its sign.

By accepting the alternative A, we assume that an external observer can see the surface

x = x∗, on which the current takes its maximum value.

We tend to introduce some exclusion principle, which requires that the surface of the

current maximum value should be only within the current zone, and would be inacces-

sible for an external observer located outside the current zone. Accepting this exclusion

principle, we have to reject the alternative A and to accept the alternative B, which

allows discontinuity of the derivatives of currents and potentials, and even discontinuity

(non-conservation) of the angular momentum of the wave. This is a hard and unpleasant

choice: since Maxwell’s time the classical field theory has always dealt only with contin-

uous solutions.

The alternative B turns a continuous and unlimitedly growing solution of the equation

(224), into periodic ripple: the amplitude of doublet wave X oscillates along the wave di-

rection, from some certain minimum value, reached at X′ = 0, up to the maximum value

(226). Wave energy, according to (225) can be either positive or negative. Equations

(224) and (225), together with condition B, give, probably, qualitatively correct descrip-

tion of the doublet wave with circular polarization. For correct quantitative description,

it is necessary to take into account the space curvature generated by the wave itself with

large amplitude of current X.

9.4.3 Stationary Two-current State

For stationary two-current state, there is an allocated intrinsic frame of reference in

which neither current nor field components depend on time. The position and the shape

of current zone boundaries (the outer boundary – pomerium and internal boundaries –

latens) do not depend on time as well. We will assume that these boundaries are common

for the two orthogonal currents of the two-current state.

The equations, describing distribution of fields in the current zone, can be borrowed from

the equations of a one-current stationary problem, by adding the terms, generated by the

presence of the second current, to equations (166):

Δa

T −2{U, Ta

} −ab

Ib

T −4π(

1

T1a

δ +2

T2a

δ

)= 0,

Δa

U −2{U, Ua

} +{Uk, Uk

a

} −{T, Ta

} −ab

Ib

U −4π(

1

U1a

δ +2

U2a

δ

)= 0.

(227)

In these equations, as before,a

T is the time components of YM-potentials in the intrinsic

frame of reference,a

U is their spatial components; {u, va

} is Yang-Mills bracket (165),

introduced before,ab

I is ,Yang-Mills inertia tensor (169).

Similarly to the one-current problem, the conditions for vector (j) (167) divergence must

be added to Yang-Mills stationary equations (227):

∇·a

U= 0.


Instead of the two orthogonality conditions (168) that were present in the one-current

problem, in the two-current problem there are three of such conditions (179), (180) im-

posing three holonomic constraint equations to the twelve components of the triplet of

potentialsa

A ν :1

T2

T −1

U ·2

U= 0,2

T3

T −2

U ·3

U= 0,3

T1

T −3

U ·1

U= 0.

(228)

These orthogonality conditions mean that Yang-Mills inertia tensorab

I is diagonal:

ab

I= 0 a �= b.

Currents1

j ν and2

j ν , and, consequently, the corresponding YM-components of potentials,

are space-like and module-limited everywhere in the current zone:

−(1

2jT

)2

≤1

T 2−1

U 2 ≤ 0,

−(1

2jT

)2

≤2

T 2−2

U 2 ≤ 0,

(229)

and the limiting values for the first and second currents are reached simultaneously, at

the boundaries of the current zone which are common for two currents.

We tend to believe that this joint achievement of the limiting current values will be pro-

vided only if on currents (and potentials) there is imposed one more holonomic constraint,

which would require the equality of pseudo-Euclidean modules of the two currents, not

only at the boundaries, but everywhere in the current zone:

1

T 2−1

U 2 =2

T 2−2

U 2. (230)

This condition36 means that Yang-Mills inertia tensorab

I has axial symmetry:

11

I =22

I .

Similarly to the one-current problem, if the equations (227) are solved and the components

of YM-triplet potentials are found, time (ρ) and spatial (j) components of YM-doublet of

currents of the two-current problem can be determined by the current equations (110):

1ρ= −

1

T;1

j= −1

U;

2ρ= −

2

T;2

j= −2

U .

(231)

36We have no existence theorem for a two-current stationary problem available, and we cannot conclude

whether the system of four holonomic constraints (228) and (230) is too tough and blocks the existence

of solutions.


In the zone, free of the currents, (i.e. inside the cavitated tubes latebrae, restricted by the

current boundary latens, and outside the outer boundary of the current zone- pomerium),

potentialsa

A ν obey the equations of a free stationary triplet field (172), without holonomic

constraints (228) and (230), and without restrictions on pseudo-Euclidean module (229).

At the boundaries of the current zone, the spatial components of currents do not have a

component which would be normal to the boundary:

1

U ·n = 0;2

U ·n = 0, (232)

where n is a unit normal vector to the surface.

At the pomerium boundary, both of the currents are isotropic:

1

T 2−1

U 2 =2

T 2−2

U 2 = 0,

and at the latens boundary, both of the currents rech their maximum value(175):

1

T 2−1

U 2 =2

T 2−2

U 2 = −(1

2jT

)2

.

For two-current stationary state we can literally reproduce everything that has been said

above for the one-current state due to the need for consideration of the Riemannian space

curvature (not reflected in the equations (227)) and due to the need for consideration of

the chiral determinacy condition (87).

The considered formulation of the problem of the two-current stationary state (if this

problem has a solution) provides a classical description of a certain particle37. We do not

know whether some real particle corresponds to this hypothetical model. In the octuplet

sector of physics there can be also set a problem of the stationary two-current state in

the triplet shell of potentials. This problem is described by the same system of partial

differential equations (227) with the same holonomic constraints (228), (230)), as the

two-current problem in the triplet sector of physics, considered above. Probably, some

real particles – charged pions and other charged ”two-quark” bosons – correspond to

stationary two-current states of the octuplet sector of physics. We can not specify any

known particles, resembling pions of the octuplet sector, in the triplet sector of physics38.

9.5 Three-current Triplet States

9.5.1 One-neutrino Three-current Triplet State

In three-current state, one of the currents –3

j ν – is neutrino. Two other currents are space-

like and orthogonal to each other. However, with variation of the ”weak Lagrangian” (49),

we will omit term η1

j ν2

j ν – ”the penalty for currents orthogonality”. We will adhere to

the minimalist position: either it is possible to construct solutions to the field equations

37Or, to be precise, a particle family. See the similar application for a one-current problem.38Except for W-bosons, but they are enormously massive in comparison to pions.


that arise with variation of the Lagrangian (49) with zero Lagranges multiplier η, or

the corresponding three-current states cannot exist in a ”pure” form, being transitional

between the states with other current formulas.

The states, containing neutrino current, cannot be stationary. However, the nonstationar-

ity of these states can manifest itself in different ways. For example, a set of three-current

triplet states includes the state with distinct current zones for currents1

j ν and2

j ν : there

is a triplet one-current zone with current1

j ν and another triplet current zone with current2

j ν ; each of these zones has its outer boundary – pomerium, and the Lagrangian (49)

describes the interaction of these two zones (two ”triplet particles”) through the third

neutrino current.

Description of these ”exchange” interactions within the framework of the classical field

theory may turn out to be incomparable to any particular experimental situation. But

the classical description of all triplet wave states that do not have a pomerium, seems

to be relevant – as relevant as the description of the singlet wave states, which include

ordinary electromagnetic wave and also a ”heavy photon” and ”maxwellian neutrino”,

considered in the article above [1]. Considering three-current triplet state as a plane

wave, we can use the weak Lagrangian (49) and Yang-Mills equations which arise from

this Lagrangian:

−�Aν + 2Aμ × ∂μAν −Aμ × ∂νAμ + IAν = 4πJν . (233)

In equation (233), the triplet of YM-currents, according to (49), has the form:

Jν =

{1

j ν ;2

j ν ; Nν

};

1

j ν2

j ν = 0; NνNν = 0.

(234)

Current equations, supplementing Yang-Mills systems of equations (233), have the fol-

lowing form for the three-current problem:

1

j ν+1

A ν = 0;2

j ν+2

A ν = 0;

λNν+3

A ν = 0.

(235)

From relations (234) and (235) follows that potentialsa

A ν obey the following constraint

relations:1

A ν2

Aν= 0;3

A ν3

Aν= 0. (236)

The potentials also obey the gauge condition by 4-divergence:

∂νa

A ν = 0. (237)


Besides this, physical variables of the three-current problem obey the differential relation

(19):

∂νJν +Aν × Jν = 0. (238)

Comparison of (238) with (237), (236) and (235) allows to fix the undetermined Lagranges

multiplier :

λ = 1. (239)

Since neutrino current N ν is present in the formation of the three-current triplet state,

the corresponding plane wave can only have a space-like wave vector kν (kνkν < 0). This

vector can be normalized to unit by scale transformation. In the wave intrinsic frame of

reference, vector kν has the form:

kν = {0; 1; 0; 0} . (240)

Subject to (237) and (240), none of YM-potentialsa

A ν has a longitudinal component:

a

A 1 = 0. (241)

Considering (241), Yang-Mills equations (233) for a plane wave with wave vector (240)

are easy to write in the form of the equation for Yang-Mills triple of three-dimensional

complex vectorsaτ=

{a

T; ia

U

}, introduced above, (

a

T is a time component of potential

a

A 0, anda

U is two-dimensional vectors of spatial components of potentiala

A, orthogonal

to the wave direction):

1

τ ′′ −(4π− 2

τ 2)

1τ −

(1τ · 3

τ)

3τ= 0,

2

τ ′′ −(4π− 1

τ 2)

2τ −

(2τ · 3

τ)

3τ= 0,

3

τ ′′ −(4π− 1

τ 2− 2τ 2

)3τ −

(1τ · 3

τ)

1τ −

(2τ · 3

τ)

2τ= 0.

(242)

Holonomic constraints for vectorsaτ (234) have the following form:

3τ 2 = 0,

1τ · 2

τ= 0. (243)

Nonholonomic constraints, resulted from Yang-Mills equations (233), have the form:

abcε

bτ · c

τ ′ = 0, (244)

It is sufficient to account the constraints (244) while forming the initial conditions for the

system of motion (242) at some initial value of longitudinal coordinate .

Energy integral for the system (242) has the form:

+1

2

∑a

(aτ ′

)2

− 1

2

(4π

(1τ 2+

2τ 2

)+

(1τ · 3

τ)2

+(

2τ · 3

τ)2

− 1τ 2· 2

τ 2

)= E. (245)


Energy E in (245) can have any sign.

While constructing solutions for the systems (242), we also have to take into account the

restrictions imposed on the currents (5) and (7):

−(1

2jT

)2

≤ 1τ 2 < 0,

−(1

2jT

)2

≤ 1τ 2 < 0,

(246)

It is also necessary to take into account the condition for chiral determinacy, imposed on

chiral determinant CD, composed of three YM-vectorsaτ (101).

Conditions (246) and (101) can be treated as unilateral (”releasing”) holonomic con-

straints, imposed on the solution of the system (242). As it is known, such unilateral

constraints generate solutions which have derivatives discontinuities.

For problem (242) it is not difficult to construct a degenerate doublet solution, supposing

that3τ≡ 0. This solution has been considered above (p. 9.4.2.4). We have not managed

to construct any other, vector3τ nondegenerate, partial solutions to the problem (242),

compatible with conditions (243) and inequalities (246). However, we also have not man-

aged to prove formally the statement of nonexistence of the three-current solutions to the

equations (242), which obey conditions (243) and inequalities (246).

9.5.2 Three-neutrino Three-current Triplet State

Let us consider such a triplet state, where all of the three currents are neutrino:

1

j ν =1

Nν ;

2

j ν =2

Nν ;

3

j ν =3

Nν ;

1

Nν

1

N ν= 0;2

Nν

2

Nν= 0;3

Nν

3

Nν= 0;1

Nν

2

Nν= 0.(247)

Upon obtaining Yang-Mills equations for this condition in the basic Lagrangian (49),

we have to omit the ”current part” of Lw and to add a ”penalty” to the basic weak

Lagrangian (49) for isotropy of all the three currents of Lad:

Lad = −1

2λ1

1

Nν

1

Nν −1

2λ2

2

Nν

2

Nν −1

2λ3

3

Nν

3

N ν , (248)

where λ1, λ2, λ3 are the Lagranges multipliers.

The term Lad (248) restores the current part in the Lagrangian, but with arbitrary

multipliers λa.

Yang-Mills field equations and the current equations can be obtained from the weak

Lagrangian (49) subject to Lad (248). Yang-Mills equations in the neutrino problem

coincide with the equations (233), at the notation of the right sides (i.e. current) on the

basis of (247). The current equation with regard to (248) have the form:

λ1

1

Nν+

1

Aν= 0;

λ2

2

Nν+

2

Aν= 0;

λ3

3

Nν+

3

Aν= 0.

(249)


From the current equations (249) and differential current condition (237), the statement

of the equality of all Lagranges multipliers λa can be obtained as a consequence:

λ1 = λ2 = λ3 (= λ) . (250)

For a plane three-neutrino wave with wave vector (240), the differential equations for

Yang-Mills triple of three-dimensional complex vectorsaτ can be obtained from the field

equations (233) and current equations (249):

1

τ ′′ +4π p1τ −

(1τ · 3

τ)

3τ= 0,

2

τ ′′ +4π p2τ −

(2τ · 3

τ)

3τ= 0,

3

τ ′′ +4π p3τ −

(1τ · 3

τ)

1τ −

(2τ · 3

τ)

2τ= 0.

(251)

In these equations p = −1/λ (λ remains undefined). Nonholonomic constraints (243) are

imposed on the solution of the system (251).

These equations (251) are structurally identical to the equations (242), but they account

the neutrino character of the currents:1τ 2 = 0;

2τ 2 = 0;

3τ 2 = 0. (252)

and the orthogonality condition of1τ and

2τ :

1τ · 2

τ= 0. (253)

It is easy to construct a partial ”three-fold degenerate” solution for the system (251),

where all three YM-potentials coincide with each other39.

1τ=

2τ=

3τ . (254)

In this solution, all three complex isotropic vectors are at the same time parallel (254)

and orthogonal (252) to each other - the geometry is quite strange for those who are used

to real vectors in the Euclidean space. Each of the vectors obeys the elementary oscillator

equation:a

τ ′′ +4πpaτ= 0, (255)

which, with p > 0, describes a sinusoidal ripple with spatial frequency√4πp, that is

stationary in the intrinsic frame of the wave.

The solution (254), (255) describes a system of three non-interacting and not noticing

each other maxwellian neutrinos, described in the article [1]. The Lagrangian parameter

p remains undefined in this solution: it is controlled by the processes which occur outside

the neutrino zone40.

39 In spite of the fact that in this solution1

A ν =2

A ν =3

A ν , we have to suggest that each of the three

YM-potentialsa

A ν and each of the three YM-currentsa

N ν can be measured independently from each

other by means of different (and still unknown) measuring devices: if you do not believe it, there is no

need for you to study the theory of Yang-Mills fields.40We did not manage to construct any solutions to the system (251), which are less trivial than (254),

(255).


