Edward s Article

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Physica A 348 (2005) 505–543

0378-4371/$ -

doi:10.1016/j

�CorrespoEcuador Exp

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Econophysics: from Game Theory andInformation Theory to Quantum Mechanics

Edward Jimeneza,b,c,�, Douglas Moyad

aExperimental Economics, Todo1 Services Inc., Miami, FL 33126, USAbGATE, UMR 5824 CNRS, France

cResearch and Development Department, Petroecuador, Paul Rivet E11-134 y 6 de Diciembre,

Quito, EcuadordPhysics Department, Escuela Politecnica Nacional, Ladron de Guevara E11-253, Quito, Ecuador

Received 11 March 2004; received in revised form 20 September 2004

Available online 27 October 2004

Abstract

Rationality is the universal invariant among human behavior, universe physical laws and

ordered and complex biological systems. Econophysics isboth the use of physical concepts in

Finance and Economics, and the use of Information Economics in Physics. In special, we will

show that it is possible to obtain the Quantum Mechanics principles using Information and

Game Theory.

r 2004 Elsevier B.V. All rights reserved.

PACS: 02.30.Cj; 02.50.Le; 03.65.Ta

Keywords: Rationality; Optimal laws; Quantum mechanic laws; Energy; Information

1. Introduction

At the moment, Information Theory is studied or utilized by multiple perspectives(Economics, Game Theory, Physics, Mathematics, and Computer Science). Our goal

see front matter r 2004 Elsevier B.V. All rights reserved.

.physa.2004.09.029

nding author. Research and Development Department, Petroecuador, Todo1 Services,

erimental Economics, Miami, USA.

dress: [email protected] (E. Jimenez).

www.elsevier.com/locate/physa

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E. Jimenez, D. Moya / Physica A 348 (2005) 505–543506

in this paper is to present the different focuses about information and to select themain ones, to apply them in Economics of Information and Game Theory.Economics and Game Theory1 are interested in the use of information and orderstate, in order to maximize the utility functions of the rational2 and intelligent3

players, which are part of an interest conflict. The players gather and processinformation. The information can be perfect4 and complete5 see Refs. [1–4], (Sorin,1992). On the other hand, Mathematics, Physics and Computer Science are allinterested in information representation, entropy (disorder measurement), optim-ality of physical laws and in the living beings’ internal order see Refs. [5–7]. Finally,information is stored, transmitted and processed by physical means. Thus, theconcept of information and computation can be formulated not only in the contextof Economics, Game Theory and Mathematics, but also in the context of physicaltheory. Therefore, the study of information ultimately requires experimentation andsome multidisciplinary approaches such as the introduction of the OptimalityConcept [8].The Optimality Concept is the essence of the economic and natural sciences [9,10].

Economics introduces the optimality concept (maximum utility and minimum risk)as equivalent of rationality and Physics understands action minimum principle, andmaximum entropy (maximum information) as the explanation of nature laws [11,36].If the two sciences have a common backbone, then they should allow certainanalogies and to share other elements such us: equilibrium conditions, evolution,uncertainty measurement and the entropy concept. In this paper, the contributionsof Physics (Quantum Information Theory)6 and Mathematics (Classical InformationTheory)7 are used in Game Theory and Economics being able to explain mixed

1Game Theory can be defined as the study of mathematical models of conflict and cooperation among

intelligent rational decision-makers.2A decision-maker is rational if he makes decision in pursuit of his own objectives. It is normal to

assume that each player’s objective is to maximize the expected utility value of his own payoff.3A player in the game is intelligent if he knows everything that we know about the game and he can

make inferences about the situation that we can make.4Perfect information means that at each time only one of the players moves, that the game depends only

on their choices, they remember the past information (utilities, strategies), and in principle they know all

possible futures of the game [1].5In games of incomplete information the state of the nature is fixed but not know to all players. In

repeated games of incomplete information, the changes in time is each player’s knowledge about the other

players’ past actions, which affects his beliefs about the (fixed) state of nature.

Games of incomplete information are usually classified according to the nature of the three important

elements of the model, namely players and payoffs (within the two-person games: zero-sum games and

non-zero-sum games), prior information on the state of the nature (within two-person games: incomplete

information on one side and incomplete information on two sides), and signaling structure (full

monitoring and state independent signals) [2,3,37].6Quantum Information Theory is fundamentally richer than Classical Information Theory, because

quantum mechanics includes so many more elementary classes of static and dynamic resources—not only

does it support all the familiar classical types, but there are entirely new classes such as the static resource

of entanglement (correlated equilibria) to make life even more interesting that it is classically.7Classical Information Theory is mostly concerned with the problems of sending Classical Information-

letters in an alphabet, speech, strings of bits—over communications channels which operate within the

laws of classical physics.

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E. Jimenez, D. Moya / Physica A 348 (2005) 505–543 507

strategy Nash’s equilibrium using Shannon’s entropy [12–15]. According to [16,p. 11]‘‘quantities of the form H ¼ �

Ppi log pi play a central role in information theory

as measures of information, choice and uncertainty. The form of H will be recognized as

that entropy as defined in certain formulations of statistical mechanics where pi is the

probability of a system being in cell i of its phase space; . . .’’In Quantum Information Theory, the correlated equilibria in two-player games

means that the associated probabilities of each-player strategies are functions of acorrelation matrix. Entanglement, according to the Austrian physicist ErwinSchrodinger, which is the essence of Quantum Mechanics, has been known forlong time now to be the source of a number of paradoxical and counterintuitivephenomena. Of those, the most remarkable one is the usually called non-localitywhich is at the heart of the Einstein–Podolsky–Rosen paradox (ERP) see Ref. [17, p.12]. Einstein et al. [18] which consider a quantum system consisting of two particlesseparated long distance.‘‘ERP suggests that measurement on particle 1 cannot have any actual influence on

particle 2 (locality condition); thus the property of particle 2 must be independent of the

measurement performed on particle 1.’’The experiments verified that two particles in the ERP case are always part of one

quantum system and thus measurement on one particle changes the possiblepredictions that can be made for the whole system and therefore for the otherparticle [5].It is evident that physical and mathematical theories have had a lot of utility for

the economic sciences, but it is necessary to highlight that Information Theory andEconomics also contribute to the explanation of Quantum Mechanics laws. Will thestrict incorporation of Classic and Quantum Information Theory elements allow thedevelopment of Economics and Game Theory? The definitive answer is yes.Economics has carried out its own developments around information theory;

especially it has demonstrated both that the asymmetry of information causes errorsin the definition of a optimal negotiation and that the assumption of perfect markets isuntenable in the presence of asymmetric information see Refs. [19,20]. The asymmetryof the information according to the formalism of Game Theory can have two causes:incomplete information and imperfect information. As we will see in the developmentof this paper, Information Economics does not even incorporate in a formal wayneither elements of Classical Information Theory nor Quantum Information concepts.The creators of Information Theory are Shannon and von Newmann see Ref. [15,

Chapter 11]. Shannon the creator of Classical Information Theory introduces theentropy as the heart of your theory, endowing it of a probabilistic characteristics. On

(footnote continued)

‘‘The key concept of Classical Information Theory is the Shannon Entropy Theory. Suppose we learn

the value of a random variable X. The Shannon Entropy of X quantifies how much information we gain,

on average, when we learn the value of X : An alternative view is that the entropy of X measures the

amount of uncertainty about X before we learn its value. These two views are complementary; we can view

the entropy either as a measure of our uncertainty before we learn the value of X ; or as measure of howmuch information we have gained after we learn the value of X : ‘‘[15, p. 500].

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the other hand, von Newmann also creator of Game Theory, uses the probabilisticelements take into account by Shannon but defines a new mathematical formulationof entropy using the density matrix of Quantum Mechanics. Both entropyformulations developed by Shannon and von Newmann, respectively, permit us tomodel pure states (strategies) and mixed states (mixed strategies). In Eisert et al.[21,35], they not only give a physical model of quantum strategies but also expressthe idea of identifying moves using quantum operations and quantum properties. Thisapproach appears to be fruitful in at least two ways. On one hand, several recentlyproposed quantum information application theories can already be conceived ascompetitive situations, where several factors which have opposing motives interact.These parts may apply quantum operations using a bipartite quantum system. Onthe other hand, generalizing decision theory in the domain of quantum probabilitiesseems interesting, as the roots of game theory are partly rooted in probability theory.In this context, it is of interest to investigate what solutions are attainable ifsuperpositions of strategies are allowed [22–24].As we have seen before, from a historical perspective we can affirm that Game

Theory and Information Theory advances are related to Quantum Mechanicsespecially regarding nanotechnology. Therefore, it is necessary to use QuantumMechanics for five reasons:

�
The origin of quantum information and its potential applications: encryption,quantum nets and correction of errors has wakened up great interest especially inthe scientific community physicists, mathematicians and economists.
�
Quantum Game Theory is the first proposal of unifying Game Theory andQuantum Mechanics with the objective of finding synergies between both.
�
In this paper we present an immediate result, product of using these synergies(possibility theorem). Possibility theorem allows us to introduce the concept ofrationality in time series.
�
A perfect analogy exists between correlated equilibria that fall inside the domainof Game Theory, and entanglement that falls inside the domain of QuantumInformation see Refs. [25,15].
�
The reason of being of this paper is to demonstrate theoretically and practicallythat Information Theory and Game Theory permit us to obtain QuantumMechanics Principles.
This paper is organized as follows. Section 1 is a revision of the existentbibliography. In Section 2, we show the main theorems of quantum games. Section 3is the core of this paper; here we present the Quantum Mechanics Principles as aconsequence of maximum entropy and minimum action principle. In Section 4 wecan see the conclusions of this research.

2. Elements of Quantum Game Theory

Let G ¼ ðK ;S; vÞ be a game to n-players, with K the set of players k ¼ 1; . . . ; n:The finite set Sk of cardinality lk 2 N is the set of pure strategies of each player

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where k 2 K ; skjk2 Sk; jk ¼ 1; . . . ; lk and S ¼ PK Sk set of pure strategy

profiles with s 2 S an element of that set, l ¼ l1; l2; . . . ; ln represent the cardinalityof S [26,4,9]. Furthermore, in physics the word strategy represents a state of thesystem.The vector function v : S ! Rn associates every profile s 2 S the vector of utilities

vðsÞ ¼ ðv1ðsÞ; . . . ; vnðsÞÞT; where vkðsÞ designates the utility of the player k facing theprofile s. In order to get facility of calculus we write the function vkðsÞ in one explicitway vkðsÞ ¼ vkðj1; j2; . . . ; jnÞ: The matrix vn;l represents all points of the Cartesianproduct

Qk2K Sk: The vector vkðsÞ is the k-column of v:

If the mixed strategies are allowed, then we have:

DðSkÞ ¼ pk 2 Rlk :Xlk

jk¼1

pkjk¼ 1

( )

the unit simplex of the mixed strategies of player k 2 K ; and pk ¼ ðpkjkÞ is the

probability vector. The set of profiles in mixed strategies is the polyhedron D withD ¼

Qk2K DðSkÞ; where p ¼ ðp1j1 ; p

2j2; . . . ; pn

jnÞ; and pk ¼ ðpk

1 ; pk2 ; . . . ; p

klnÞT: Using the

Kronecker product � it is possible to write:8

p ¼ p1 � p2 � � pk�1 � pk � pkþ1 � � pn ;

pð�kÞ ¼ p1 � p2 � � pk�1 � lk � pkþ1 � � pn ;

lk ¼ ð1; 1; . . . ; 1ÞT; ½lk�lk ;1;

ok ¼ ð0; 0; . . . ; 0ÞT; ½ok�lk ;1 :

The n-dimensional function u : D ! Rn associates with every profile in mixedstrategies the vector of expected utilities uðpÞ ¼ ðu1ðp; vðsÞÞ; . . . ; unðp; vðsÞÞÞT; whereukðp; vðsÞÞ is the expected utility for each player k: Every uk

jk¼ uk

jkðpð�kÞ; vðsÞÞ

represents the expected utility for each player’s strategy and the vector uk is noteduk ¼ ðuk

1 ; uk2 ; . . . ; u

knÞT:

uk ¼Xlk

jk¼1

ukjkðpð�kÞ; vðsÞÞpk

jk;

u ¼ v0p ;

uk ¼ ðlk � vkÞpð�kÞ :

The triplet ðK ;D; uðpÞÞ designates the extension of the game G with the mixedstrategies. We get Nash’s equilibrium (the maximization of utility) if and only if, 8k;p; the inequality ukðpnÞXukððpkÞ

n; pð�kÞÞ is respected. Moreover, we can write theoriginal problem as a decision one where ukðpnÞ � ukððpkÞ

n; pð�kÞÞ:Another way to calculate the Nash’s equilibrium, [27, 9, pp. 96–104], is

equaling the values of the expected utilities of each strategy when it is

8We use bold fonts in order to represent vector or matrix.

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possible.

uk1ðp

ð�kÞ; vðsÞÞ ¼ uk2ðp

ð�kÞ; vðsÞÞ ¼ ¼ ukjkðpð�kÞ; vðsÞÞ ;

Xlk

jk¼1

pkjk¼ 1 8k ¼ 1; . . . ; n ;

s2k ¼Xlk

jk¼1

ukjkpð�kÞ; vðsÞ� �

� uk� 2

pkjk¼ 0 :

If the resulting system of equations does not have solution ðpð�kÞÞ� then we

propose the minimum entropy theorem. This method is expressed as MinpðP

k HkðpÞÞ;where s2kðp

�Þ standard deviation and HkðpnÞ entropy of each player k.

s2kðpnÞps2kððp

kÞn; pð�kÞÞ

or

HkðpnÞpHkððp

kÞn; pð�kÞÞ

Theorem 1 (Minimum entropy theorem). The game entropy is minimum only in mixed

strategy Nash’s equilibrium. The entropy minimization program MinpðP

k HkðpÞÞ; is

equal to standard deviation minimization program MinpðPkskðpÞÞ; when ðukjkÞ has

Gaussian density function or multinomial logit see proof in Ref. [14].

Theorem 2 (Minimum dispersion). The Gaussian density function f ðxÞ ¼ jfðxÞj2

represents the solution of one differential equation given by ðx�hxi2s2x

þ qqxÞfðxÞ ¼ 0 related

to minimum dispersion of a lineal combination between the variable x, and a Hermitian

operator ð�i qqxÞ see proof in Ref. [14].

3. Information Theory towards Quantum Mechanics

In this section, a presentation of Information Theory is made. Moreover, a studyabout how this subject, along with the variational principles of TheoreticalMechanics, is used as a frame to construct Quantum Mechanics Principles. Also,a relation of equivalence between variation of information and mass–energy isfound.