10. Composite Wave Singlet-triplet States

10.1 One-current Composite Singlet-Triplet Neutrino State

(Maxwell-Yang-Mills Neutrino – mymino)

10.1.1 The mymino Lagrangian and Field Equations

The simplest singlet-triplet state contains only one isotropic current Nν which is simul-

taneously located in the two sectors of physics. This current is the singlet current and

the third component of YM-triplet of currents:

Jν = N ν ;3

J ν = −Nν ; NνNν = 0. (256)

Two other components of YM-triplet of currents in the neutrino state do not exist. We

will call the state (256) ”Maxwell-Yang-Mills neutrino” or mymino. Using the basic

Lagrangian of the singlet-triplet theory (17) and taking into account (256), the Lagrangian

mymino can be presented as follows:

Lmym = − 1

2 pS pTNνZ

ν − 1

16π(WμνW

μν +Wμν ·Wμν)−λ

4 pS pTNνNν . (257)

In the Lagrangian (257), potential Zν is a linear combination of singlet potential Wν and

the third component of YM-potential3

W ν :

Zν = pT Wν − pS3

W ν . (258)

Potential Zν appears in (257), since current N ν interacts with both singlet potential Wν

and the third component of YM- triplet of potentials.

Field tensors Wμν and Wμν are determined by the relations (14) and (15).

The last term in (257) was introduced as a ”penalty” for the isotropy of current Nν . In

this term λ is the undefined Lagranges factor. The denominator 4pSpT is introduced into

this term for the relative easiness of further calculations.

Occurrence of the last term in (257) with the Lagranges multiplier λ, which is necessary

from the mathematical point of view for the problem of conditional extremum of the

action functional, causes certain concern in terms of physical interpretation of the the-

ory. It is equivalent to the occurrence of some external field (”Higgs-like-field”), affecting

current directly so as to ensure that it is isotropic. The external field itself is not affected

by physical fields and currents occurring in the problem.

Probably, it should be assumed that the neutrino state mymino, described by the La-

grangian (257), can not occupy the whole four-dimensional space – time, but can exist

only in a finite and non-stationary four-dimensional area which is a buffer between the

areas filled with space-like currents Jν and Jν and the areas of free fields Wν and Wν

which contain no currents Jν and Jν . Such ”multi-zone” problem would describe, in clas-

sical language, the process of neutrino exchange between some of the massive particles.

The formulation of such ”multi-zone” problem is not considered in the article. We treat


the mymino state as a model problem, extending the scope of the Lagrangian (257) appli-

cation all over the space-time continuum. In such model problem, the ”Lagrange’s field”

λ becomes an uninterpreted external parameter, a kind of a ”constraint force”. Varying

the action functional with the Lagrangian (257) by current Nν and potentials Wν , we

obtain the following equations that describe the mymino state:

λN ν + Zν = 0 (”current equation”), (259)

∂μWμν =

2π

pSNν (”maxwellian equations”), (260)

∂μWμν +Wμ ×Wμν = −2π

pTNν 3

e (”Yang-Mills equations”). (261)

In equation (261)3e is the unit vector of the third YM- direction with YM-components

{0; 0; 1}.Mymino problem solution must satisfy, besides the recorded field equations, the following

conditions:

∂νNν = 0 (current conservation), (262)

∂νWν = 0, ∂νW

ν = 0 (zero 4-divergence gauge of potentials). (263)

Besides these differential equations, the algebraic orthogonality condition, resulting from

missing of currents1

J ν ,2

J ν and the differential condition (19) for Yang-Mills triplet of

currents, are imposed on the solution of the mymino problem:

Nνa

Wν= 0, (a = 1, 2) ,

or, subject to (259):

Zνa

Wν= 0, (a = 1, 2) . (264)

By means of current equation (259) and the definition of compound Z-field (258), current

Nν and maxwellian singlet Wν can be excluded from consideration, reducing the problem

of mymino to the solution of the system of nonlinear wave equations for YM-triplet Wν

and singlet-triplet compound potential Zν :

−�Zν +2πq

pSpTZν = −pTpS

(3

h ν + pT3b

Ib

W ν

);

−�a

W ν = pT

(a

h ν + pTab

Ib

W ν +2πq

p2TZν

3a

δ

); (a = 1, 2, 3) .

(265)

where q = −1

λ;

hν = Wμ × (2∂μWν − ∂νWμ) , (266)


hν is an auxiliary object (YM-vector and Lorentz vector), vanishing for plane waves.

All additional algebraic and differential conditions of the problem, in terms of four po-

tentials Zν , Wν , entering into (265), take the following form:

∂νZν = 0, (a)

∂νWν = 0, (b)

ZνZν = 0, (c)

Zνa

Wν (a = 1, a = 2) . (d)

(267)

10.1.2 Mymino as a Plane Wave

As for the other wave problems, considered above, we shall try the solution for a mymino

problem in the form of a plane wave. For such solution, all the problem variables depend

only on a single scalar argument φ (”wave phase”):

φ = kμxμ,

where kμ is a wave vector.

Wave equations (265) for the plane wave take the following form:

k2S (Z

ν)•• +2πq

pSpTZν = −pTpS

(3

h ν + pT3b

Ib

W ν

);

k2S

(a

W ν

)••= pT

(a

h ν + pTab

Ib

W ν +2πq

p2TZν

3a

δ

); (a = 1, 2, 3) .

(268)

kν (Zν)• = 0,

kν (Wν)• = 0.

(269)

In these equations the ”dot” symbol, following the variable, means a wave phase φ dif-

ferentiation; k2S is a Lorentz invariant of wave vector kν :

kνkν = −k2S. (270)

Wave vector kν in the mymino wave, containing isotropic vectors Nν and Zν , can be only

space-like.

As in the other wave problems, we shall omit the differentiation symbol in the equations

(269), assuming that there are no external constant fields:

kνZν = 0,

kνWν = 0.

(271)

Vectors hν for the plane wave, according to the definition of hν (266), have the form:

hν = 2kμWμ × (Wν)• − kνWμ × (Wμ)• . (272)

In view of (271), the first term in (272) vanishes. The convolution of the equations (268)

with wave vector demonstrates, subject to (271), that the second term in (272) should

also vanish:

Wμ × (Wμ)• = 0. (273)


As in other wave problems, the equations (273) are nonholonomic constraints, imposed

on the initial conditions of the problem.

Vanishing of vector hν simplifies the mymino equation (268):

k2S (Z

ν)•• +2πq

pSpTZν = −pSp2T

3b

Ib

W ν ;

k2S

(a

W ν

)••= p2T

ab

Ib

W ν +2πq

pTZν

3a

δ .

(274)

While solving wave equations (274), it is convenient to use the mymino intrinsic frame of

reference, in which wave vector has the only non-zero component directed along axis :

kν = kS {0; 1; 0; 0} . (275)

In this frame φ = kSx and in field equations (274) it is convenient to move from phase φ

differentiation to the differentiation of longitudinal coordinate :

(Zν)• =1

kS(Zν)′;

(a

W ν

)•=

1

kS

(a

W ν

)′,

where the prime stands for coordinate differentiation. The wave equations take the form:

(Zν)′′ +2πq

pSpTZν = −pSp2T

3b

Ib

W ν ;(a

W ν

)′′= p2T

ab

Ib

W ν +2πq

pT

3a

δ Zν .

(276)

From conditions (271) follows that in the mymino intrinsic frame of reference, the longi-

tudinal components of potentials disappear:

Z1 = 0;a

W 1 = 0.

The simplest partial solution of a mymino problem can be constructed by assuming that

potentials1

W ν and2

W ν are missing, and potentials Zν and3

W ν are isotropic. Trying the

solution for the problem in the form of linear-polarized wave:

Zν = {T; 0; T; 0} ;3

W ν =

{3

T; 0;3

T; 0

},

(277)

we find that all holonomic and nonholonomic constraints for the solution of the form

(277) are satisfied, Yang-Mills inertia tensor is identically zero, and the dynamic system

(276) turns into a couple of linear differential equations:

T′′ +2πq

pSpTT = 0; (278)

3

T ′′ =2πq

pTT. (279)


From (278) follows that, with q > 0:

T = Q · sin(x

√2πq

pSpT+ α

), (280)

where Q and α are arbitrary constants.

Since the solution of a mymino problem should depend only on phase kSx, from (210)

can be concluded that the pseudo-Euclidean module of the wave number of a mymino

wave has a specified value:

k2S =

2πq

pSpT. (281)

But due to the Lagranges multiplier q indeterminacy, the relation (281) can hardly be

subjected to any experimental trial.

By choosing for3

T the solution to the equation (279), bounded at infinity, we can present

the partial solution of the mymino problem, constructed here, in the following form:

1

W ν = 0;2

W ν = 0;

Zν = Qξνf; Nν = Qqξνf;

Wν = QpT ξνf;

3

W ν = −QpSξνf.

(282)

In formulas (282) f-phase wave function is f = sin (kSx+ α); ξν = {1; 0; 1; 0} is isotropic4-vector, which specifies 4-orientation of the linear-polarized mymino wave, Q is the

arbitrary amplitude multiplier.

From formulas (282) follows that the linear combination of potentials Wν and3

W ν ,

identified with electromagnetic potential Aν :

Aν = pSWν + pT

3

W ν ,

is identically zero in the mymino wave.

On seeing the formulas (282), the reader may exclaim in disbelief: can it really be the

neutrino? Perhaps, one has just to get used to this description: there is no other classical

picture for neutrino in physics. Formulas (282) give a complete description of the object

in its intrinsic frame of reference. Similarly to the maxwellian neutrino, described in

the article [1], mixed singlet-triplet neutrino (282) in the intrinsic frame is the current

and field ripple that does not change in time. According to the mathematical model,

constructed here, we can give the following description of mymino: Maxwell-Yang-

Mills neutrino is isotropic current Nν, which couples two potentials Wν and3

W ν from the two sectors of physics- singlet and triplet; the constraint of

these three isotropic vectors N ν, Wν and3

W ν is so strong that it ”switches

off” two other components of YM-triplet of potentials and ”switches off”

the nonlinearity which is inherent to the triplet sector of physics; this closely

coupled triple of vectors generates one physical object - mymino wave .

Within the framework of the classical theory, constructed here, it would be incorrect to


speak about the ”interaction of neutrino with field Zν”, although this is a conventional

way now to describe some of the weak interaction processes. Indeed, the term of the

form NνZν is present in the mymino Lagrangian (257), which allows to speak about the

interaction of current and field, but this is the interaction which one physical object,

not two objects, distinguishable from each other, corresponds to: field Zν is a part of

neutrino rather than its environment.

While calculating the components of singlet Wμν and triplet Wμν field tensor, we can

find that in the mymino intrinsic frame of reference, the corresponding ”electric” fields

W0ν and3

W 0ν are longitudinal (only the components W01 and3

W 01 are nonzero), and

”magnetic” fields are transverse (only components W12 and3

W 12, corresponding to z-

component of the magnetic field vectors, are nonzero in the ”magnetic” part of the field

tensor). Calculation of the corresponding energy – momentum tensor components shows

that in the mymino intrinsic frame of reference, the three-dimensional vector of energy

flux density is directed along axis y, as well as the three-dimensional current in the wave

– that is, orthogonally to the three-dimensional vector of the mymino wave. The three-

dimensional velocity of energy flux is equal to the velocity of light.

Undoubtedly, in this phase of construction of physics of currents and fields, the distance

between uncommon, but quite clear picture of the classical relativistic neutrino presented

here, and, for example, phenomenological Fermi theory of β-decay – quantum and non-

relativistic theory, seem almost infinite. However, the same physical object, neutrino,

appears in both theories, and the distance between such different portraits of neutrino

has to be once overcome.

10.1.3 Non-linear Mymino Model.Restrictions on Pseudo-Euclidean Potential Modules

It is possible to consider a mymino model, which is a little bit more general than model

(277). In model (277), all four potentials Zν ,a

W ν , appearing in the problem, are isotropic

vectors. However, both holonomic (264) and nonholonomic (273) constraints of the prob-

lem will be identically satisfied if we choose a more general form of potentialsa

W ν in the

wave intrinsic system, while preserving the form of Zν , according to (277):

a

W ν =

{a

T; 0;a

T; V

}; a = 1, 2, 3. (283)

For wave form (283) three YM-potentialsa

W ν have the same z-component in the intrinsic

frame of reference, and the same square of pseudo-Euclidean module, which is equal to

−V2. Yang-Mills inertia tensor for wave (283) is determined by the value of V2:

I = V2

⎛⎜⎜⎜⎜⎝2 −1 −1

−1 2 −1

−1 −1 2

⎞⎟⎟⎟⎟⎠ . (284)


Substitution of YM-tensor (284) into field equations (276) will provide the following

system of equations for five unknown functions T,a

T, V:

T′′ + k2ST −pSp2T V2

(1

T +2

T −23

T

)= 0;

1

T ′′ +p2T V2

(2

T +3

T −21

T

)= 0;

2

T ′′ +p2T V2

(3

T +1

T −22

T

)= 0;

3

T ′′ − pSk2ST +p2T V2

(1

T +2

T −23

T

)= 0;

V′′ = 0.

(285)

The last equation (285), except for trivial solution V = 0, embedded into the linear

mymino model (277), (278), (279), has a linearly growing solution:

V = ax+ b. (286)

If in (286) a �= 0, constant b in (286) can be attached a zero value by simple shifting of

the reference point along axis x.

Let us first consider solution (285) with a = 0. This solution per se is of no physical

interest, since the state V = b = const corresponds to imposition of some external field

on mymino wave (277). But this solution allows to see the instability of mymino in such

external field.

At V = b = const, the first four equations of system (285) turn into the system of linear

homogeneous equations with a constant coefficient. Trying functions T anda

T in the form

of const · epx from (285), we can obtain a characteristic equation relative to eigenvalues

of p:

p = ±μ√u+ 2,

where μ = pT b, and u satisfies the equation:

(u− 1) (u+ 2)(u2 + u

(1 + κ2

)− 2− κ2

(1− 2p2S

))= 0, (287)

with κ = kS/μ.

Polynomial root (287) u1 = 1 demonstrates exponential instability of mymino wave:

p1 = ±μ√3. The second root u2 = −2 generates the possibility of linear growth of

instability: p2 = 0 (a double root). The remaining two roots u± of polynomial (287) have

the form:

u± + 2 =1

2

(3− κ2 ±

(3 + κ2

) √1− 8κ2p2S

(3 + κ2)2

). (288)

In the asymptotic behavior of weak external perturbation (μ→ 0; κ→∞), roots (288)

describe a mymino wave (p− ∝ ±ikS), which long-wave oscillations(p+ ∝ ±iμpS

√2)are

imposed on. In the asymptotic behavior of strong external perturbation (μ→∞; κ→ 0),


roots u± ”duplicate” roots u1 and u2

(p+ ∝ ±μ

√3; p− ∝ 0

).

This analysis demonstrates that, with a = 0 in (286), the components of potentials T

anda

T unrestrictedly grow along the longitudinal wave coordinate, but pseudo-Euclidean

modules of 4-potentialsa

W ν remain constant, and we have no mechanism for suppressing

the growth of T anda

T.