3.1. Information Theory elements

Let n be, a binary word with symbols wi 2 ð1; 0Þ: The total number of wordswhich can be built with symbols in n is 2n:Each word can be an instruction, a number, an alphanumeric character, and so

on. Therefore, the number 2n indicates the total amount of information, which is:

N ¼ 2n ; (1)

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n can be a very large number so that we choose (due to Shannon [16]) to countlogarithmically. We define the amount of information:

I i ¼ �log2 Pi ; (2)

where Pi is the probability of finding the word i.Mean information is defined

Im ¼ �X

i

Pi log2 Pi

or equivalently

Im ¼ �X

i

Pi ln Pi : (3)

Now, our purpose is to find the maximum amount of information, which impliesthat:

dIm ¼ 0 (4)

with constrainXi

Pi ¼ 1 : (5)

In order to maximize the Eq. (3), we will use Lagrange’s multipliers

dIm ¼ �X

i

½ln Pi þ 1�dPi ¼ 0 ;

lX

i

dPi ¼ 0 : ð6Þ

Using the Eq. (6), we can obtainXi

½ln Pi þ 1þ l�dPi ¼ 0 ; (7)

whose solution is

Pi ¼ e�ð1þlÞ (8)

i.e., each word has the same probability to be chosen. A process of this type is calleda random process. This process allows to save or load information in an efficient oroptimal manner. Therefore, computers use random access memories (RAM).ConsideringX

i

Pi ¼X

i

e�ð1þlÞ ¼ 2ne�ð1þlÞ ¼ 1 ;

we have

Pi ¼1

2n (9)

and

l ¼ n ln 2� 1 : (10)

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Using logarithms of base 2, the amount of information is

I ¼ �log2 P ¼ �log21

2n ¼ n bits : (11)

Consequently, the mean information is

Im ¼ �X

i

Pi log2 Pi (12)

and when the process is completely random we can write

Im ¼ �X

i

1

2n log21

2n ¼X

i

n

2n ¼n2n

2n ¼ n bits : (13)

Example 1. Let us consider a two-state system

Im ¼ �P log2 P � ð1� PÞ log2ð1� PÞ : (14)

The maximum is obtained when P ¼ 12and Im ¼ 1 bit, see Fig. 1.

If we have a multiplet of spins then they have a degeneracy of 2s þ 1:We can formwords of length 2s þ 1 symbols; therefore, the total number of words must be:

N ¼ 2ð2sþ1Þ (15)

and the amount of information will be 2s þ 1 bits.

200 400 600 800 1000 1200

100

200

300

400

500

600

700

800

900

Fig. 1. Behavior of a two-state system described by (14).

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Because of the Mixed State Spin Theory, spin would naturally fluctuate randomly.In a quantum computer, we would not have to program complex random numbergeneration algorithms (which in fact are quasi-random algorithms).

3.2. Measurement of physical magnitudes

Let us consider an experiment, where we obtain data ui for i ¼ 1; 2; . . . ; n: Themean value of ui is given by

u ¼X

i

uiPi : (16)

We are interested in the error. Evidently, Dui ¼ ui � u has mean value

Dui ¼X

i

ðui � uÞPi ¼ 0 ;

which is not interesting at all. A valuable and interesting definition, in this case, is thequadratic mean value

Du2i ¼X

i

ðui � uÞ2PiX0 (17)

that is higher from or at least equal to zero.Now, we are faced with the problem of how to measure a physical parameter

exactly, when we want to have the minimum quadratic possible error (minimumstandard deviation or minimum dispersion theorem) and maximum information (ormaximum average). The maximum average is according to von Newmann’sformalization of Game Theory see Ref. [26].We have to maximize Im under the constrains

Pi Pi ¼ 1 and Du2i : The function

Du2i must take its minimum value according to minimum dispersion theorem.

Theorem 3. Gaussian Density Function permits both maximize Im and minimize Du2i :

Proof. Let it be the next maximization program

maxPi

Im ¼ maxPi

�X

i

Pi ln Pi

!;

subject to

1 ¼X

i

Pi

and

Du2i ¼X

i

Du2i Pi

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evaluating and making zero first derivatives to find the max value.

dIm ¼ �X

i

ðln Pi þ 1ÞdPi ¼ 0 ;

aX

i

dPi ¼ 0 ;

bX

i

Du2i dPi ¼ 0 ; ð18Þ

which brings us to

dIm ¼ �X

i

ðln Pi þ 1þ aþ bDu2i ÞdPi ¼ 0 (19)

and it has a solution

Pi ¼ Ae�bDu2i : (20)

In order to determine the integration constants A and b; let us consider thefollowing continuous distribution:Z 1

�1

Ae�bDu2 dDu ¼ 1 ;Z 1

�1

Du2Ae�bDu2 dDu ¼ s2u : ð21Þ

Integrating we found

P ¼1ffiffiffiffiffiffi2p

psu

e�Du2

2s2u (22)

Eq. (22) is a Gaussian density function which also follows minimum dispersiontheorem see Ref. [14]. &

3.3. Information Theory and uncertainty relations

The action is the most important magnitude in Physics. It satisfies Hamiltonprinciple, see Ref. [28, p. 38].

dS ¼ 0 : (23)

This equation deserves additional analysis.Let E be a physical state of a particular system

E ¼ ðqi; _qi; tÞ ; (24)

where qi is a generalized reference system of coordinates, _qi generalized velocities, t

the instant they are determined, and i every generalized coordinates.Let us define the configuration space C ¼ fqig: A point in that space represents the

system’s physical configuration. Considering time evolution, we observe that thispoint describes a trajectory in the configuration space. If we fix instants t1 and t2

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where t24t1 between qiðt2Þ and qiðt1Þ; then the system’s time evolution describes itsstory between ½t1; t2�:Let us define a state system dynamic function

L ¼ Lðqi; _qi; tÞ : (25)

The Lagrangian L is different for each possible trajectory. Let us postulate that toknow this function allows us to know the set of differential equations that describesthe system’s real story, because using ‘‘virtual work method’’ we can built L.It is clear that for every trajectory between the points Aðqiðt1ÞÞ and Bðqiðt2ÞÞ in the

configuration space, the action integral has the next form

S ¼

Z t2

t1

Lðqi; _qi; tÞdt ; (26)

which will take a different numeric value in function of the chosen trajectory. S iscalled the system’s action.What we now is only the initial A and final B configurations of the system.

Therefore, there exist an infinite number of trajectories that begin in A and reach B,but there is only one that is real. The question is how to find it.Let us suppose that the real trajectory has an action S. Any other trajectory will be

a virtual one even though it may be infinitesimally close. Trajectories like those havean action S þ dS: This way, the real trajectory satisfies that:

dS ¼ 0 : (27)

Eq. (27) represents an optimum in the functional field of all trajectories, and also isstationary because t2 and t1 are fixed numbers and the d operator is timeindependent.This brings us to deduce that Lðqi; _qi; tÞ satisfies the Euler–Lagrange equation

qL

qqi

�d

dt

qL

q _qi

� �¼ 0 ; (28)

where it can be shown that (see Ref. [28])

L ¼ T � U (29)

being T the kinetic energy and U the potential energy. When the systems are non-conservative, Eq. (28) takes the form

qL

qqi

�d

dt

qL

q _qi

� �¼

qFq _qi

; (30)

whereF is the Rayleigh dissipation function. Eq. (30) can be deduced from Eq. (28)considering the universe as a whole.In fact, total energy is conserved in the universe. Furthermore, we can

consider a subsystem described by a Lagrangian L. Let LAB be, the Lagrangian ofinteraction between the borders of the system and let L0 be, the Lagrangian of therest of the universe. Because the Lagrangian is scalar, it satisfies the additionproperty. Therefore, the Lagrangian of the whole universe will be the sum of L;LAB

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and L0

LU ¼ L þ LAB þ L0 ; (31)

where LU is the Lagrangian of the universe, and it satisfies

qLU

qqi

�d

dt

qLU

q _qi

� �¼ 0 ; (32)

qL

qqi

�d

dt

qL

q _qi

� �¼ �

qqqi

ðL0 þ LABÞ �d

dt

qq _qi

ðL0 þ LABÞ (33)

the right term represents an interaction with the rest of the universe and we will call itgeneralized force Qi

qL

qqi

�d

dt

qL

q _qi

� �¼ Qi ; (34)

Qi can represent dissipative processes as well as energy sources, so more generally(see Ref. [28])

Qi ¼qFq _qi

(35)

with

F ¼ 12

f i _qi2 � _qi

~Qi : (36)

The first right term represents dissipative processes and the other represents energysources or external energy supplies. In short, it is clear that isolated systems in theuniverse cannot exist. In fact, let us consider a so-called ‘‘thermodynamically isolatedsystem’’, which means that its energy is constant. Due to Hamilton–Jacobi equation,although its energy is constant, its action is not

E ¼ �qS

qt¼ const: (37)

Therefore, the system is interchanging action with the rest of the universe.The Action is a physical magnitude which all theoretical physicists, except R.

Feynman, have not considered of importance. But, it is the most fundamentalphysical magnitude because it is the origin of all other magnitudes, for instance, theenergy (37) or any others,

pi ¼qS

qqi

; (38)

where pi and qi are conjugate variables.It is known that interactions travel at maximum speed (of light). This type of

interaction with our system such as radiation, heat then

E ¼ pc (39)

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but

E ¼ �qS

qtand ~p ¼ rS : (40)

Because~c is perpendicular to rS ¼ cte and parallel to~p; the Eq. (39) can be written as

�qS

qt¼ r S~c

or equivalently

r S~c þqS

qt¼ 0 (41)

which is a conservational law. Therefore, if we consider a region R in the universe,action will be a random input/output flux in this region, because natural phenomenasatisfies the condition of maximum mean information.Also, Eq. (39) can be written as

E2

c2� p2 ¼ 0 ; (42)

1

c2qS

qt

� �2� ðrSÞ2 ¼ 0 : (43)

What enables us to define the Lagrangian density

L ¼1

c2qS

qt

� �2� ðrSÞ2 ; (44)

which satisfies the following Euler–Lagrange equation

1

c2q2Sqt2

¼ r2S (45)

and

Sð~x; tÞ ¼ S0eið~k~x�otÞ (46)

whose mean value is

S ¼ 0 (47)

and its quadratic mean value is

SSn ¼ jS0j2a0 : (48)

This completely justifies Eq. (56).On the other hand, Lagrange equation has the next form

qL

qqi

�d

dt

qL

q _qi

� �¼ 0 ;

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where

L ¼X

i

1

2mi _qi

2 � UðqiÞ (49)

satisfies the condition

q2L

q _qi2X0 : (50)

In the book of Elsgostz [29], we can see the Eq. (50) which is considered as acondition dS ¼ 0 implies a strong minimum.Hamilton principle is another optimal principle that natural laws must satisfy, and

the origin of the next physical magnitudes.

H ¼ �qS

qt; pi ¼

qS

qqi

: (51)

In a free particle system, classical energy and momentum are conserved; H ¼ E

and qi ¼ x:

E ¼ �qS

qt;

S ¼ �Et þ W ðxÞ ; (52)

qS

qx¼dW

dx¼ p (53)

and

W ¼

Zp dx þ S0 (54)

therefore, S ¼ px � Et þ S0 and

DS ¼ px � Et : (55)

From Eq. (22), considering all given arguments and the fact that action is not adirectly measurable quantity, we postulate

PðSÞ ¼1ffiffiffiffiffiffi2p

pse�

DS2

2s2 (56)

which contains all the information of the system.From Hamilton’s equation, where there is an interaction we can write.

_p ¼ �qH

qx¼ �

qE

qx: (57)

Considering p; x; E; t independent conjugate variables from Eq. (57) we have

dpdx ¼ �dE dt ; (58)

Z p

0

dp

Z x

0

dx ¼ �

Z E

0

dE

Z t

0

dt ;

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px ¼ �Et (59)

and

DS ¼ px � Et ¼ 2px (60)

px is an area on the phase space (see Fig. 2) and it is a Poincare invariant thereforeit is conserved.

px ¼ DpDx :

From this equation we obtain

DS ¼ 2DpDx :

Now let us maximize information

dIm

dPi

¼X

i

Pi ln Pi ¼ 0 (61)

with the constrains

dDx2 ¼X

i

Dx2i dPi ¼ 0 ;

50 100 150 200 250 300 350 400 450

50

100

150

200

250

300

350

A

A

∆p

∆ x

xx

p

Fig. 2. Evolution of a Poincare invariant.

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dDp2 ¼X

i

Dp2i dPi ¼ 0 ;

dDS2 ¼X

i

DS2i dPi ¼ 0 ;Xi

dPi ¼ 0 : ð62Þ

Using Lagrange multipliers we obtain:

dIm ¼X

i

½ln Pi þ 1þ lþ aDx2i þ bDp2i � gDS2i �dPi ¼ 0 : (63)

Let us represent the universe as a set (see Fig. 3). Let us consider a region R wherewe perform physical measurements.Information satisfies addition property, so we have

ImU ¼ ImR þ ImRU ; (64)

where ImU is the total amount of information of the universe, ImR is the informationin the place where the measurement is performed, and ImRU represents theinformation of the rest of the universe.Because the total information is optimal, then

DImU ¼ 0 ; (65)

50 100 150 200 250

50

100

150

200

250

U

R

Fig. 3. The universe and a region R where we perform physical measurements.

ARTICLE IN PRESS


which brings us to

DImR þ DImRU ¼ 0 (66)

so

DImR ¼ �DImRU ; (67)

DX

i

Pi ln Pi ¼ �DX

j

jai

Pj ln Pj ; (68)

Xi

½ln Pi þ 1�DPi ¼ �X

j

½ln Pj þ 1�DPj ;

Xi

ln Pi þX

i

DPi ¼ �X

j

ln Pj �X

j

DPj (69)

and taking into accountXi

Pi ¼ 1 ;

Xj

Pj ¼ 1 ; (70)

Xi

DPi ¼ 0 ;

Xj

DPj ¼ 0 (71)

and Xi

ln Pi ¼ �X

j

ln Pj ; (72)

which will be satisfied only ifXi

ln Pi ¼ �X

j

ln Pj ¼ 0 (73)

in other words

Pi ¼ Pj ¼ 1 (74)

The last equations indicate that the perturbation induced by an experiment will befelt in every point in space instantaneously, which produces a break in therandomness of the universe, which has no sense because it contradicts the SpecialRelativity Theory. Therefore, it must exist an additional term of information

ImU ¼ ImR þ ImRU þ Imd (75)

ARTICLE IN PRESS


from here

0 ¼ DImR þ DImRU þ DImd : (76)

The term

DImRU ¼ 0 (77)

and

DImR þ DImd ¼ 0 : (78)

In this way, we can observe the randomness of the rest of the universe and continuesatisfying the theory of relativity because information cannot be transmitted fasterthan the speed of light.To sum up, if we compute the probabilities in the whole universe and consider that

space and time are globally homogeneous,we can infer that probability does notdepend on xi; pi; ti;Ei: Therefore

aDx2i þ bDp2i ¼ gDS2i (79)

with

Pi ¼ e�ð1þlÞ: (80)

The Eq. (79) which implies that statistical interaction processes cover all phase space.