The solutions to problem (285), with V = ax, a �= 0, will also grow unrestrictedly along

the longitudinal coordinates of the wave. But for such solutions we have a mechanism for

suppression T anda

T growth if we decide to extend the introduced earlier restric-

tions of pseudo-Euclidean module of YM-currents (7) on YM-potentialsa

W ν ,

and assume that

V2 ≤ j2T. (289)

For each value of longitudinal coordinate x, at which the equality sign is reached in (289),

we have to change the signs for V′, T′ anda

T ′41. In accordance with the determination

of potential (258), singlet potential W for wave (283) has the form:

Wν =1

pT

{T + pS

3

T; 0; T + pS3

T; pSV

},

and, correspondingly, its pseudo-Euclidean module is determined by value V:

WνWν = −p2Sp2T

V2. (290)

Assuming that triplet potentials, as well as triplet currents, have restricted pseudo-

Euclidean module (283), following the same logic, we have to assume that singlet potential

Wν also has module restriction – the same as singlet current (96):

−WνWν ≤ j2S. (291)

Comparison of (289), (290) and (291) allows to relate the limiting values of currents jSand jT:

jS pT = pSjT,

or

jT = tgθw · jS, (292)

where θw is Weinberg angle.

It should be noted that in nonlinear mymino wave (285), in contrast to linear wave (282),

electromagnetic potential Aν is nonzero.

41 If you do not like this appalling violent procedure, which restricts the growth of solutions in mymino

wave, you have to come up with some other procedure – perhaps no less appalling – or even abandon the

attempt to construct the classical theory of neutrino and, the classical version of singlet-triplet physics,

on the whole.


10.2 Mixed Singlet-triplet Two-current One-neutrino State

10.2.1 The Lagrangian and Field Equations

Let us consider such a mixed state, where along with the singlet-triplet neutrino current

Nν there is one more YM-triplet current, for example,1

J ν :

Jν = Nν ;3

J ν = −Nν ; NνNν = 0;2

J ν = 0;1

J ν = jν ; jνjν < 0. (293)

The state Lagrangian (293), in accordance with general form of the basic Lagrangian of

the singlet-triplet theory (17), can be written in the following form:

L = − 1

8p2Tjνjν −

1

2pTpSNνZν −

1

2pTjν

1

Wν −1

16π(WμνW

μν +Wμν ·Wμν)− λ

4pTpSN νNν .

(294)

The last term in the Lagrangian (294) is a ”penalty” for neutrino character of current

Nν ; λ is the Lagranges multiplier; Zν is the linear combination of potentials W and3

W ν

(258).

The following field equations result from the Lagrangian form (294):

λN ν + Zν = 0;

jν + 2pT1

W ν = 0; (”current equations”) (295)

∂μWμν =

2π

pSNν ; (”Maxwell equations”) (296)

∂μWμν + pT Wμ ×Wμν =

2π

pT

(jν

1e −Nν 3

e); (”Yang-Mills equations”) (297)

In equations (297)1e and

3e are the corresponding YM-unit vectors with YM-components

1e= {1; 0; 0}; 3

e= {0; 0; 1}.From current equations (295) follows that the combined potential Zν , similarly to current

Nν , is isotropic:

ZνZν = 0. (298)

The gauge condition by divergence of all potentials can be imposed on the problem under

consideration:∂νZ

ν = 0;

∂νWν = 0.

(299)

The algebraic relations, which the solution of the field equations (295), (296), (297)

must obey, follow from the differential condition for YM-triplet currents (19) and current

equations (295):

Zν2

W ν = 0;1

W ν2

Wν= 0;1

Wν

(2pT

3

W ν − qZν

)= 0.

(300)


In the last equation (300) it is indicated that q = −1/λ.

Relations (298) and (300) specify the system of holonomic constraints, imposed on the

solution of the field equations of the problem.

Extracting currents jν , Nν and singlet potential Wν through current equations, we can

reduce the problem under consideration to the system of wave equations relative to four

potentials Zν and Wν :

�Zν + k2SZ

ν + pSp2T

3b

Ib

W ν = −pSpT3

h ν ;

�Wν −(4π

1e

1

W ν + k2SpSZ

ν 3e

)− p2T IW = pTh.

(301)

In the equations (301), parameter kS is determined by formula (281), and vector hν – by

relation (266). Nonholonomic constraint (273) is imposed on YM-potentials Wν .

10.2.2 Solution in the Form of a Plane Wave

Let us consider the solution of the problem (301) in the form of a plane wave. In such

solution potentials Zν and Wν depend only on a single argument, the wave phase φ =

kνxν . Since in the problem (301) there is isotropic vector Zν , wave vector kν can only be

space-like. In the wave intrinsic system, wave vector kν has only one nonzero component

in longitudinal coordinate x. D’Alembert operator in (301) is reduced to the operator of

the twofold differentiation in the longitudinal coordinate. Vector hν in the plane wave

vanishes. Field equations (301) turn into the system of ordinary differential equations:

Zν ′′ + k2SZ

ν +pSp2T

3b

Ib

W ν = 0;1

W ν ′′ − 4π1

W ν −p2T1b

Ib

W ν = 0;2

W ν ′′ −p2T2b

Ib

W ν = 0;3

W ν ′′ − k2SpSZ

ν −p2T3b

Ib

W ν = 0.

(302)

In equations (302), the prime stands for longitudinal coordinate x derivative.

Differential conditions (299), as well as in other problems of the waves with space-like

wave vector, mean that no potential has a longitudinal component:

Z1 = 0;a

W 1 = 0; a = 1, 3. (303)

Considering the wave with linear polarization, we can assume, without restricting gener-

ality, that isotropic vector Zν in the wave intrinsic system has the form:

Zν = T {1; 0; 1; 0} , (304)

where T is some desired function of longitudinal coordinate x.

Both holonomic and nonholonomic constraints, imposed on solution (302), will be satisfied


if we try YM-potentialsa

W ν in the following form:

1

W ν =

{1

T; 0;1

T; V

},

2

W ν =2

T {1; 0; 1; 0} ,3

W ν =3

T {1; 0; 1; 0} .

(305)

In five-component solution (304), (305), functions T,1

T,2

T,3

T and V are desired ones.

The choice of five-component structure of the solution is not the only possible method to

satisfy all the constraint equations. However, this partial choice of the solution structure

allows seeing the fundamental problem of unrestricted growth of the solutions to equations

(302).

YM-inertia tensor I for wave (305) has the form:

I = V2

⎛⎜⎜⎜⎜⎝0 0 0

0 1 0

0 0 1

⎞⎟⎟⎟⎟⎠ . (306)

Substitution of tensor (306) into field equations (302) for wave (304), (305) gives the

following system of differential equations:

V′′ − 4πV = 0,1

T ′′ − 4π1

T = 0,2

T ′′ − p2T V22

T = 0,3

T ′′ − p2T3

T −k2SpST = 0,

T′′ + k2ST + pSp

2T V2

3

T = 0.

(307)

The possible, but uninteresting situation of choice of the trivial solutions to the first three

equations of system (307) is: V = 0;1

T= 0,2

T= 0, returns us to the mymino wave consid-

ered above. Except for this degenerate situation, system (307) has no solutions restricted

at the growing of longitudinal coordinate x. The first two equations of the system (307),

linear and independent from the rest of the equations (307), have solutions, exponentially

growing with growth of x. Substitution of these growing solutions as coefficients of other

equations of the system generates unrestricted growth of the rest of the problem variables

(307).

10.2.3 Two Methods of Allowing for Current Restriction.Neutrino Instability

Is it possible to cope with this non-physical solution growth?

It is possible to take into account the restriction on pseudo-Euclidean module of current


jν (7). From current equations (295) and (305) follows that

jν jν = 4p2T1

W ν1

Wν= −4p2T V2,

consequently,

V2 ≤(

jT2pT

)2

. (308)

The growing solution of the system (307) is applicable only for such values of longitudinal

coordinate x while inequality (308) is satisfied.

At obtaining the equality in (308) with some x = xe, we have to reverse the signs of

all the derivatives of potentials V′, T′,1

T ′,2

T ′,3

T ′, while maintaining the continuity of

the potentials themselves and to continue the solution further along the longitudinal

coordinate x, with x > xe under the new initial conditions. This barbaric method

generates a restricted solution with discontinuities of derivatives in the form of some

ripple of potentials and currents, frozen in the intrinsic system of the wave. For correct

numerical description of this ripple, the curvature of space, caused by the wave itself

(307), should be taken into account.

The second, and probably, the deeper interpretation of inequality (308) consists of refusal

of the plane wave picture, covering the whole space-time. At obtaining the equality in

(308), the plane wave vanishes, the formation of some cavitated latebrae zones, not

containing current jν , takes place, and then the formation of the pomerium’s system,

specifying the outer boundaries of current zones, takes place. In other words, the two-

current wave turns into one or a few massive triplet particles, described in part 9.3.6,

surrounded by neutrino fields. However, it would be naive to expect that such an intimate

and, undoubtedly, quantum process of massive particle’s production permits the adequate

numerical description in terms of the classical Lagrangians.

Comparison of the equations, which describe mymino neutrino wave (10.1) with the

system of equations (307), describing the two-current singlet-triplet wave, allows to make

the following assumption.

Mymino neutrino wave (282), stable in itself, is unstable against the buildup of the two-

current state: no matter how small initial perturbations are (i.e. no matter how small

”inoculating” values of current1

J ν components are), the solution of the neutrino problem

turns fast into the solution of two-current singlet-triplet problem due to the exponential

growth of1

T and V, but this two-current wave without latens and pomerium quickly turns

into a massive YM-particle having inner latebrae voids, inner latens boundaries and outer

pomerium boundary. Within the framework of this picture, the neutrino production in

any process must be accompanied by the production of a massive YM-particle with a

very small time delay (probably, with delay of about r0/c ∝ 10−36 seconds)42.

42According to beliefs common for modern physics, such massive ”satellite” of neutrino is W-boson. We

do not know whether identification of W-boson with massive one-current particle is acceptable (p.9.3.6.)

W-boson is treated as massive quantum of free (zero-current) YM-field. The particle described in p.


10.3 Mixed Singlet-triplet Three-current One-neutrino State

10.3.1 The Lagrangian and Field Equations

Let us turn to the study of such a mixed state, where, besides neutrino singlet-triplet

current Nν , there are two more YM-currents:

Jν = N ν ;3

J ν = −N ν ; NνNν = 0;1

J ν1

Jν< 0;2

J ν2

Jν< 0;1

J ν2

Jν= 0. (309)

The last equality in the description of this state (309) results from current3

J ν isotropy

and the condition of the inner normalization of YM-triplet currents (32).

The Lagrangian of the state (309), in accordance with the type of base Lagrangian of

ST-theory (17), can be denoted in the form:

L =− 1

8p2T

(1

J ν1

Jν +2

J ν2

Jν

)− 1

2pT

(1

J ν1

Wν +2

J ν2

Wν

)− 1

2pTpSNνZν−

− 1

16π(WμνW

μν +Wμν ·Wμν)− λ

4pTpSN νNν .

(310)

In (310), Zν is the linear combination of potentials Wν and3

W ν (258) introduced before;

λ is the Lagranges multiplier. The last term in (310) is the ”penalty” for the current’s

neutrino character. Besides this ”penalty”, the Lagrangian (310) must contain one more

term – ”penalty” for currents1

J ν and2

J ν orthogonality. For the reasons expressed above,

in p. 9.4.1, we do not include this penalty into the Lagrangian (310), supposing that the

condition of currents1

J ν and2

J ν orthogonality is ”natural” for this problem.

Field equations, following from the Lagrangian (310), have the following form:

λN ν + Zν = 0;1

J ν + 2pT1

W ν = 0;2

J ν + 2pT2

W ν = 0; (”current equations”)

(311)

∂μWμν =

2π

pSNν ; (”Maxwell equations”)


2π

pT

(1

J ν 1e +

2

J ν 2e −N ν 3

e

). (”Yang-Mills equations”)

(312)

In equations (312)ae is YM-unit vectors.

From current equations (311) follows that the mixed potential Zν is isotropic, and YM-

potentials1

W ν and2

W ν are orthogonal to each other (under the orthogonality of the

corresponding currents (309)):

ZνZν = 0;1

W ν2

Wν= 0.(313)

9.3.6, consists of mixture of YM-current and YM-field, similarly to massive leptons, which are a mixture

of singlet current and singlet field.


Differential conditions for the currents of YM-triplet (19) and current equations (311)

generate algebraic conditions for potentials Zν anda

W ν :

1

Wν

(2pT

3

W ν − qZν

)= 0;

2

Wν

(2pT

3

W ν − qZν

)= 0.

(314)

In equation (314) it is denoted that q = −1/λ.

Relations (313) and (314) are holonomic constraints, which the solution of the field equa-

tions (312) obeys.

Extraction of currents and singlet potential from field equations (312) leads to the fol-

lowing system of wave equations relative to potentials Zν anda

W ν :

�Zν + k2SZ

ν + pSp2T

3b

Ib

W ν = 0;

�Wν − 4π

(1

W ν 1e +

2

W ν 2e

)− k2

SpSZν 3e −p2T IWν = pTh

ν .(315)

In system (315), describing the three-current state, all notations are the same as in the

similar system of equations (301), describing the two-current state.

10.3.2 Solution in the Form of a Plane Wave

For plane wave hν ≡ 0, all variables of the problem depend on longitudinal coordinate x,

and wave equations take the form which is similar to equations (302), introduced for the

two-current problem:

Zν ′′ + k2SZ

ν + pSp2T

3b

Ib

W ν = 0;1

W ν ′′ − 4π1

W ν − p2T1b

Ib

W ν = 0;2

W ν ′′ − 4π2

W ν − p2T2b

Ib

W ν = 0;3

W ν ′′ − k2SpSZ

ν − p2T3b

Ib

W ν = 0.

(316)

All the potentials of Zν anda

W ν , have zero longitudinal potential Z1 anda

W 1.

Solution to problem (316) has to satisfy the four holonomic constraints (313), (314) and

the three nonholonomic constraints (273). We do not know whether there are any solu-

tions to the system of differential equations (316), burdened with so many constraints,

anyway, we cannot construct a solution of some simple structure, which would automati-

cally account all the constraint equations. If solutions do exist, they grow unrestrictedly

along the wave direction. Taking into account the restriction on the growth of pseudo-

Euclidean current module, as described in p. 10.2.2, it is possible to construct restricted

solutions having discontinuities of the variables.


10.4 Mixed Singlet-triplet Three-current Neutrinoless State

(the left-hand three-current state)

10.4.1 The Lagrangian of the Left-hand Three-current State

Let us consider such a mixed triplet-singlet state, where the same space-like current lν

(lνlν < 0) is present both as singlet current J and the third component of YM-triplet of

currents3

J ν :

Jν = lν ;3

J ν = lν . (317)

This current state is compatible with formulas of current decomposition (22). Within

the framework of this decomposition, current lν is the ”left-hand” current. The right-

hand current rν and neutrino current Nν are missing from the state (317). The algebraic

conditions of coupling of currents Jν and3

J ν (20) in state (317) turn into identity.