Theorem 4. Plank’s constant has a statistical nature. Furthermore, s ¼ _; DpDxX _2:

Proof. The action distribution function, which is local, due to Eq. (56), is

e�gDS2 ¼ e�aDx2e�bDp2 : (81)

Using the conditions

s2 ¼

R1�1

DS2 e�gDS2 dDSR1�1

e�gDS2 dDS¼

R1�14Dp2Dx2 e�aDx2 dDx e�bDp2 dDpR1�1

e�aDx2 dDx e�bDp2 dDp; (82)

s2 ¼ 4

R1�1

Dx2 e�aDx2 dDxR1�1

e�aDx2 dDx

R1�1

Dp2 e�bDp2 dDpR1�1

e�bDp2 dDp; (83)

s2 ¼ 4s2xs2p

) sxsp ¼s2; (84)

we obtain the minimum Heisenberg’s uncertainty relation.

) s ¼ _ ; (85)

sxsp represents the minimum area in phase space, more generally speaking, anyother area will satisfies

DpDxX_

2: (86)

ARTICLE IN PRESS


We can conclude this because dDx2 ¼ 0 and Dx2X0 are true only for a minimum.The same arguments are valid for Dp2: &

More general, any action can be written as (79).

DS2 ¼ ðpx � EtÞ2 ¼ p2x2 þ E2t2 � 2pExt : (87)

We can perform a coordinate rotation, so that we obtain

DS2 ¼ l1x2 þ l2t2 (88)

We can see that action is a quadratic function of x; t: In fact, it is proportional tothe relativistic interval

DS2 / c2t2 � x2

which represents a wave front.

3.4. Quantum mechanics

The analysis we have made is very general and we want to insist in this idea in adifferent way. In order to do that, it is necessary to recall

E2 ¼ p2c2 þ m20c4 ; (89)

E ¼ �c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þ m2

0c2

q¼ �cP ; (90)

which has the same form of a photon equation. Also we know

DS ¼ ~p ~x � Et (91)

which shows that at t ¼ 0; DS ¼ ~p ~x represents an equation of a plane when DS ¼

cte: Every point in this plane have the same time, t ¼ 0: This is a characteristic of awave front. Moreover, DS ¼ px � Et ¼ cte bring us to

p _x � E ¼ 0) _x ¼E

p¼ vf ; (92)

where vf is the phase speed.Or also, xdp � tdE ¼ 0

_x ¼dE

dp¼ vg (93)

is the group speed. Given that E ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2c2 þ m2

0c4

qdE

dp¼

pc2

E: (94)

If we multiply Eqs. (92) and (93) then we have

vgvf ¼E

p

p

Ec2 ¼ c2 ; (95)

which represents a relation of dispersion of an electromagnetic wave in Optics.

ARTICLE IN PRESS


Therefore, it is possible to think about action waves. We can propose the followingintegral equation

e�DS2

2s2 ¼

ZAðp;E; k;oÞf ðkx � otÞdk do : (96)

On the other hand, we do not consider any relation between k and o; we will findit from the Eq. (96).

Theorem 5. Optimal wave superposition (Eq. (96)), which satisfies Gaussian density

function (Eq. (22)), permit us to obtain Plank’s and De Broglie’s equations.

Proof. Let dc be the derivative on space coordinates.

�dc

DS2

2s2

� �e�

DS2

2s2

� ¼

ZAðp;E; k;oÞdc f dk do ;

�dc

DS2

2s2

ZAðp;E; k;oÞf dk do ¼

ZAðp;E; k;oÞdc f dk do :

From hereZAðp;E; k;oÞ fdc

DS2

2s2þ dcf

� �dk do ¼ 0

this implies the differential equation

fdc

DS2

2s2þ dcf ¼ 0 ; (97)

DS2

2s2þ ln f ¼ ln A

) f ¼ Ae�DS2

2s2 ; (98)

where f is an exponential function on x and t

f ¼ Ae�Fðkx�otÞ : (99)

From where

Fðkx � otÞ ¼DS2

2s2� ln A : (100)

Expanding Fðkx � otÞ in Taylor’s series we have

Fðkx � otÞ ¼ Fð0Þ þdFdu

��0

ðkx � otÞ þ1

2

d2Fdu2

��0

ðkx � otÞ2

þ ¼DS2

2s2� ln A : ð101Þ

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All terms are zero, except Fð0Þ

Fð0Þ ¼ � ln A ;

a2

2ðkx � otÞ2 ¼

ðpx � EtÞ2

2s2(102)

and we obtain

p ¼ ask ¼ sk0 ;

E ¼ aso ¼ so0 : (103)

The integral equation is

e�DS2

2s2 ¼1

a2

ZAðp;E; k0;o0Þ e�

ðk0x�o0 tÞ2

2 dk0 do0 ; (104)

k0 and o0 are dummy variables, then we can do k0! k and o0 ! o

e�DS2

2s2 ¼1

a2

ZAðp;E; k;oÞe�

ðkx�otÞ2

2 dk do

therefore a ¼ 1 and

p ¼ sk ;

E ¼ so ; ð105Þ

which are De Broglie’s and Plank’s equations, respectively. Again we obtain that

s ¼ _ : & (106)

The new integral equation is

e�ðpx�EtÞ2

2s2 ¼

ZAðp;E; k;oÞe�

ðkx�otÞ2

2 dk do

this implies that

Aðp;E; k;oÞ ¼ dE

s� o

� �d

p

s� k

� (107)

and obviously

e�ðpx�EtÞ2

2s2 ¼

Zd

E

s� o

� �d

p

s� k

� e�

ðkx�otÞ2

2 dk do : (108)

Let us analyze every term in Eq. (108)

dE

s� o

� �¼

1

2ps

Z 1

�1

e�iðE�soÞ

s t dt ;

dp

s� k

� ¼

1

2ps

Z 1

�1

eiðp�skÞ

s x dx ð109Þ

ARTICLE IN PRESS


from there

1

ð2psÞ2

Zei

ðpx�EtÞs e�i

ðskx�sotÞs dxdt ¼ Aðp;E; k;oÞ : (110)

Now we are able to define the wave functions

Cp;E ¼1

2psei

ðpx�EtÞs ;

Cp0 ;E0 ¼1

2pse�i

ðp0x�E0tÞs : ð111Þ

Or considering only p (and the fact that s ¼ _)

Cp ¼1ffiffiffiffiffiffiffiffi2p_

p eipx_ ; (112)

ZCpCn

p0 dx ¼ dðp � p0Þ ; (113)

which is the Dirac’s condition of normalization. If we consider only E; we have

CE ¼1ffiffiffiffiffiffi2p

p_e�i

Et_ ; (114)

ZCECn

E0 dt ¼ dðE � E0Þ ; (115)

which is a energy conservation law. From Eq. (112) we see that

�i_qqx

Cp ¼ pCp (116)

and from Eq. (114) we obtain

i_qqt

CE ¼ ECE : (117)

We see the way in which all concepts of Quantum Mechanics appear naturally.In general, the wave function is

C ¼ Aei DS_ : (118)

Given that

DS ¼X

i

piqi � Et ; (119)

where pi and qi are physical conjugate observables.

�i_qCqqi

¼ piC (120)

in this way, every physical (pi) observable has associated a Hermitian operator(pi ¼ �i_ q

qqi), such that its mean values are the eigenvalues of that operator.

ARTICLE IN PRESS


Let us consider a photon

E ¼ pc ;

E2

c2� p2 ¼ 0 (121)

given that E ¼ � qSqtand p ¼ qS

qxthen

1

c2qS

qt

� �2�

qS

qx

� �2¼ 0 : (122)

This bring us to define the Lagrangian density

L ¼qS

qx

� �2�1

c2qS

qt

� �2: (123)

From Euler–Lagrange equation

qqx

qLqSqx

� �" #

þqqt

qLqSqt

� �" #

¼ 0 ; (124)

q2Sqx2

�1

c2q2Sqt2

¼ 0 ; (125)

where we obtain

S ¼ S0eiðkx�otÞ (126)

the energy

E ¼qS

qt¼ �ioS0e

iðkx�otÞ : (127)

The square modulus

jEj2 ¼ o2jS0j2 : (128)

The average is

hjEj2i ¼1ffiffiffiffiffiffi2p

ps

Z 1

�1

o2jS0j2e�

DS20

2s2 dDS0 ¼ o2_2 (129)

i.e.

Efot �on ¼

ffiffiffiffiffiffiffiffiffiffiffiffihjEj2i

q¼ o_ : (130)

What we measure, in fact, is the mean quadratic value of the random fluctuations ofenergy.Given that

DS202s2

¼DE2

2s2Eþ

Dt2

2s2t; (131)

ARTICLE IN PRESS


we find

sEst ¼s2¼

_

2(132)

or in general

DEDtX_

2; (133)

which is the other Heisenberg’s uncertainty equation. Eq. (131) explains the reasonthere is a Gaussian dispersion of energy in a monochromatic LASER and not aDirac’s delta (see Fig. 4).

Remark 1. Every eigenvalue of a physical observable is a mean quadratic value of itsrandom fluctuations.

pi ¼

ffiffiffiffiffiffiffiffiffiffiffihjp2i ji

q(134)

if we measure its conjugate observables

DS2

2s2¼

Dp2i2s2p

þDq2i2s2q

(135)

and

DpiDqiX_

2; (136)

200 400 600 800 1000 1200

100

200

300

400

500

600

700

800

900

E=hω0

E=(ω)

ω0

ω

Fig. 4. Spectrum of dispersion of a minimum uncertainty LASER.

ARTICLE IN PRESS


hp2i i ¼

Rp2i e

�DS2

2s2 dDSRe�

DS2

2s2 dDS¼

Rp2i e

�Dp2

i

2s2p dDp

Re�

Dp2i

2s2p dDp

(137)

on the other hand,

hpii ¼ 0 : (138)

Therefore, pi ¼ffiffiffiffiffiffiffiffihp2i i

pis the only one magnitude which we can measure. Quantum

Mechanics consider it as the eigenvalues of the observable p. Because p is a number,it has standard deviation zero.

s2p ¼ hp2i i � hpii2 ¼

ZCnpi

2Cd3x �

ZCnpiCd

3x

� �2;

s2p ¼ p2i � p2i ¼ 0 : (139)

Copenhagen school concludes that eigenvalues do not have standard deviation, thenthey have to find other arguments (for example, the uncertainty principle) to explainthe standard deviation observed in experiments. That is not scientifically acceptedbecause Occam’s razor principle (see [30,31]) exhorts to introduce the minimumnumber of arbitrary postulates in order to choose the best of the competing theories.We justify that in experimental measurements, a maximum will be obtained in pi ¼ffiffiffiffiffiffiffiffi

hp2i ip

and around this value there will be a statistical dispersion.

3.4.1. Spontaneous decay transitions

Optimal configuration are those configurations that satisfies the condition ofmaximum of mean information quantity. Let us consider a number of atoms in theirstationary states.Let E0 be a stationary state, i.e., which satisfies that

Imop ¼ ImðE0Þ :

Let us suppose a transition E0 ! E0 þ DE

ImðE0 þ DEÞ ¼ ImðE0Þ þdIm

dE

��E0

DE þ1

2

dI2mdE2

��E0

DE2 þ OðDE3Þ ;

where OðDE3Þ is negligible. If DE ¼ 10 eV ; then, DE is of order of 10�18 J; therefore,DE2 is of the order of 10�36 J2; which is a very small quantity.

dIm

dE

��E0

¼ 0 ðmaximumÞ ;

�a2 ¼dI2mdE2

��E0

o0 ðmaximumÞ ;

ARTICLE IN PRESS


ImðE þ DEÞ ¼ ImðE0Þ �a2

2DE2 :

Therefore, the amount of information decreases from its maximum value, which isnot a stable condition. The system will evolve in such a way that it will try to removethe energy DE in order to reach the maximum amount of information,

from DIm ¼ �a2

2DE2 ;

to DIm ¼ 0 : (140)

Now, let us suppose that the probability varies in �51 around P ¼ 12where the

maximum of the distribution is produced. The variation on P in a small quantities �is equivalent to study the replicator dynamics see Ref. [32].

Im ¼ P ln P þ ð1� PÞ ln P ;

Im ¼1

2þ �

� �ln

1

2þ �

� �þ

1

2� �

� �ln

1

2� �

� �¼ �4�2 : (141)

In the next sections we will show that (Eq. (187))

DIm ¼ �2DSffiffiffiffiffiffi2p

ps

� �2(142)

therefore

� ¼ �DSffiffiffiffiffiffi2p

ps

(143)

but � ¼ DNN; where N is the number of excited atoms and DS ¼ DEDt:

DN

N¼ �

DEDtffiffiffiffiffiffi2p

ps:

In order to provide these arguments with physical sense, we define the decay speed as

dN

dt¼ �

NDEffiffiffiffiffiffi2p

ps

or

dN

N¼ �

DE dtffiffiffiffiffiffi2p

ps; (144)

lnN

N0

� �¼ �

DEtffiffiffiffiffiffi2p

ps; (145)

N ¼ N0e� DEtffiffiffi

2pp

s : (146)

ARTICLE IN PRESS


Defining

t ¼

ffiffiffiffiffiffi2p

ps

DE; (147)

N ¼ N0e�t

t : (148)

Therefore, the excited atoms will decrease exponentially. We cannot know whichatom will be affected by a transition E0 þ DE ! E0 emitting a photon during theprocess, the only thing we can say is that the number of atoms will evolve as is statedin (148).

t measures the mean time of de-excitation of the population of atoms

hti ¼

R10 te�

tt dtR1

0 e�t

t dt¼ t ; (149)

because of DE ¼ _o and s ¼ _; we have

t ¼

ffiffiffiffiffiffi2p

p

o�1

o(150)

this is the expected result. This theoretical argumentation gives a solid base toFermi’s study (calculation of the speed of transition of an excited atom).

3.4.2. Schrodinger’s equation

Let a quantum mechanical system be confined in a potential U. We can expect thatthe wave function will be scattered an infinite number of times in the potential.Forming a wave package which is described by

Cð~x; tÞ ¼

Z 1

�1

Að~kÞeið~k~x�otÞ d3k (151)

with

o ¼ oðkÞ (152)

and

p ¼ _k; E ¼ _o (153)

these equations are the momentum and total energy. Applying the Laplace operatorin Eq. (151)

r2Cð~x; tÞ ¼

Z 1

�1

�k2Að~kÞeið~k~x�otÞ d3k ;

�_2r2C2m0

¼

Z 1

�1

p2

2m0Að~kÞeið

~k~x�otÞ d3k : (154)

ARTICLE IN PRESS


In the same way

qCqt

¼

Z 1

�1

�ioAð~kÞeið~k~x�otÞ d3k ;

i_qCqt

¼

Z 1

�1

EAð~kÞeið~k~x�otÞ d3k ð155Þ

but

E ¼p2

2m0þ U ; (156)

i_qCqt

¼

Z 1

�1

p2

2m0Að~kÞeið

~k~x�otÞ d3k þ

Z 1

�1

Uð~xÞeið~k~x�otÞ d3k :

Thus, we obtain Schrodinger’s equation.

i_qCqt

¼ �_2

2m0r2Cþ Uð~xÞC (157)

because Uð~xÞ depends on the position, the integral is calculated respect to k. Eq.(157) is the non-relativistic Schrodinger’s equation.