In state (317) two other YM-currents,1

J ν and2

J ν , must be present, since, due to the

conditions of the inner normalization of YM-triplet currents (32), their scalar product is

nonzero:

21

J ν2

Jν= lνlν . (318)

The substitution of formulas (317) into the base Lagrangian of the ST-theory (17) allows

us to write the Lagrangian of the left-hand three-current states in the following form:

L = Lcur + Lint + Lf , (319)

where the current Lagrangian Lcur contains pseudo-Euclidean squares of all three currents

of state:

Lcur = −lνlν

8pSpT− 1

8p2T

(1

J ν1

Jν +2

J ν2

Jν

). (320)

The interaction Lagrangian Lint is convenient to be written as follows:

Lint = −1

2pSpTlνQν −

1

2pT

(1

J ν1

Wν +2

J ν2

Wν

), (321)

where Qν is the linear combination of potentials Wν and3

W ν :

Qν = pTWν + pS

3

W ν .

The field Lagrangian Lf in (319) has an ordinary form (13).

We have to note that usually in Weinberg-Salam theory it is not potential Qν itself that

is used, but its decomposition into electromagnetic potential Aν

(Aν = pTW

ν + pS3

W ν

)and ”neutral” potential Zν

(Zν = pTW

ν − pS3

W ν

):

Qν = 2pSpTAν −

(p2S − p2T

)Zν ,

or Qν = sin2ϑW · Aν − cos2ϑW · Zν .(322)


If we use decomposition (322), the interaction Lagrangian (321) can be denoted as:

Lint = −lνAν + ctg2ϑW · lνZν −1

2pT

(1

J ν1

Wν +2

J ν2

Wν

), (323)

Expression (323) allows us to speak, using the generally accepted ”jargon” of theoreti-

cal physics, about the fact that the left-hand current lν ”interacts” with electromagnetic

potential Aν (the first term in (323)), and ”interacts” with neutral field Zν (the second

term in (323)). At full formal equality of (321) and (323), formula (321) seems more

convenient and compact: the interaction Lagrangian (321) contains only one scalar prod-

uct, which includes current and combined potential Qν – it is the one that the left-hand

current lν ”interacts” with43.

While recording the three-current state Lagrangian (319), it is convenient to introduce

rescaled currents, assuming that

mν =1

2pSpTlν ;

1

j ν =1

2pT

1

J ν ;2

j ν =1

2pT

2

J ν ,(324)

and also the conventional YM-triplet of currents jν with zero third YM-component:

jν =

{1

j ν ;2

j ν ; 0

}. (325)

In notations (324), (325), the three-current state Lagrangian takes the form which is

compact enough:

L = −1

2mνmν −

1

2jνjν −mνQν − jν ·Wν −

1

16π(WμνW

μν +Wμν ·Wμν) . (326)

Weinberg coefficients pS and pT , which are missing from the Lagrangian (326) an explicit

form, of course, cannot be excluded from the description of mixed singlet-triplet state by

any scale transformation. They remain in the determination of potential Qν (322), in the

expression for YM-field tensor Wμν (15) and in normalizing condition (318), which has

the following form in the notations (324):

21

j ν2

jν= p2S mνmν . (327)

43Acknowledging the acceptability of using the stable term ”interaction Lagrangian”, rooted due to his-

torical reasons, we tend to consider the term ”interaction” (of currents and fields) itself, as an archaic

and inaccurate one. In three-current state, current lν and potential Qν , while interweaving, form an

integrated physical object – a ”three-current state”, and it would be incorrect to speak about ”inter-

action” of current and field – as well as, by reasons, cited in p. 10.1.2, it is incorrect to speak about

interaction of neutrino current Nν and neutral potential Zν , which form an integrated physical object

– Maxwell-Yang-Mills neutrino. The term ”interaction” is appropriate only as a term that allows to

provide approximate description of quasi-stationary states that have two or more current zones where

each of them has its own pomerium, and these zones are located far enough from each other.


Condition (327) is a holonomic constraint, imposed on functional arguments of the La-

grangian (326). Generally speaking, accounting of this constraint requires additional term

Lad for the base Lagrangian of the three-current problem (326):

Lad = η

(2

1

j ν2

jν −p2S mνmν

), (328)

with undetermined Lagrange’s multiplier η.

By introducing additional term (328) into the Lagrangian, with η �= 0, we actually

introduce some ”Higgs-like” field η which stimulates us to satisfy the condition (327).

It is reasonable to try to construct a theory of three-current states on the basis of the

Lagrangian (326), supposing that condition (327) is ”natural”, i.e. can be also satisfied

with η = 0. If non-trivial solutions to the three-current problem do not exist with η = 0,

three-current states may exist only as non-stationary transition states between other

states.

10.4.2 Field Equations of the Left-hand Three-current State

The independent functional arguments of the Lagrangian are currents mν and jν , and

potentials Wν and Wν . By varying the Lagrangian (326) by these arguments, we obtain

field equations of the left-hand three-current state:

mν +Qν = 0;1

j ν+1

W ν = 0;2

j ν+2

W ν = 0. (”current equations”)

(329)

∂μWμν = 4πpTm

ν . (”Maxwell equations”) (330)

∂μWμν + pT Wμ ×Wμν = 4π

(1

j ν1e +

2

j ν2e +pSm

ν 3e

). (”Yang-Mills equations”) (331)

General gauge conditions by 4-divergence of potentials should be added to field equations

(329), (330), (331):

∂μWμ = 0; ∂μW

μ = 0. (332)

Conditions (332) are combined with the conditions for conservation of all the three cur-

rents of the problem:

∂νmν = 0; ∂ν

1

j ν = 0; ∂ν2

j ν = 0. (333)

and, in their turn, conditions (333), along with general differential condition for triplet

current, generate the system of holonomic constraints imposed on the solution of the field

equations of the left-hand three-current state:

pSmν

1

Wν=1

j ν3

Wν ;

pSmν

2

Wν=2

j ν3

Wν ;1

Wν

2

j ν =2

Wν

1

j ν .

(334)


As a result of adding currents coupling (327) to expressions (334), we obtain a complete

system of equations describing the left-hand three-current state under consideration in

terms of currents and potentials. Exclusion of currents and singlet potential Wν from this

system of equations results in more compact description containing only four potentialsa

Wν and Qν :

−�Qν + 4πQν + pSpT3

h ν + pSp2T

3b

Ib

W ν = 0;

−�1

W ν + 4π1

W ν + pT1

h ν + p2T1b

Ib

W ν = 0;

−�2

W ν + 4π2

W ν + pT2

h ν + p2T2b

Ib

W ν = 0;

−�3

W ν + 4πpSQν + pT

3

h ν + p2T3b

Ib

W ν = 0,

(335)

where vector hν , introduced by formula (266), vanishes for plane waves.

The system of four field equations (335) is supplemented by conditions (332) and holo-

nomic constraints (327) and (334). With regard to current equations (329), one of equa-

tions (334) transforms into an identity, and the remaining constraint equations take the

form:

21

W ν2

Wν= p2SQν Qν ;

1

W ν Rν = 0;2

W ν Rν = 0;

where

Rν = pSQν−3

Wν . (336)

10.4.3 Left-hand Three-current state as a plane wave

Solution to the system of wave equations (335) can be tried in the form of a plane wave

with wave vector kν = {ω; k}. If vector kν is time-like(ω2 > k2

), in the intrinsic wave

system k = 0 and all potentials and currents in this system depend only on time, and ac-

cording to (332) time components of potentials vanish. Let us introduce the conventional

three-dimensional notations for potentials in the intrinsic system of the plane wave:

Qν = {0; U} ;a

W ν =

{0;

a

U

}. (337)

In notations (337), wave equations (335) in the intrinsic wave system take the form of a

system of four ordinary differential equations:

U•• + 4πU+ pSp2T

3b

Ib

U= 0;1

U •• + 4π1

U +p2T1b

Ib

U= 0;2

U •• + 4π2

U +p2T2b

Ib

U= 0;3

U •• + 4π pS U+ p2T3b

Ib

U= 0.

(338)


In these equations (338), the dot stands for time derivative in the wave intrinsic system

and, as usual:ab

I= −a

U ·b

U +ab

δc

U ·c

U, (339)

Equations of holonomic constraints (335) in three-dimensional notations (337) take the

form:

21

U ·2

U= p2S U2;

1

U ·R = 0;2

U ·R = 0;

(340)

where

R = pS U−3

U . (341)

Nonholonomic constraints (338) are imposed on solution of the system:

abcε

b

U ·c

U • = 0. (342)

Coupling equations (342), as well as in other problems studied above, are the restrictions

imposed on the initial conditions of the system (338).

Let us use the term lethos (left-hand threecurrent oscillator) to denote the object obeying

the system of differential equations (338), holonomic constraints (340) and nonholonomic

constraints (342). We do not have a theorem which would guarantee the existence of the

object lethos.

It is possible to demonstrate two partial solutions to this problem:

1. If we suppose that√2

1

U=√2

2

U=3

U= pSU, (343)

all constraint equations will be satisfied, non-linear terms in equations (338) will vanish,

and we will obtain the equation of harmonic oscillator for determination of vector U:

U•• + 4πU = 0. (344)

From equation (344) follows that vector U executes harmonic oscillations in the intrinsic

system with frequency ωS =√4π. So, Lorentz-invariant pseudo-Euclidean square of the

wave vector is equal to 4π (kνkν = ω2S = 4π).

The singlet-triplet wave lethos of the form (343), (344) is the three-current analogue of

the considered in the article [1] one-current singlet object – ”heavy photon”.

2. We may take into account that three-current wave (338), similarly to other triplet and

singlet-triplet current waves, in contrast to free (zero-current) Yang-Mills waves, does not

have a large-scale invariance in the wave amplitude (or energy) – it is possible to dis-

tinguish the behavior of small – and large amplitude waves. For small-amplitude waves

there is appropriate a procedure of problem linearization, with which in wave equations

(338) the non-linear terms of the formab

Ib

U are omitted as small quantities of the third

amplitude order (free Yang-Mills waves are essentially-nonlinear and do not allow the


linearization procedure).

In such linearized formulation of the problem we may not subject the variables of the

problem to rigid condition (343), and, in this way, remove the strong geometric degen-

eration of wave (condition (343) transforms the three-dimensional object lethos into an

uninteresting one-dimensional structure).

All vectors U anda

U of the problem in such approximate linearized formulation execute

synchronous harmonic oscillations with frequency ωS =√4π. Vectors U and

3

U are or-

thogonal to the plane stretched on vectors1

U and2

U. Relation 21

U ·2

U= p2S U2 links the

amplitudes of synchronously oscillating motions of vectors U,1

U and2

U.

In this approximate solution, in contrast to the exact, but geometrically degenerate solu-

tion (343), chiral determinant CD

(1

U,2

U,3

U

)is not identically zero, and we can submit

the solution to the condition of chiral determinacy .

System of equations (338) describes the left-hand three-current wave with time-like wave

vector . It is possible to consider a similar wave with space-like wave vector. The solution

of such problem (similar to the other wave problems with a space-like vector, considered

above) in the intrinsic wave system is some ”frozen transverse ripple” of currents and

potentials with unrestricted amplitude growth along the longitudinal wave coordinate.

The solution growth can be restricted by imposing the requirement of chiral determinacy

of the solution and the condition of restriction of pseudo-Euclidean module of currents.

10.5 Mixed Singlet-triplet Four-current One-neutrino State

(the Left-hand Four-current State)

10.5.1 Left-hand Four-current State Lagrangian

Let us consider such a singlet-triplet state, in which one couple of orthogonal currents

– left-hand current lν and neutrino current Nν , form both a singlet current Jν and the

third component of YM-triplet of currents3

J ν :

Jν = lν +Nν ;3

J ν = lν −N ν ; lνNν = 0; NνNν = 0; lνlν < 0. (345)

Besides this couple of currents in the state under consideration, there must also be avail-

able two other YM-currents1

J ν and2

J ν , since their scalar product is nonzero:

21

J ν2

Jν= lνlν . (346)

Substitution of relations (345) into the base Lagrangian of the ST-theory (17) results in

the following Lagrangian expression of the left-hand four-current problem:

L = −1

2mνmν−

1

2jνjν−mνQν−nνZν−jν ·Wν−

1

16π(WμνW

μν +Wμν ·Wμν)− 1

2λnν nν .

(347)


In the Lagrangian notation (347) there have been used the same rescaled currents mν

and jν (324), (325), which were used in the left-hand three-current Lagrangian notation

(326). The linear combinations of singlet and triplet potential Qν (322) and Zν (258),

introduced above, are also used here. Current nν in the Lagrangian (347) is a rescaled

neutrino current Nν :

nν =1

2 pTNν . (348)

The last term in (347) is a ”penalty” for neutrino character of current nν ; λ is the

Lagranges multiplier of this penalty.

The algebraic constraints, imposed on currents (345), (346) in these notations take the

form:

nν nν = 0; mν nν = 0; 21

j ν2

jν= p2Smνmν , (349)

mν mν < 0;1

j ν1

jν< 0;2

j ν2

jν< 0. (350)

From three constraints (349) in the Lagrangian (347) only the condition of current nν

isotropy is included as a ”penalty” term. The Lagrangian extension (347) of the term of

the form μmν nν (”penalty” for condition of orthogonality of currents mν and nν , μ is

the Lagrange’s multiplier) results in such current equations, with which the conditions

mν nν = 0 and mν mν < 0 turn out to be incompatible if μ �= 0.

10.5.2 Field Equations of the Left-hand Four-current State

From the Lagrangian of the form (347) , the following field equations describing the

left-hand four-current state, follow:

mν +Qν = 0;

λnν + Zν = 0;

a

j ν+a

W ν = 0; a = 1, 2. (”current equations”) (351)

∂μWμν = 4πpT (mν + nν) . (”Maxwell equations”) (352)

∂μWμν + pT Wμ ×Wμν = 4π

(jν + pS (m

ν − nν)3e). (”Yang-Mills equations”) (353)

By extracting the currents from Maxwell and Yang-Mills equations (351), and by extract-

ing auxiliary potentials Qν and Zν , the problem of the left-hand four-current state can

be reduced to the system of wave equations relative to potentials Wν and Wν :

−�Wν + 4πp2T (1− q)Wν + 4π pSpT (1 + q)3

W ν = 0;

−�a

W ν + 4πa

W ν+a

h ν+ab

Ib

W ν = 0, a = 1, 2;

−�3

W ν + 4πp2S (1− q)3

W ν + 4πpSpT (1 + q)Wν+3

h ν + p2T3b

Ib

W ν = 0,

(354)


where q = −1/λ, vector h

ν is determined in (266),ab

I is Yang-Mills inertia tensor.