3.4.3. Quantum mechanics postulates

1. We have seen that to every physical observable corresponds a Hermitianoperator. The measured eigenvalues are the eigenvalues of this operator.

piC ¼ piC (158)

with

pi ¼ i_qqqi

(159)

also

pi ¼

ffiffiffiffiffiffiffiffihp2i i

q: (160)

2. Every physical system is described by a complex function C which satisfiesSchrodinger’s equation.

i_qCqt

¼ �_2

2mr2Cþ Uð~xÞC : (161)

3. The wave function C contains all the information of the physical system(because it is a function of the action of the system) and its square modulus is theprobability density of measuring an interaction in the element d3x: In fact,multiplying (161) by Cn; and taking the complex conjugate of (161) and multiplyingit by C; we have:

i_Cn qCqt

¼ �_2

2mCnr2Cþ Uð~xÞCnC ;

ARTICLE IN PRESS


�i_CqCn

qt¼ �

_2

2mCr2Cn þ Uð~xÞCnC :

Subtracting them

i_Cn qCqt

þ i_CqCn

qt¼ �

_2

2mðCnr2C�Cr2CnÞ : (162)

i_qqt

ðCnCÞ ¼ �_2

2mðCnr2C�Cr2CnÞ

or

qqt

jCj2 ¼i_

2mr ðCnrC�CrCnÞ : (163)

Integrating respect the volume and applying Gauss’ theorem, we have

d

dt

ZjCj2 d3x ¼

i_

2m

Is

ðCnrC�CrCnÞ ~nds : (164)

The system is confined in a potential Cð�1Þ ¼ 0; so, the value of the integral in theinfinite is zero

d

dt

ZU

jCj2 d3x ¼ 0 ; (165)

where U represents the whole space. Therefore, this integral does not depends ontime, it is a constant whose value we decide is 1.Z

U

jCj2 d3x ¼ 1 : (166)

Therefore, jCj2 is a density probability. On the other hand, if we define

~J ¼ �i_

2mðCnrC�CrCnÞ (167)

for a volume V limited by a surface A.

d

dt

ZV

jCj2 d3x þ

IA

~J ~nds ¼ 0 (168)

or

qqt

jCj2 þ r ~J ¼ 0 ; (169)

which is the law of conservation of probability. Because probability is conserved,information will be conserved too.Conservation of the amount of information is associated with physical

symmetries. If a symmetry is broken, the amount of information decreases andmass is generated, as we will see later in this paper.4. Given that wave functions are eigenfunctions of a Hermitian operator, they

form a Hilbert space and if they are not degenerated then the space is complete. Any

ARTICLE IN PRESS


other function will be expressed as a linear combination of the elements of the base

C ¼X

i

CiCi : (170)

This is what is called superposition principle.5. The mean value of a physical magnitude is

f ¼X

i

f iPi ;

where Pi is the probability of measure f i: Given that

f iCi ¼ f iCi (171)

and ZCn

i Cj d3x ¼ dij (172)

also

Pi � CiCn

j dij (173)

therefore

f ¼X

i

f iCiCn

j dij ¼X

i;j

f iCiCn

j

ZCn

j Ci d3x ; (174)

f ¼X

i;j

CiCn

j

ZCn

j fCi d3x ;

f ¼

Z Xi;j

Cn

j fCi d3xCiC

n

j ¼

Z Xj

Cn

j Cn

j fX

i

CiCi d3x ; (175)

f ¼

ZCn fCd3x : (176)

This is called expected value or mean value. All of this is in perfect agreement withLandau’s hypothesis (see Ref. [33]).

3.5. Information and matter

Theorem 6. If symmetry is broken then the variation of information implies the

variation of action according to Eq. (187).

Proof. The distribution function of action is

F ðSÞ ¼1ffiffiffiffiffiffi2p

pse�

Ds2

2s2

ARTICLE IN PRESS


and the probability

PðjSjpS0Þ ¼

Z S0

�S0

F ðsÞdDS

whose diagrams are represented in Fig. 5.F ðSÞ ¼ 1ffiffiffiffi

2pp

sreaches its maximum value when DS ¼ 0 which corresponds to P ¼ 1

2:

So

PðþcÞ ¼ Pð�cÞ ¼ 12

(177)

considering that DS / Ds (the relativistic interval) and Ds2 ¼ c2Dt2 � Dx2 ¼ 0 forlight, then DS ¼ 0 corresponds to the action for light; therefore

v ¼ �c : (178)

When matter exists DSa0 and Pa 12 : The symmetry is broken.

Let us consider:

0oDS

s

��51 ; (179)

PðDSÞ ¼1ffiffiffiffiffiffi2p

ps

Z 0

�1

e�DS2

2s2 dDS þ1ffiffiffiffiffiffi2p

ps

Z DS

0

e�DS2

2s2 dDS ;

200 400 600 800 1000 1200

100

200

300

400

500

600

700

800

900

F(S)

P(S)

0

0

12

∆S

∆S

1

Fig. 5. Distribution function and accumulated probability of action.

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PðDSÞ ¼1

2þ

DSffiffiffiffiffiffi2p

ps: (180)

In the same way

Pð�DSÞ ¼1

2�


ps;

PðDSÞ þ Pð�DSÞ ¼ 1 (181)

and the amount of information is

�Im ¼1

2þ


ps

� �ln

1

2þ


ps

� �þ

1

2�


ps

� �ln

1

2�


ps

� �: (182)

Let us analyze the term

ln1

2þ


ps

� �¼ ln

1

21þ

2DSffiffiffiffiffiffi2p

ps

� �� ¼ ln

1

2

� �þ ln 1þ


ps

� �

¼ ln1

2

� �þ2DSffiffiffiffiffiffi2p

ps; ð183Þ

ln1

2�


ps

� �¼ ln

1

2

� ��2DSffiffiffiffiffiffi2p

ps: (184)

From here

�Im ¼1

2þ


ps

� �ln

1

2

� �þ2DSffiffiffiffiffiffi2p

ps

� �þ

1

2�


ps

� �ln

1

2

� ��2DSffiffiffiffiffiffi2p

ps

� �;

�Im ¼ ln1

2

� �þ4DS2

2ps2: (185)

The optimal information is

�Imop ¼1

2ln

1

2

� �þ1

2ln

1

2

� �¼ ln

1

2

� �(186)

from here we obtain

DIm ¼ �4DS2

2ps2(187)

and ffiffiffiffiffiffiffiffiffiDIm

p¼ i


ps

or

iDS

s¼

ffiffiffiffiffiffiffiffiffiffiffipDIm

2

r: & (188)

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Mass is generated due to the interaction

_p ¼ �dE

dx) px ¼ �Et ) DS ¼ px � Et ¼ 2px : (189)

For a wave that travels at light speed, we have

DS ¼ 0 ¼ px � Et ;

Cp light ¼ Aeipx_ ; (190)

DCp light ¼ iDðpxÞ

_Cp luz (191)

and the dispersion relation

DCp mass ¼ iDð2pxÞ

_Cp ¼ i

DS

_Cp (192)

with

iDS

_¼

ffiffiffiffiffiffiffiffiffiffiffipDIm

2

r: (193)