Potentials Wν and Wν have zero 4-divergences and obey the holonomic constraints fol-

lowing from the conditions (349):

p2TWνWν = p2S

3

W ν3

Wν ;

pS3

W ν3

Wν= pTWν

3

Wν ;

pTWνWν = pSW

ν3

Wν ;

21

W ν2

Wν= 4p2Tp2SW

νWν .

(355)

It is obvious that equations (355) are not independent.

Besides the constraints (355), the potentials obey two auxiliary constraints following from

the condition for YM-currents (19):

a

W νRν = 0; a = 1, 2, (356)

where

Rν = pSpT (1 + q)Wν −(p2T + qp2S

) 3

Wν . (357)

10.5.3 Left-hand Four-current State as a Plane Wave

Having an intention to construct the solution to the system of wave equations (354) in

the form of a plane wave, we have to consider only transverse waves with space-like vector

kν . (The presence of isotropic vectors nν and Zν in the problem excepts the existence of

a wave with a time-like wave vector). In the plane wave, vectora

h ν vanishes. In the wave

intrinsic system, D’Alembert operator � is reduced to the second longitudinal coordinate

derivative. By using complex three-dimensional YM-vectors τ ,aτ instead of 4-potentials

Wν anda

W ν , the system of equations (354) can be reduced to the system of ordinary

differential equations:

− τ ′′ + 4πp2T (1− q) τ + 4πpSpT (1 + q)3τ= 0;

− aτ ′′ + 4π

aτ +

bτ ×

(aτ × b

τ)= 0; a = 1, 2

− 3τ ′′ + 4πp2S (1− q)

3τ +4πpSpT (1 + q) τ+

bτ ×

(3τ × b

τ)= 0.

(358)

Coupling equations (355) and (356) for vectors τ ,aτ take the form:

2p2Tτ · τ = p2S3τ · 3

τ

pS3τ · 3

τ= pTτ ·3τ ;

pTτ · τ = pSτ ·3τ ;

1τ · 2

τ= 4p2Tp2Sτ · τ ,

(359)


andaτ ·ρ = 0; a = 1, 2, (360)

where

ρ = pSpT (1 + q) τ −(p2T + qp2S

) 3τ . (361)

Besides holonomic constraints (359), (360), nonholonomic constraints, which are sufficient

to take into account in the initial conditions for (358), are also imposed on the system of

motion (358):abcε

bτ · c

τ ′ = 0; a = 1, 2, 3. (362)

The system of motion (358) is so overloaded with auxiliary algebraic conditions (359),

(360), that the very fact of the existence of solutions is doubtful. But if solutions (358),

satisfying the constraints (359), (360), do exist, they grow unrestrictedly along the lon-

gitudinal coordinate of the wave. To restrict the solution growth, like in the other wave

problems, it is possible to use the condition of chiral determinacy of the solution and the

condition of restriction of pseudo-Euclidean module of currents.

10.6 Total Four-current Wave State

The base Lagrangian of four-current state is determined by the formula (17):

L = − 1

8p2SJνJν −

1

8p2TJν · Jν −

1

2pSJνWν −

1

2pTJν ·Wν −

1

16π(WμνW

μν +Wμν ·Wμν) .

(363)

Additional algebraic conditions, imposed on the four-current state, are determined by

formulas (20) and (32):3

J ν

(3

Jν −Jν)

= 0;

21

J ν2

Jν=3

J ν3

Jν .

(364)

Like in other wave problems, considered above, at the derivation of the field equations,

we will directly use the base Lagrangian (363), without ”extending” it with additional

conditions up to the ”effective” Lagrangian (36). By doing so, we interpret the conditions

(364) as ”natural constraints” for some of the four-current states and as the conditions

for forbidding the existence of other states which are incompatible with the constraints

(364).

The Lagrangian (363) generates the following current and field equations of the four-

current state:Jν + 2pSW

ν = 0;

Jν + 2pTWν = 0;

(”current equations”).

(365)

∂μWμν =

2π

pSJν , (”Maxwell equations”) (366)



2π

pTJν . (”Yang-Mills equations”) (367)

By extracting currents from field equations (366) and (367), by means of current equations

(365), it is possible to reduce the problem of four-current state to the system of wave

equations relative to the potentials Wν and Wν :

−�Wν + 4πWν = 0; (368)

−�Wν + 4πWν + pThν + p2T I ·Wν = 0, (369)

where, as previously,

hν = Wμ × (2∂μWν − ∂νWμ) ;

ab

I=a

Wμb

Wμ −ab

δc

Wμ

c

Wμ.

The solutions to wave equations (368) and (369) obey the conditions of zero 4-divergence

of potentials:

∂μWμ = 0; ∂μW

μ = 0.

Wave equations (368) and (369) are independent from each other, but their solutions

obey the constraints following from (364) and (365):

pT3

W ν3

Wν= pS3

W νWν ; (370)

21

W ν2

Wν=3

W ν3

Wν . (371)

By constructing solution to the four-current problem in the form of a plane wave with

time-like wave vector, we can reduce equations (368) and (369) to the system of ordinary

differential equations relative to four three-dimensional vectors U,a

U in the wave frame

of reference (the dot next to the letter stands for a time derivative in the wave intrinsic

system):

U•• + 4πU = 0; (372)

a

U •• + 4πa

U +p2Tab

Ib

U= 0, (a = 1, 2, 3) . (373)

In the wave intrinsic system there are no time components of potentials:

Wν = {0; U} ;a

W ν =

{0;

a

U

}.

Yang-Mills inertia tensor has the form:

ab

I=ab

δc

U ·c

U −a

U ·b

U .

Coupling equations ((370), (371) take the form:

pT3

U 2 = pS3

U ·U; (374)


21

U ·2

U=3

U 2. (375)

The system of equations (372), (373) with conditions (374), (375) has one obvious and

rather primitive solution of the following form:

1

U=1√2V;

2

U=1√2V;

3

U= V; U =pTpS

V. (376)

For solution (376) all four vectors U,a

U are parallel, the coupling equations (374) and

(375) are satisfied, non-linear terms in (377) vanish:

ab

Ib

U= 0, (377)

vector V is a harmonic vector, oscillating at frequency√4π:

V•• + 4πV = 0. (378)

Four-current wave (376), (378) disappoints with its primitiveness it is just a four-time

replicated ”heavy photon” considered in the article [1]. In this wave, Yang-Mills’ non-

linearity vanishes due to relation (377). But the constraint condition, coordinating the

dynamics of a nonlinear triplet oscillator (374) with the dynamics of a linear singlet os-

cillator (374), is so rigid that, apparently, there are no other four current states besides

the state (376), (378): the four currents of the singlet-triplet theory are too ”tight” in

the same point of the four-dimensional space-time continuum. This fact causes trouble

and doubt of the usefulness of the four-current Lagrangian itself.

If we afford to neglect the condition (374), intermixing currents of the singlet and triplet

sectors, there appears possibility in the triplet sector to construct the waves of a more

complicated form than a linear wave (376). For example, it is possible to construct a

solution, where vector3

U is orthogonal to vectors1

U and2

U which coincide with each

other:1

U=2

U= u {1; 0; 0};3

U= u {0; 0;√2}. (379)

For the wave of the form (379) the condition of triplet auto-normalization (375) is satis-

fied, and Yang-Mills inertia tensor has the form:

I = u2

⎛⎜⎜⎜⎜⎝3 −1 0

−1 3 0

0 0 2

⎞⎟⎟⎟⎟⎠ . (380)

Substitution of (379) and (380) into Yang-Mills equations (373) gives Duffing equation

for the amplitude function u:

u•• + 4πu+ p2Tu3 = 0. (381)


The system described by (379), (381) can be named Duffing-Yang-Mills’ oscillator (duy-

mos). The object duymos, undoubtedly, belongs to Yang-Mills’ mathematics, however

it does not exist as an object of Maxwell-Yang-Mills’ singlet-triplet physics, in view of

impossibility to construct a solution to Maxwell’s equations (372) which would match it

by (374).

Summary

In the present article we have made an extensive attempt of a consistent interpretation of

the singlet-triplet theory as a sector of the classical field theory. Each field (both singlet

and triplet) is treated as a dyad consisting of a current and a potential. Particles and

waves emerge as solutions to field equations. There have been considered a number of

wave states, – in contrast to electrodynamics, there are about two dozens of wave types in

the singlet-triplet theory. Numerical investigations of some wave states will be provided

in the other articles of this series. It is unknown how to compare these objects of the

theory with some objects of experimental physics that allow observation.

The most vulnerable point of the theory, developed here, is the absence of theorems of the

existence of stationary problems solutions. It is these problems that provide a classical

description of massive elementary particles. If stationary states exist, the corresponding

solutions can be constructed numerically. If solutions to stationary problems do not exist,

the current part of the Lagrangian has to be complicated with addition of, for example,

an expression, quadratic in the tensor of currents Gμν , Gμν , into the Lagrangian of each

sector of physics:

Gμν = ∂μJν − ∂νJμ, (singlet current tensor)

Gμν = ∂μJν − ∂νJμ + pTJμ × Jν (triplet current tensor).

However, this Lagrangian extension dramatically changes the current equations of the

theory, and to some extent makes the postulate of space-like character of any currents

that appear in the classical field theory, less convincing.

Hopefully, among the readers of this article there will appear a mathematician able to

formulate correctly and prove the existence theorem for stationary states.

The second vulnerable point of the theory is the method of accounting the chirality prob-

lem suggested in the article (soft chiralization procedure). This method forms unsmooth

solutions with derivative discontinuities. Not every physicist will agree to pay such a

price for the chiral definitiveness of solutions to field equations.

References

[1] Temnenko V.A., Physics of currents and potentials. I. Classical electrodynamics withnon-point charge. – Electronic Journal of Theoretical Physics, 11, No. 31, 2014. –pp. 221–256.

[2] Landau L.D., Lifshitz E.M., The classic theory of fields. 4th ed. Butterworth -Heinemann, 1975.


[3] Yang C.N., Mills R.L., Conservation of isotopic spin and isotopic gauge invariance.- Phys. Rev., 1954, 96, 1. pp. 191–195.

[4] Straumann N., On Paulis invention of non-abelian Kalutza-Klein theory in 1953.Arhiv: gr-qc/002054v1, 15 Dec. 2000.

[5] Kane G., Modern elementary particle physics. Perseus Books, Addison-Wesley Pubc.Co., Inc., 1987.

[6] Dirac P.A.M., A new classical theory of electrons. - Proc. Roy. Soc. A, 1951, vol. 209,pp. 291–296.

[7] Glashow S.L., Partial symmetries of weak interactions. Nuclear Physics, 1961, 22,pp. 579-588.

[8] Rouse Ball W.W., Histoire des mathematiques. Paris, Libraire Scientifique A.Hermann., 1906, pp. 104-105.

[9] Chaunu P., La cuvilization de l’Europe des lumieres. Paris, Arthad, 1971.


From Quantum Mechanics to Intelligence

Michail Zak∗

Jet Propulsion Laboratory California Institute of Technology, Pasadena, CA 91109,USA

Received 06 September 2014, Accepted 19 December 2014, Published 10 January 2015

Abstract: The challenge of this work is to connect quantum mechanics with the concept of

intelligence. By intelligence we understand a capability to move from disorder to order without

external resources, i.e. in violation of the second law of thermodynamics. The objective is to find

such a mathematical object described by ODE that possesses such a capability. The proposed

approach is based upon modification of the Madelung version of the Schrodinger equation by

replacing the force following from quantum potential with non-conservative forces that link to

the concept of information. A mathematical formalism suggests that a hypothetical intelligent

particle, besides the capability to move against the second law of thermodynamics, acquires

such properties like self-image, self-awareness, self- supervision, etc. that are typical for Livings.

However since this particle being a quantum-classical hybrid acquires non-Newtonian and non-

quantum properties, it does not belong to the physics matter as we know it: the modern physics

should be complemented with the concept of the information force that represents a bridge to

intelligent particle. It has been suggested that quantum mechanics should be complemented by

the intelligent particle as an independent entity, and that will be the necessary step to physics of

Life. At this stage, the intelligent particle is introduced, as an abstract mathematical concept

that is satisfied only mathematical rules and assumptions, and its physical representation is still

an open problem.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Quantum Mechanics; Intelligence; Thermodynamics; Quantum Information;

Madelung equation

PACS (2010): 03.65.-w; 03.67.-a; 05.70.-a; 47.35.-i; 07.05.Mh

1. Introduction

The recent statement about completeness of the physical picture of our Universe made in

Geneva raised many questions, and one of them is the ability to create Life and Mind out

of physical matter without any additional entities. The main difference between living and



non-living matter is in directions of their evolution: it has been recently recognized that

the evolution of livings is progressive in a sense that it is directed to the highest levels of

complexity if the complexity is measured by an irreducible number of different parts that

interact in a well-regulated fashion. Such a property is not consistent with the behavior

of isolated Newtonian systems that cannot increase their complexity without external

forces. That difference created so called Schrodinger paradox: in a world governed by the

second law of thermodynamics, all isolated systems are expected to approach a state of

maximum disorder; since life approaches and maintains a highly ordered state – one can

argue that this violates the Second Law implicating a paradox,[1].

But livings are not isolated due to such processes as metabolism and reproduction:

the increase of order inside an organism is compensated by an increase in disorder outside

this organism, and that removes the paradox. Nevertheless it is still tempting to find a

mechanism that drives livings from disorder to order. The purpose of this paper is to

demonstrate that moving from a disorder to order is not a prerogative of open systems:

an isolated system can do it without help from outside. However such system cannot

belong to the world of the modern physics: it belongs to the world of living matter, and

that lead us to a concept of an intelligent particle – the first step to physics of livings.

In order to introduce such a particle, we start with an idealized mathematical model

of livings by addressing only one aspect of Life: a biosignature, i.e. mechanical

invariants of Life, and in particular, the geometry and kinematics of intelligent

behavior disregarding other aspects of Life such as metabolism and reproduction. By

narrowing the problem in this way, we are able to extend the mathematical formalism of

physics’ First Principles to include description of intelligent behavior. At the same time,

by ignoring metabolism and reproduction, we can make the system isolated, and it will

be a challenge to show that it still can move from a disorder to the order.

2. Starting with Quantum Mechanics

The starting point of our approach is the Madelung equation that is a hydrodynamical

version of the Schrodinger equation

∂ρ

∂t+∇ • ( ρ

m∇S) = 0 (1)

∂S

∂t+ (∇S)2 + F −

�2∇2√ρ2m√ρ

= 0 (2)

Here ρ and S are the components of the wave functionψ =√ρeiS/�, and � is the Planck

constant divided by 2π. The last term in Eq. (2) is known as quantum potential.

From the viewpoint of Newtonian mechanics, Eq. (1) expresses continuity of the flow

of probability density, and Eq. (2) is the Hamilton-Jacobi equation for the action S of

the particle. Actually the quantum potential in Eq. (2), as a feedback from Eq. (1) to

Eq. (2), represents the difference between the Newtonian and quantum mechanics, and

therefore, it is solely responsible for fundamental quantum properties.


The Madelung equations (1), and (2) can be converted to the Schrodinger equation

using the ansatz √ρ = Ψexp(−iS/h) (3)

where ρ and S being real function.

In order to associate quantum potential with the concept of information, recall that

information is an indirectly observed quantity that is defined via entropy as a measure of

unpredictability: For a random variable X with n outcomes, the Shannon information

denoted by H(X), is

H(X) = −n∑

i=1

ρ(xi) logb ρ(xi) (4)

In our further applications we will use the continuous version of this formula

H(X) = −∞∫

−∞

ρ(x) ln ρ(x)dx (5)

Actually our approach is based upon a modification of the Madelung equation, and in

particular, upon replacing the quantum potential with a different Liouville feedback, Fig.1

Figure 1 Classic Physics, Quantum Physics and Physics of Life.

In Newtonian physics, the concept of probability ρis introduced via the Liouville

equation∂ρ

∂t+∇ • (ρF) = 0 (6)

generated by the system of ODE

dv

dt= F[v1(t), ...vn(t), t] (7)

where v is velocity vector.

It describes the continuity of the probability density flow originated by the error

distribution

ρ0 = ρ(t = 0) (8)


in the initial condition of ODE (8).

Let us rewrite Eq. (2) in the following form

dv

dt= F[ρ(v)] (9)

where v is a velocity of a hypothetical particle.

This is a fundamental step in our approach: in Newtonian dynamics, the probability

never explicitly enters the equation of motion, [2,3]. In addition to that, the Liouville

equation generated by Eq. (9) is nonlinear with respect to the probability density ρ

∂ρ

∂t+∇ • {ρF[ρ(V)]} = 0 (10)

and therefore, the system (9),(10) departs from Newtonian dynamics. However although

it has the same topology as quantum mechanics (since now the equation of motion is

coupled with the equation of continuity of probability density), it does not belong to it

either. Indeed Eq. (9) is more general than the Hamilton-Jacoby equation (2): it is not

necessarily conservative, and F is not necessarily the quantum potential although further

we will impose some restriction upon it that links F to the concept of information, [3].

The relation of the system (9), (10) to Newtonian and quantum physics is illustrated in

Fig.1.

Remark. Here and below we make distinction between the random variable v(t) and

its values V in probability space.

Prior to considering a specific form of the force F, we will make a comment concerning

the normalization constrain satisfaction∫V

ρdV = 1 (11)

in which V is the volume where Eqs. (9) and (10) are defined.

Turning to Eq. (10) and integrating it over the volume V

∂

∂t

∫V

ρdV = −∫V

dV∇ • {ρF[ρ(V)]} = −∮Φ

dΦ∇ • (ρF) = 0 (12)

if

ρ = 0, |F| <∞ at Φ (13)

where Φ is the surface bounding the volume V.

Therefore, if the normalization constraint (9) is satisfied at t = 0, it is satisfied for all

the times.

3. Information Force Instead of Quantum Potential

In this section we propose the structure of the force F that plays the role of a feed-

back from the Liouville equation (10) to the equation of motion (9). Turning to one-

dimensional case, let us specify this feedback as

F = c0 +1

2c1ρ−

c2ρ

∂ρ

∂v+

c3ρ

∂2ρ

∂v2(14)


c0 > 0, c1 > 0, c3 > 0 (15)

Then Eq.(9) can be reduced to the following:

v = c0 +1

2c1ρ−

c2ρ

∂ρ

∂v+

c3ρ

∂2ρ

∂v2(16)

and the corresponding Liouville equation will turn into the following PDE

∂ρ

∂t+ (c0 + c1ρ)

∂ρ

∂V− c2

∂2ρ

∂v2+ c3

∂3ρ

∂V 3= 0 (17)

This equation is known as the KdV-Bergers’ PDE. The mathematical theory behind

the KdV equation became rich and interesting, and, in the broad sense, it is a topic of

active mathematical research. A homogeneous version of this equation that illustrates

its distinguished properties is nonlinear PDE of parabolic type. However a fundamental

difference between the standard KdV-Bergers equation and Eq. (17) is that Eq. (17)

dwells in the probability space , and therefore, it must satisfy the normalization

constraint ∞∫−∞

ρdV = 1 (18)

However as shown in [4], this constraint is satisfied: in physical space it expresses conser-

vation of mass, and it can be easily scale-down to the constraint (18) in probability space.

That allows one to apply all the known results directly to Eq. (17). However it should

be noticed that all the conservation invariants have different physical meaning: they are

not related to conservation of momentum and energy, but rather impose constraints upon

the Shannon information.

In physical space, Eq. (17) has many applications from shallow waves to shock waves

and solitons. However, application of solutions of the same equations in probability

space is fundamentally different. In the next sections we present two phenomena that

exist neither in Newtonian nor in quantum physics.

Remark . Another version of an intelligent particle that is based upon a feedback

different from that expressed by Eq. (14) was considered in [2,3,8].

4. Emergence of Randomness

In this section we discuss a fundamentally new phenomenon: transition from determinism

to randomness in ODE that coupled with their Liouville PDE.

In order to complete the solution of the system (16), (17), one has to substitute the

solution of Eq. (17):

ρ = ρ(V, t) at V = v (19)

into Eq.(16). Since the transition from determinism to randomness occurs at t → 0, let

us turn to Eq. (17) with sharp initial condition

ρ0(V ) = δ(V ) at t = 0, (20)


Then applying one of the standard analytical approximations of the delta-function, one

obtains the asymptotic solution

ρ =1

t√πe−

V 2

t2 at t→ 0 (21)

Substitution this solution into Eq. (14) shows that

O(c0 +12c1ρ) =

1t, O( c2

ρ∂ρ∂v) = 1

t2,

and O( c3ρ

∂2ρ∂v2

) = 1t4

at t→ 0, v �= 0(22)

i.e.

c0 +1

2c1ρ <<

c2ρ

∂ρ

∂v<<

c3ρ

∂2ρ

∂v2at t→ 0, v �= 0 (23)

and therefore, the first three terms in Eq. (16) can be ignored

v =c3ρ

∂2ρ

∂v2at t→ 0, v �= 0 (24)

or after substitution of eq. (21)

v =4c3v

2

t4at t→ 0, v �= 0 (25)

Eq. (35) has the following solution (see Fig. 2)

v =t3

4c3 + Ct3at t→ 0, v �= 0 (26)

where C is an arbitrary constant.

This solution has the following property: the Lipchitz condition at t→ 0 fails

∂v

∂v=

8c3v

t4=

8c3t3

t4(4c3 + Ct3)→∞ at t→ 0, v �= 0 (27)

and as a result of that, the uniqueness of the solution is lost. Indeed, as follows from

Eq. (36), for any value of the arbitrary constant C, the solutions are different, but they

satisfy the same initial condition

v → 0 at t→ 0 (28)

Due to violation of the Lipchitz condition (27), the solution becomes unstable. That

kind of instability when infinitesimal errors lead to finite deviations from basic motion

(the Lipchitz instability) has been discussed in [2,3,5]. This instability leads to unpre-

dictable shift of solution from one value of Cto another. It means that appearance of any

specified solution out of the whole family is random, and that randomness is controlled

by the feedback (14) from the Liouville equation (17). Indeed if the solution (26) runs

independently many times with the same initial conditions, and the statistics is collected,

the probability density will satisfy the Liouville equation (17), Fig.3.

Remark . It should be emphasized that with the probability density defined by Eq.

(20), the point v = 0 must be excluded from consideration since at this point Eq. (16)

is meaningless.


Figure 2 Family of random solutions describing transition from determinism to stochastisity.

Figure 3 Stochastic process and probability density.

5. Departure from Newtonian and Quantum Physics

In this section we will derive a distinguished property of the system (16),(17) that is

associated with violation of the second law of thermodynamics i.e. with the capability of

moving from disorder to order without help from outside. That property can be predicted

qualitatively even prior to analytical proof: due to the nonlinear term in Eq. (17), the

solution form shock waves and solitons in probability space, and that can be interpreted

as “concentrations” of probability density, i.e. departure from disorder. In order to

demonstrate it analytically, let us turn to Eq. (17) at

c1 >> |c2|, c3 (29)

and find the change of entropy H

∂H∂t

= − ∂∂t

∞∫−∞

ρ ln ρdV = −∞∫−∞

1c1ρ(ln ρ+ 1)dV =

∞∫−∞

1c1

∂∂V

(ρ2) ln(ρ+ 1)dV

= 1c1[∞|−∞

ρ2(ln ρ+ 1)−∞∫−∞

ρdV ] = − 1c1

< 0(30)

At the same time, the original system (16), (17) is isolated: it has no external interactions.

Indeed the information force Eq. (14) is generated by the Liouville equation that, in turn,

is generated by the equation of motion (16). Therefore the solution of Eqs. (16), and

(17) can violate the second law of thermodynamics, and that means that this class of


dynamical systems does not belong to physics as we know it. This conclusion triggers the

following question: are there any phenomena in Nature that can be linked to dynamical

systems (16), (17)? The answer will be discussed bellow.

Thus despite the mathematical similarity between Eq.(17) and the KdV-Bergers equa-

tion, the physical interpretation of Eq.(17) is fundamentally different: it is a part of the

dynamical system (16),(17) in which Eq. (17) plays the role of the Liouville equation gen-

erated by Eq. (16). As follows from Eq. (30), this system being isolated has a capability

to decrease entropy, i.e. to move from disorder to order without external resources. In

addition to that, the system displays transition from deterministic state to randomness

(see Eq. (27)).

This property represents departure from classical and quantum physics, and, as shown

in [2,3], provide a link to behavior of livings. That suggests that this kind of dynamics

requires extension of modern physics to include physics of life.

Remark. The system (16), (17) displays transition from deterministic state to ran-

domness (see Eq. (27))., and this property can be linked to the similar property of the

Madelung equation, although strictly speaking, Eq.(1) is a “truncated” version of the

Liouvile equation: it does not include the contribution of the quantum potential. Nev-

ertheless the origin of randomness in quantum mechanics is the same as in the system

(16), (17) as demonstrated in [3,8,9].

6. Hypothetical Particle with a Diffusion Feedback

In this Section we concentrate on a specific form of the system (16), (17) by choosing

the Liouville feedback (12) in the form

F = −σ2 ∂

∂vln ρ, (31)

to obtain the following equation of motion

v = −σ2 ∂

∂vln ρ, (32)

The feedback (31) is a particular case of the feedback (14) when

c0 = 0, c1 = 0, c2 > 0, c3 = 0 (33)

This equation should be complemented by the corresponding Liouville equation (in this

particular case, the Liouville equation takes the form of the Fokker-Planck equation)

∂ρ

∂t= σ2 ∂

2ρ

∂V 2(34)

Here v stands for a particle velocity, and σ2is the diffusion coefficient.


6.1 Emergence of Randomness

In this sub-section we describe the random solution not only at t→ 0, but also in whole

time interval.

if σ2 = const. (35)

the solution of Eq. (34) subject to the sharp initial condition

ρ =1

2σ√πt

exp(− V 2

4σ2t) (36)

describes diffusion of the probability density, and that is why the feedback (31) can be

called a diffusion feedback.

Substituting this solution into Eq. (32) at V = v, one arrives at the differential

equation with respect to v(t)

v =v

2t(37)

and therefore,

v = C√t (38)

whereC is an arbitrary constant. Since v = 0 at t = 0 for any value of C, the solution (38)

is consistent with the sharp initial condition for the solution (36) of the corresponding

Liouvile equation (34). The solution (38) describes the simplest irreversible motion: it is

characterized by the “beginning of time” where all the trajectories intersect (that results

from the violation of Lipcsitz condition at t =0, Fig.3), while the backward motion

obtained by replacement of t with (−t) leads to imaginary values of velocities. One can

notice that the probability density (36) possesses the same properties.

It is easily verifiable that the solution (36) has the same structure as the solution (27).

Further analysis of the solution (38) demonstrates that this solution is unstable since

dv

dv=

1

2t> 0 (39)

and therefore, an initial error always grows generating randomness. Initially, at t =0,

this growth is of infinite rate since the Lipchitz condition at this point is violated

∂v

∂v→∞ at t→ 0 (40)

This type of instability has been introduced and analyzed in [5]. The unstable equilibrium

point (v = 0) has been called a terminal repeller, and the instability triggered by the

violation of the Lipchitz condition – non-Lipchitz, or terminat instability. The basic

property of the non- Lipchitz instability is the following: if the initial condition is infinitely

close to the repeller, the transient solution will escape the repeller during a bounded time

while for a regular repeller the time would be unbounded. Indeed, an escape from the

simplest regular repeller can be described by the exponentv = v0et. Obviously v → 0 if

v0 → 0, unless the time period is unbounded. On the contrary, the period of escape from


the terminal repeller (38) is bounded (and even infinitesimal) if the initial condition is

infinitely small, (see Eq. (40)).

Considering first Eq. (38) at fixedC as a sample of the underlying stochastic process

(36), and then varying C, one arrives at the whole ensemble characterizing that process,

(see Fig. 3). The curves that envelope the cross-sectional blue areas at t∗ = const present

the probability density distribution at fixed times. One can verify that, as follows from

Eq. (36), [6], the expectation and the variance of this process are, respectively

v = 0, v = 2σ2t (41)

The same results follow from the ensemble (38) at−∞ ≤ C ≤ ∞. Indeed, the first

equality in (41) results from symmetry of the ensemble with respect to v = 0 ; the second

one follows from the fact that

v ∝ v2 ∝ t (42)

It is interesting to notice that the stochastic process (35) is an alternative to the following

Langevin equation, [6]

v = Γ(t), Γ = 0, Γ = σ (43)

that corresponds to the same Fokker-Planck equation (33). Here Γ(t) is the Langevin

(random) force with zero mean and constant variance σ.

Thus, the emergence of self-generated stochasticity is the first basic non-Newtonian

property of the dynamics with the Liouville feedback.

6.2 Second Law of Thermodynamics

In order to demonstrate another non-Newtonian property of the systems considered

above, let us start with the dimensionless form of the Langevin equation for a one-

dimensional Brownian motion of a particle subjected to a random force

v = Γ(t), < Γ(t) >= 0, < Γ(t)Γ(t′) >= 2σδ(t− t′), [Γ] = 1/s (44)

Here v is the dimensionless velocity of the particle (referred to a representative velocity

v0), and Γ(t) is the Langevin (random) force per unit mass,σ >0 is the noise strength.

The representative velocity v0 can be chosen, for instance, as the initial velocity of the

motion under consideration.

The corresponding continuity equation for the probability density ρ is the following

Fokker-Planck equation

∂ρ

∂t= σ

∂2ρ

∂V 2,

∞∫−∞

ρdV = 1 (45)

Obviously without external control, the particle cannot escape the Brownian motion.