For the wave that travels at speed of light, we have to consider the statistical weight12; which is the probability

PðþcÞ ¼ Pð�cÞ ¼ 12;

DCp light ¼1

2iDS

_Cp ; (194)

DCp light ¼ iDSL

_Cp (195)

with

DS ¼ 2DSL : (196)

Note that the angular momentum of the interaction that generates mass is

DS

Df¼ 2

DSL

Df; (197)

Lz grav ¼ 2_ ; (198)

which is the spin of the graviton.Let us consider the first proposed problem in Feynman’s book ‘‘Addition of paths

and Quantum Mechanics’’ (see Fig. 6).A free Dirac’s particle satisfies the following Hamiltonian

H ¼ cða pÞ þ m0c2b (199)

ARTICLE IN PRESS

50 100 150 200 250 300 350 400 450 500 550

50

100

150

200

250

300

ct

x=ctx=-ct

0

A

x

C

Fig. 6. Contribution of virtual Dirac’s particles to the process of generation of mass.


with

(200)

and

(201)

where 1 is the identity matrix and si Pauli’s spinors. Calculating a speed in onedirection, for instance, x; we have

i

_½H; x� ¼ _x ; (202)

i

_½cða pÞ þ m0c

2b; x� ¼i

_½cða pÞ; x� ;

ci

_½axpx þ aypy þ azpz; x� ¼ c

i

_½axpx; x�

¼ ci

_ax½px; x� ¼ c

i

_axð�i_Þ ¼ axc ; ð203Þ

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_x ¼ axc which applied to a state function gives

_xC1

C2

� �¼ cax

C1

C2

� �: (204)

Let us calculate the eigenvalues of ax

C1

C2

� �¼ l

C1

C2

� �(205)

given that axy¼ ax:

C1

C2

� �¼ 0 : (206)

Its determinant is

l21� sx2¼ 0; sx

2¼ 1 ; (207)

ðl2 � 1Þ1 ¼ 0 ; (208)

l ¼ �1

therefore

_xC ¼ �cC (209)

and the eigenvalue of speed is �c:Feynman calls that particle ‘‘Dirac’s particle’’ and gives it the propagator i DSL

_ ; so

DCp ¼ iDSL

_Cp

He divides the segment OA in n parts and build different paths for the propagator,this is shown in Fig. 6, which represents a particle propagating in the x-axis with þc

or �c:Actually, this is a very clever application of Huygens principle because every

change in the direction represents the point from a wave is propagating, as it isindicated in Fig. 6 in point C.The propagator which goes from O to A is

KðA; 0Þ ¼Xn

p¼0

n

p

� �iDSL

_

� �p

Kð0Þ : (210)

The number of changes of direction is N ¼ 2n; because of the sum

Xn

p¼0

n

p

� �¼ 2n ; (211)

which is deduced from the binomial

ð1þ xÞn ¼Xn

p¼0

n

p

� �xp (212)

ARTICLE IN PRESS


for x ¼ 1: The derivative of the binomial is

nð1þ xÞn�1 ¼Xn

p¼0

pn

p

� �xp�1

so that

n2n�1 ¼Xn

p¼0

pn

p

� �: (213)

The action gained in p changes of direction is

Sp ¼ 2pn

p

� �DSL (214)

in order to preserve the mirror symmetry between x and �x in 0. The total gainedaction is

Stotal ¼Xn

p¼0

2pn

p

� �DSL ¼ 2n2n�1 ¼ n2n : (215)

The mean action trough ct axis is

S ¼Stotal

N¼

n2n

2n DSL ¼ nDSL (216)

therefore

DSL ¼S

n(217)

and

KðA; 0Þ ¼ 1þiS

_n

� �n

Kð0Þ (218)

When n ! 1

KðA; 0Þ ¼ Kð0ÞeiS_ (219)

with

S ¼ �m0c2Dtp ; (220)

where Dtp is the proper time.Therefore, particles does not exist, they are dispersion processes of waves.Consider an observer moving with speed v; then

S ¼ �m0c2

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

v2

c2

rDt ; (221)

where Dt is not the proper time.

ARTICLE IN PRESS


For v5c

S � �m0c2 þ

m0v2

2

and

v ¼xA � x0

tA � t0;

Dt ¼ tA � t0 :

Therefore,

KðA; 0Þ ¼ Kð0Þe�iðm0c2=_ÞDte

im0 ðxA�x0Þ

2

2_ðtA�t0 Þ (222)

and by ‘‘smoothing’’ the function because m0c2

_ ¼ o is a high frequency

KðA; 0Þ ¼ Kð0Þei

m0 ðxA�x0 Þ2

2_ðtA�t0 Þ (223)

or

KðA; 0Þ ¼ Kð0Þe�

ðxA�x0 Þ2

2i_ðtA�t0 Þ

m0

� �with

s2 ¼i_ðtA � t0Þ

m0

� �(224)

given thatZ 1

�1

KðA; 0Þdx ¼ 1 (225)

Kð0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p_ðt � t0Þi

m0

s¼ 1 (226)

and

Kð0Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim0

2p_iðt � t0Þ

r;

which gives the Feynman’s non-relativistic propagator

KðA; 0Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim0

2p_iðt � t0Þ

rei

m0 ðxA�x0Þ2

2_ðtA�t0 Þ : (227)

4. Conclusions

1. The Universe is structured in optimal laws. Random processes are those thatmaximize the mean information and are strongly related to symmetries, therefore, to

ARTICLE IN PRESS


conservational laws. Random processes have to do with optimal processes tomanage information, but they do not have anything to do with the contents of it, thisis the anthropic principle: Laws and physical constants are designed to produce lifeand conscience see Ref. [34].2. Every physical process satisfies that the action is a locally minimum. It is the

most important physical magnitude, after information, because through it is possibleto obtain the energy, angular momentum, momentum, charge, etc. Every physicaltheory must satisfy the necessary (but not sufficient) condition dS ¼ 0; because it isan objective principle.3. We have demonstrated that the concepts of information and the principle of

minimum action dS ¼ 0 leads us to develop the concepts of Quantum Mechanicsand to explain the spontaneous decay transitions. Also, we have understood that allphysical magnitudes are quadratic mean values of random fluctuations.4. Information connects every thing in nature, each phenomenon is an expression

of totality. Moreover, in quantum gravity, this fact is explained by Wheeler–DeWittequation.5. Mass appears were certain symmetries are broken, or equivalently, when the

mean information decreases. A very small decrease in information producesenormous amounts of energy. Also, we have shown that there exist a relationbetween the amount of information and the energy of a system (Eq. (187)).Information, mass and energy are conservative quantities which are able totransform one into another.6. A interdisciplinary scope it is possible, only, if we suppose that General System

Theory is valid because games and information can be seen as particular cases ofcomplex systems. Quantum Mechanics is obtained in the form of a generalizedcomplex system where entities, properties and relations conserve his primitive formand add some other ones related with nature of the physical system.

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vol. I, Wiley, New York, 1992 (Chapter 5).

[3] F. Forges, Repeated Games of Incomplete Information: Non-Zero-Sum, Handbook of Game

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[4] H. Varian, Analyse Microeconomique, Troisieme edition, De Boeck Universite, Bruxelles, 1995.

[5] S. Braunstein, Quantum Computing, Wiley-Vch, Weinheim, Germany, 1999.

[6] N. Boccara, Modeling Complex Systems, Springer, Heidelberg, 2004.

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