Let us now introduce a new force (referred to unit mass and divided by v0) as a

Liouville feedback

f = σ exp√D

∂

∂vln ρ, [f ] = 1/s (46)


Here D is the dimensionless variance of the stochastic process D(t) =∞∫−∞

ρV 2dV ,

Then the new equation of motion takes the form

v = Γ(t) + σ exp√D

∂

∂vln ρ, (47)

and the corresponding Fokker-Planck equation becomes nonlinear

∂ρ

∂t= σ(1− exp

√D)

∂2ρ

∂V 2,

∞∫−∞

ρdV = 1 (48)

Obviously the diffusion coefficient in Eq. (48) is negative. Multiplying Eq. (48) by V 2,

then integrating it with respect to V over the whole space, one arrives at ODE for the

variance D

D = 2[σ(1− exp√D)] (49)

Thus, as a result of negative diffusion, the variance D monotonously vanishes regardless

of the initial value D(0). It is interesting to note that the time T of approaching the

point D =0 is finite

T =1

2σ

∞∫0

dD

exp√D − 1

=π

6σ(50)

This terminal effect is due to violation of the Lipchitz condition, at D = 0, [5].

Let us review the structure of the force (46): it is composed only out of the probability

density and its variance, i.e. out of the components of the conservation equation (47); at

the same time, Eq. (47) itself is generated by the equation of motion (46). Consequently,

the force (45) is not an external force. Nevertheless, it allows the particle to escape from

the Brownian motion using its own “internal effort”. It would be reasonable to call the

force (45) an information force since it links to information rather than to energy.

Thus, we came across the phenomenon that violates the second law of thermodynamics

when the dynamical system moves from disorder to order without external interactions

due to a feedback from the equation of conservation of the probability to the equation of

conservation of the momentum. One may ask why the negative diffusion was chosen to

be nonlinear. Let us turn to a linear version of Eq. (49)

∂ρ

∂t= −σ2 ∂

2ρ

∂V 2,

∞∫−∞

ρdV = 1 (51)

and discuss the negative diffusion in more details. As follows from the linear equivalent

of Eq. (49)

D = −2σ , i.e. D = D0 − 2σt < 0 at t > D0/(2σ)(52)


Thus, eventually the variance becomes negative, and that disqualifies Eq. (52) from

being meaningful. As shown in [3], the initial value problem for this equation is ill-posed:

its solution is not differentiable at any point. Therefore, a negative diffusion must be

nonlinear in order to protect the variance from becoming negative, Fig.4.

Figure 4 Negative diffusion.

It should be emphasized that negative diffusion represents a major departure from

both Newtonian mechanics and classical thermodynamics by providing a progressive evo-

lution of complexity against the Second Law of thermodynamics.

Next we will demonstrate again that formally the dynamics introduced above does not

belong to the Newtonian world; nevertheless its self-supervising capability may associate

such a dynamics with a potential model for intelligent behavior. For that purpose we will

turn to even simpler version of this dynamics by removing the external Langevin force

and simplifying the information force:

v = σ√D

∂

∂vln ρ, (53)

∂ρ

∂t= −σ

√D

∂2ρ

∂V 2,

∞∫−∞

ρdV = 1 (54)

Removal of the Langevin forces makes the particle isolated. Nevertheless the particle has

a capability of moving from disorder to order. For demonstration of this property we

will assume that the Langevin force was suddenly removed at t = 0 so that the initial

variance D0 > 0. Then

D = −2σ√D (55)

whence

D = (√

D0 − σt)2 (56)

As follows from Eq. (56), as a result of internal, self-generated force

F = σ√D

∂

∂vln ρ, (57)


the Brownian motion gradually disappears and then vanishes abruptly:

D → 0, D → 0,dD

dD→∞ at t→

√D0

σ(58)

Thus the probability density shrinks to a delta-function at t =√D0

σ. Consequently, the

entropy H(t) = −∫V

ρ ln ρdV decreases down to zero, and that violates the second law of

thermodynamics, Fig. 5.

Figure 5 Vanishing Brownian motion.

6.3 Violation of the First Law of Thermodynamics

Let us turn to the general case described by Eq. (9). As follows from this equation, the

particle under consideration possesses only kinetic energy

W = v2/2

However this energy is not conserved although the particle is isolated. Indeed,

dW = v · F[ρ(v)]dt (59)

i.e. change of the kinetic energy is equal to the work done by the self-generated infor-

mation force F[ρ(v)]. But in contradistinction to dissipative systems, this work can be


positive, i.e. an information force can increase the kinetic energy of the particle. In

particular, that would happen in case of negative diffusion.

The significance of Eq. (59) is fundamental: it relates the change of energy to change

of information.

Since the particle velocity as a solution of Eq. ((9) is random, its kinetic energy (59) is

also random, and it is more convenient to operate with the average value W =∞∫−∞

ρWdV .

Then for a particular case of the information force (31), with reference to Eq. (5), Eq.

(59) can be written in the form of conservation energy/information

W + σ2H = const (60)

and that make relation between energy and information for the intelligent particle more

transparent.

7. Hypothetical Particle with Soliton Feedback

In this section we introduce the structure of the force F that is a particular case of the

feedback (14) at

c2 = 0 (61)

i.e.

F = c0 +1

2c1ρ+

c3ρ

∂2ρ

∂v2(62)

Then Eq.(16) can be reduced to the following:

v = c0 +1

2c1ρ+

c3ρ

∂2ρ

∂v2(63)

and the corresponding Liouville equation will turn into the following PDE

∂ρ

∂t+ (c0 + c1ρ)

∂ρ

∂V+ c3

∂3ρ

∂V 3= 0 (64)

that is a celebrated Korteweg-de Vries (KdV) equation.

However a fundamental difference between the standard KdV equation and Eq. (64)

is that Eq. (64) dwells in the probability space , and therefore, it must satisfy the

normalization constraint ∞∫−∞

ρdV = 1 (65)

But since the KdV equation has the conservation invariants, [7]

∞∫−∞

ρdV = Const., (66)


∞∫−∞

ρ2dV = Const., etc. (67)

the constraint (65) becomes a particular case of the invariant (66); consequently, if the

normalization constraint is satisfied at t = 0, it is satisfied all the time. That allows one

to apply all the known result directly to Eq. (64). However it should be noticed that

the conservation invariants (66) and (67) have different physical meaning: they are not

related to conservation of momentum and energy, but rather impose constraints upon the

Shannon information.

We will start the analysis of the equation (64) with consideration of its linear version

when c1 = 0∂ρ

∂t+ c0

∂ρ

∂V+ b

∂3ρ

∂V 3= 0 (68)

The first applications of linear (parabolic) version of KdV equation appear in models of

shallow water waves [4]. The equation is also conservative, and its solution is represented

by a train of traveling waves

ρ(v, t) = Aeikv−ωt (69)

where ω is the frequency, and k is the wave number. For KdV equation, these two

constants are connected by the following dispersion relation

ω = c0k − bk3 (70)

If the initial profile ρ = u(v, 0) is represented as a sum of the Fourier harmonics, then

each of this harmonic will propagate with the phase speed

C = ω/k. (71)

Comparing equations (70) and (71), one can see that each Fourier harmonics will prop-

agate with different phase speed that depends upon its wave number k. Therefore any

initial profile eventually is dispersed over the whole positive subspace, Fig.6.

Figure 6 Linear dispersion of initial profile.


An important property of the linear version of the KdV equation is the dependence

of its solution on the initial conditions for all times.

Let us assume now that

b = 0, c0 = 0 (72)

We get the equation∂ρ

∂t+ c1ρ

∂ρ

∂V= 0 (73)

Unlike the previous versions of the KdV equation, this is a nonlinear PDE of hyperbolic

type. It appears in models of free particles flow, traffic jam, etc. This is the simplest

equation that describes formation of shock waves. Its closed analytical solution can be

written only in an implicit form, and here we will analyze it only qualitatively. We

will start our analysis with studying a propagation of an initial profile ρ = ρ(v, 0). As

follows from Equation (73), the higher values of ρ propagate faster than lower ones. As a

result, the moving front becomes steeper and steeper, and finally a strong discontinuity

representing a shock emerges, see Fig.7.

Figure 7 Formation of shock waves in probability space.

Since closed form solution of Eq. (73) is not available, we will continue with the

solution for large time. The rationale for that is the assumption that eventually the

solution tends to a stationary shape as a result of a balance between dispersion and

shock wave formation. Therefore we will seek the solution in the form of a stationary

motion

ρ(v, t) = f(v − Ut) = u(ζ) at t→∞ (74)

Substituting Eq.(74) into Eq.(73) one obtains

ρ(v, t) = f(v − Ut) = u(ζ) at t→∞ (75)

Integrating this equation with respect to ζand setting the arbitrary constant to zero, one

arrives at the ODE in its final form

b∂2ρ

∂ζ2+ (c0 − U)ρ+

c12ρ2 = 0 (76)

The solution of this equation is a soliton moving with the speed U

ρ = aSech2[

√c1a√12b

(v − Ut)] (77)


where

U = c0 +1

3c1a (78)

see Fig. 8. It should be emphasized that the soliton (77) does not depend upon initial

conditions, and consequently it can be considered as a static attractor in probability

space . This means that in physical space, a solution of Eq. (63) eventually approaches a

stochastic attractor. The analytical form of this solution at t→ 0 was derived in Section

4, (see Eq. (26), and Fig. 2).

Figure 8 Soliton as an attractor of KdV solution.

It should be emphasized that the dynamics system (63), (64) is isolated, but despite

of that, its entropy decreases in the course of the soliton wave formation.

8. Origin of Intelligence

8.1 Relevance to Model of Intelligent Particle

The proposed model illuminates the “border line” between living and non-living systems.

The model introduces an intelligent particle that, in addition to Newtonian properties,

possesses the ability to process information. The probability density can be associated

with the self-image of the intelligent particle as a member of the class to which this

particle belongs, while its ability to convert the density into the information force - with

the self-awareness (both these concepts are adopted from psychology). Continuing this

line of associations, the equation of motion (such as Eqs (16) or (32)) can be identified

with a motor dynamics, while the evolution of density (see Eqs. (17) or (34) –with a

mental dynamics. Actually the mental dynamics plays the role of the Maxwell sorting

demon: it rearranges the probability distribution by creating the information potential

and converting it into a force that is applied to the particle. One should notice that men-

tal dynamics describes evolution of the whole class of state variables (differed from each

other only by initial conditions), and that can be associated with the ability to generalize

that is a privilege of intelligent systems. Continuing our biologically inspired interpreta-

tion, it should be recalled that the second law of thermodynamics states that the entropy


of an isolated system can only increase. This law has a clear probabilistic interpretation:

increase of entropy corresponds to the passage of the system from less probable to more

probable states, while the highest probability of the most disordered state (that is the

state with the highest entropy) follows from a simple combinatorial analysis. However,

this statement is correct only if there is no Maxwell’ sorting demon, i.e., nobody inside

the system is rearranging the probability distributions. But this is precisely what the

Liouville feedback is doing: it takes the probability density ρ from Equation (17), creates

functionals and functions of this density, converts them into the information force and

applies this force to the equation of motion (16). As already mentioned above, because

of that property of the model, the evolution of the probability density can become non-

linear, and the entropy may decrease “against the second law of thermodynamics”, Fig.7.

Actually the proposed model represents governing equations for interactions of intelligent

agents. In order to emphasize the autonomy of the agents’ decision-making process, we

will associate the proposed models with self-supervised (SS) active systems. By

an active system we will understand here a set of interacting intelligent agents capable

of processing information, while an intelligent agent is an autonomous entity, which ob-

serves and acts upon an environment and directs its activity towards achieving goals.

The active system is not derivable from the Lagrange or Hamilton principles, but it is

rather created for information processing. One of specific differences between active and

physical systems is that the former are supposed to act in uncertainties originated from

incompleteness of information. Indeed, an intelligent agent almost never has access to the

whole truth of its environment. Uncertainty can also arise because of incompleteness and

incorrectness in the agent’s understanding of the properties of the environment. That is

why quantum-inspired SS systems represented by the particles under consideration are

well suited for representation of active systems, and the hypothetical particle introduced

above can be associated with the term “intelligent” particle. It is important to empha-

size that self-supervision is implemented by the feedback from mental dynamics, i.e. by

internal force, since the mental dynamics is generated by intelligent particle itself.

8.2 Comparison with Control Systems

In this sub-section we will establish a link between the concepts of intelligent control and

phenomenology of behavior of intelligent particle.

Example One of the limitations of classical dynamics, and in particular, neural net-

works, is inability to change their structure without an external input. As will be shown

below, an intelligent particle can change the locations and even the type of the attractors

being triggered only by information forces i.e. by an internal effort. We will start with a

simple dynamical system

v = 0, v = 0 at t = 0 (79)

and than apply the following control

F = −kv + a¯v − σ∂

∂vln ρ, (80)


where

¯V =

∞∫−∞

ρ(V − V )2dV , V =

∞∫−∞

ρV dV , (81)

and k, a, σ are constant coefficients.

Then the controlled version of the motor dynamics (79) is changed to

v = −kv + a¯v − σ∂

∂vln ρ (82)

while F represents the information forces that play the role of internal actuator.

Let us notice that the internal actuator (80) is a particular case of the information

force (14) at

c0 = −kv + a¯v, c1 = 0, c2 = σ, c3 = 0 (83)

For a closure, Eq. (82) is complemented by the corresponding Liouville equation

∂ρ

∂t= kV

∂ρ

∂V− a ¯V

∂ρ

∂V+ σ

∂2ρ

∂V 2, (84)

to be solved subject to sharp initial condition

ρ0(V ) = δ(V ) at t = 0, (85)

As shown in Section 4, the solution of Eq.(82) is random, (see Eq. (26) and Fig. 2)

while this randomness is controlled by Eq. (84). Therefore in order to describe it, we

have to transfer to the mean values vand ¯v. For that purpose, let us multiply Eq.(84)

by V .Then integrating it with respect to V over the whole space, one arrives at ODE for

the expectation v(t)˙v = −kv + a¯v (86)

Multiplying Eq.(84) by V 2, then integrating it with respect to V over the whole space,

one arrives at ODE for the variance ¯v(t)

˙v = −2k¯v + 2av ¯v + 2σ (87)

Let us find fixed points of the system (86) and (87) by solving the system of algebraic

equations:

0 = −kv + a¯v (88)

0 = −2k¯v + 2av ¯v + 2σ (89)

By selecting

σ =k3

2a2(90)

we arrive at the following single fixed point

v∗ =k

2a, ¯v∗ =

k2

2a2(91)


In order to establish whether this fixed point is an attractor or a repeller, we have to

analyze stability of the homogeneous version of the system (86), (87) linearized with

respect to the fixed point (91)˙v = −kv + a¯v (92)

˙v = −k¯v + k2

av (93)

Analysis of its characteristic equation shows that it has non-positive roots:

λ1 = 0, λ2 = −2k < 0 (94)

and therefore, the fixed point (91) is a stochastic attractor with stationary mean and

variance. However the higher moments of the probability density are not necessarily

stationary: they can be found from the original PDE (84).

Thus as a result of a mental control, an isolated dynamical system (79) that prior

to control was at rest, moves to the stochastic attractor (91) having the expectation

v∗and the variance ¯v∗.The distinguished property of the particle introduced above definitely fits into the

concept of intelligence. Indeed, the evolution of intelligent living systems is directed

toward the highest levels of complexity if the complexity is measured by an irreducible

number of different parts that interact in a well-regulated fashion. At the same time, the

solutions to the models based upon dissipative Newtonian dynamics eventually approach

attractors where the evolution stops while these attractors dwell on the subspaces of lower

dimensionality, and therefore, of the lower complexity (until a “master” reprograms the

model). Therefore, such models fail to provide an autonomous progressive evolution of

intelligent systems (i.e. evolution leading to increase of complexity). At the same time, a

self-controlled particle can create its own complexity based only upon an internal effort.

Thus the actual source of intelligent behavior of the particle introduced above is a

new type of force - the information force - that contributes its work into the Law of

conservation of energy. However this force is internal: it is generated by the particle itself

with help of the Liouvile equation. The machinery of the intelligence is similar to that

of control system with the only difference that control systems are driven by external

actuators while the intelligent particle is driven by a feedback from the Liouvile equation

without any external resources.

c. Application to performance of human mind has been introduced and

discussed in [7]. The proposed model deals with rules of mind activity rather than with

it’s content, i.e.with grammar, but not semantic. The model is represented by a system

of differential equations with terminal attractors and repellers, [5], that provide the capa-

bility to process discrete-event-based flow of information. This system has master-slave

architecture: the master equations describe dynamics of the corresponding intelligent

particle (in the form similar to Eqs. (32) and (34)), while the slave equations present

dynamics of Boolean functions that capture the mind activity such as mathematical logic

and linguistic. The Boolean dynamics consists of a sequence of two-step cycles during

which a new values of the uncorrelated Boolean functions are exposed. It should be


recalled that in terms of mathematical logic, each Boolean function is characterized by

the truth table, which is simply a list of all the values of the function at 2n values of its

arguments. If these arguments are listed in a certain order, every Boolean function has

a unique truth table, and as a consequence, the same logical meaning. In this context,

the dynamics spontaneously generates new uncorrelated logical statements that expose a

choice for making common sense decisions.

9. Comparison with Quantum Mechanics

9.1 Mathematical Viewpoint

The model of intelligent particle is represented by a nonlinear ODE (9) and a nonlinear

parabolic PDE (10) coupled in a master-slave fashion: Eq. (10) is to be solved inde-

pendently, prior to solving Eq. ((9). The coupling is implemented by a feedback that

includes the probability density and its space derivatives, and that converts the first order

PDE (the Liouville equation) to the second or higher order nonlinear PDE. As a result of

the nonlinearity, the solutions to PDE can have attractors (static, periodic, or chaotic)

in probability space. The solution of ODE (9) represents another major departure from

classical ODE: due to violation of Lipchitz conditions at states where the probability den-

sity has a sharp value, the solution loses its uniqueness and becomes random. However,

this randomness is controlled by the PDE (10) in such a way that each random sample

occurs with the corresponding probability, Fig.3.

9.2 Physical Viewpoint

The model of intelligent particle represents a fundamental departure from both Newtonian

and quantum mechanics. The fundamental departure of all the modern physics is the

violation of the first and the second laws of thermodynamics,(see Eqs.(60), (30), (58),

Figs. 5 and 7). However the model has some similarity to quantum mechanics, and these

similarities are outlined below.

α.Superposition. In quantum mechanics, any observable quantity corresponds to an

eigenstate of a Hermitian linear operator. The linear combination of two or more eigen-

states results in quantum superposition of two or more values of the quantity. If the

quantity is measured, the projection postulate states that the state will be randomly

collapsed onto one of the values in the superposition (with a probability proportional to

the square of the amplitude of that eigenstate in the linear combination). Let us compare

the behavior of the model of intelligent particle from that viewpoint. As follows from Eq.

(38), all the particular solutions intersect at the same point v = 0 at t = 0, and that

leads to non-uniqueness of the solution due to violation of the Lipcshitz condition (see

Eq. (40). Therefore, the same initial condition v = 0 at t = 0 yields infinite number

of different solutions forming a family (38); each solution of this family appears with a

certain probability guided by the corresponding Fokker-Planck equation. For instance,


in case of Eq. (38), the “winner” solution is v ≡ 0 since it passes through the maxima

of the probability density (36). However, with lower probabilities, other solutions of the

family (38) can appear as well. Obviously, this is a non-classical effect. Qualitatively,

this property is similar to those of quantum mechanics: the system keeps all the solutions

simultaneously and displays each of them “by a chance”, while that chance is controlled

by the evolution of probability density (36).

β. Entanglement. Quantum entanglement is a phenomenon in which the quantum

states of two or more objects have to be described with reference to each other, even

though the individual objects may be spatially separated. This leads to correlations

between observable physical properties of the systems. For example, it is possible to

prepare two particles in a single quantum state such that when one is observed to be

spin-up, the other one will always be observed to be spin-down and vice versa, this

despite the fact that it is impossible to predict, according to quantum mechanics, which

set of measurements will be observed. As a result, measurements performed on one

system seem to be instantaneously influencing other systems entangled with it.

Qualitatively similar effect can be found in the model of intelligent particle. In order

to demonstrate that, we start with Eqs.(32) and (34) and generalize them to the two-

dimensional case

v1 = −a11∂

∂v1ln ρ− a12

∂

∂v2ln ρ, (95)

v2 = −a21∂

∂v1ln ρ− a22

∂

∂v2ln ρ, (96)

∂ρ

∂t= a11

∂2ρ

∂V 2+ (a12 + a21)

∂2ρ

∂V1∂V2

+ a22∂2ρ

∂V2

, (97)

As in the one- dimensional case, this system describes diffusion without a drift

The solution of Eq. (97) has a closed form

ρ =1√

2π det[aij]texp(− 1

4tb′ijViVj), i = 1, 2. (98)

Here

[b′ij] = [aij]−1 , a11 = a11, a22 = a22, a12 = a21 = a12 + a21, aij = aji, b

′ij = b′ji, (99)

Substituting the solution (98) into Eqs. (95) and (96), one obtains

v1 =b11v1 + b12v2

2t(100)

v2 =b21v1 + b22v2

2t, bij = b′ij aij (101)

Eliminating t from these equations, one arrives at the ODE in configuration space

dv2dv1

=b21v1 + b22v2b11v1 + b12v2

, v2 → 0 at v1 → 0, (102)

This is a classical singular point treated in textbooks on ODE.


Its solution depends upon the roots of the characteristic equation

λ2 − 2b12λ+ b212 − b11b22 = 0 (103)

Since both the roots are real in our case, let us assume for concreteness that they are of

the same sign, for instance,λ1 = 1, λ2 = 1. Then the solution of Eq. (102) is presented

by the family of straight lines

v2 = Cv1, C = const. (104)

Substituting this solution into Eq. (100) yields

v1 = Ct12(b11+Cb12) v2 = CCt

12(b11+Cb12) (105)

Thus, the solutions of Eqs. (95) and (96) are represented by two-parametrical families of

random samples, as expected, while the randomness enters through the time-independent

parameters C and C that can take any real numbers. Let us now find such a combination

of the variables that is deterministic. Obviously, such a combination should not include

the random parameters C or C. It easily verifiable that

d

dt(ln v1) =

d

dt(ln v2) =

b11 + Cb122t

(106)

and therefore,

(d

dtln v1)/(

d

dtln v2) ≡ 1 (107)

Thus, the ratio (107) is deterministic although both the numerator and denominator are

random,(see Eq.(106). This is a fundamental non-classical effect representing a global

constraint. Indeed, in theory of stochastic processes, two random functions are considered

statistically equal if they have the same statistical invariants, but their point-to-point

equalities are not required (although it can happen with a vanishingly small probability).

As demonstrated above, the diversion of determinism into randomness via instability (due

to a Liouville feedback), and then conversion of randomness to partial determinism (or

coordinated randomness) via entanglement is the fundamental non-classical paradigm.

γ . Decoherence. In quantum mechanics, decoherence is the process by which quantum

systems in complex environments exhibit classical behavior. It occurs when a system

interacts with its environment in such a way that different portions of its wavefunction

can no longer interfere with each other.

Qualitatively similar effects are displayed by the intelligent particle. In order to

illustrate that, let us turn to Eqs. (32), (34), and notice that, as soon as the feedback (31)

disappears, the system becomes classical, i.e. fully deterministic, while the deterministic

solution is a continuation of the corresponding “chosen” random solution.

δ . Uncertainty Principle. In quantum physics, the Heisenberg uncertainty principle

states that one cannot measure values (with arbitrary precision) of certain conjugate

quantities that are pairs of observables of a single elementary particle. These pairs include


the position and momentum. Similar (but not identical) relationship follows from Eq.

(38):

vv = C2/2 (108)

i.e. the product of the velocity and the acceleration is constant along a fixed trajectory.

In particular, at t = 0, v and v can not be defined separately.

ε. Wave–particle duality. In physics, wave–particle duality is a conceptualization

that all objects in our universe exhibit properties of both waves (such as non-locality)

and of particles (such as quantization of some of their properties). As shown by Max

Born, the wave associated with the electron is not a tangible ’matter wave’, but one that

determines the probability of scattering of the electron in different directions. Similar

“duality” follows from the model of intelligent particle. Indeed, Eq. (32) describes the

“trajectories” of particles, while Eq. (34) represents the wave of probability that captures

the particle “scattering”.

η.Interference of probabilities. In Newtonian physics, the probability is introduced

via the Liouville equation describing the continuity of the probability density flow. This

equation is linear with respect to the probability density, and therefore, according to the

superposition principle, the probabilities are combined by summation: when an event

can occur in several alternative ways, the probability of the event is the sum of the

probabilities for each way considered separately, i.e.

ρ = ρ1 + ρ2 (109)

In quantum physics, the probability is introduced via the Schrodinger equation that is

linear with respect to probability amplitude, i.e. with respect to the square root of the

probability density. Therefore, when an event can occur in several alternative ways, the

probability amplitude of the event is the sum of the probability amplitudes for each way

considered separately

The probability interference in quantum mechanics follows from the linearity of the

Schrodinger equation with respect to the probability amplitudes ψi as state variables.

Due to linear superposition of these amplitudes, the following rule can be formulated

ψ = ψ1 + ψ2, ρi = |ψi|2, ρ = |ψ1 + ψ2|2 �= ρ1 + ρ2 (110)

and this phenomenon is known as interference of probabilities: the probabilities are

combined as the intensities of waves.

The situation with interference of probabilities in the model of intelligent particle is

more complex: it depends upon the type of information forces. Indeed, in the diffu-

sion and the integral feedbacks cases, Eqs.(32), and (34) are linear with respect to the

probability density, and the probabilities are combined according to Eq. (109). i.e. with-

out interference. But in the shock/soliton feedback, the Liouville equation is nonlinear

with respect to the probability density, and consequently, the probabilities interfere, (see

Eqs.(64) and (73)). However, this interference is different from the quantum one and it

will be discussed below.


Indeed, following [4] and reinterpreting confluence of shock waves in physical space to

confluence of densities in probability space obtain the rule of combining the probabilities

ρ =ρ1f1 + ρ2f2f1 + f2

, ρ2 > ρ1 (111)

where

fi = exp(−ρiV

2σ+

ρ2i t

4σ), i = 1, 2. (112)

This means that when an event can occur in several alternative ways, the probability of

the event is the sum of nonlinear combinations of the probabilities for each way considered

separately.

The concept of interference of probabilities in more details is considered in [8].

10. Discussions and Conclusion

The discovery of the Higgs boson and the following from it completeness of the physical

picture of our Universe roused many questions, and one of them is the ability to create Life

and Mind out of physical matter without any additional entities. The primary objective

of this paper is to presents amathematical answer to the ancient philosophical question,

“How mind is related to matter” in connection with this outstanding accomplishment in

physics. The paper is inspired by analysis of the Madelung equation and discovery of the

origin of randomness in quantum mechanics, [3,9]. It turns out that replacement of the

quantum potential by the information force, while preserving some quantum properties,

introduces fundamental changes in the first and the second laws of thermodynamics, and

that leads to a mathematical model that captures behavior of livings. The idea of an

intelligent particle has been introduced as a first step of physics of life since it does not

include such properties as metabolism and reproduction. Instead it concentrates attention

to intelligent behavior. At the same time, by ignoring metabolism and reproduction, we

can make the system isolated, and it will be a challenge to show that it still can move

from a disorder to the order.

Thus the paper introduces and discusses a possible extension of modern physics to

include a concept of intelligent particle as the first step to physics of Life since all attempts

to create livings from non-living matter failed. It has been proven that there exists a

fundamentally new type of dynamical systems (represented by intelligent particles) that

can evolve from disorder to order without external forces thereby violating the second

law of thermodynamics. It has been demonstrated that these systems belong neither to

Newtonian, nor to quantum mechanics. Their departure from Newtonian mechanics is

due to a feedback from the underlying Liouville equation to the equations of motion that

represents an additional (internal) information force. Topologically this feedback shifts

intelligent particles towards quantum mechanics. However since the information force

is different from forces produced by quantum potential, the intelligent particles are not

quantum, and they can be identified as quantum-classical hybrids. Therefore intelligent

particles dwell in an abstract mathematical world rather than in the physical world, as


we know it. This means that intelligent particles, in principle, cannot be composed out of

physical particles. It also means that their behavior can be computed, but not simulated

using Newtonian or quantum resources.

Since the model of intelligent particle fits well into the mathematical formalism of

modern physics, it can be consider as a new branch of quantum mechanics, and that

rouses a belief that intelligent particle is not only a mathematical abstraction, but a

reality as well.

In this context, we will comment on the recent statement made by Stephen Hawk-

ing on December 2, 2014, in which he warns that artificial intelligence (AI) could end

mankind. Based upon our work, part of which is presented in this paper, it can be

stated that machines composed only out of physical components and without any digital

devices being included, CANNOT, in principle, overperform a human in creativity, re-

gardless of the level of technology, since such machines cannot violate the Second Law of

thermodynamics. Therefore in our opinion, the danger of AI for mankind is exaggerated.

References

[1] Schrodinger, E. What is life, Cambridge University Press, 1944

[2] Zak, M., 2013, Information: Forgotten Variable in Physics Models, EJTP 10, No. 28,199–220,

[3] Zak, M., 2013, Particle of life: mathematical abstraction or reality? , Nova publishers,NY.

[4] G. Whitham, Linear and Nonlinear Waves Wiley-Interscience, New York, 1974.

[5] Zak, M., ”Terminal Model of Newtonian Dynamics,” Int. J. of Theoretical Physics,No.32, 159-190, 1992.

[6] Risken,H., 1989, The Fokker-Planck Equation, Springer, NY.

[7] Zak, M., 2014, Toward quantum-inspired model of mind. Journal of QuantumInformation Science, 2014, 4, 22-43,

[8] Zak,M., 2014, Interference of probabilities in dynamics, AIP Advances 4, 087130.

[9] Zak, M., 2014, The Origin of Randomness in Quantum Mechanics, EJTP 11, No. 31(2014) 1–16

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