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IC/80/15B
INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS
GEON-TYPE SOLUTIONS
OF THE NON-LINEAE HEISENBERG-KLEIN-GORDON EQUATION
Eckehard W. Mlelke
and
INTERNATIONAL Reinhard Scherzer
ATOMIC ENERGY
AGENCY
UNITED NATIONS
EDUCATIONAL,
SCIENTIFIC
AND CULTURAL
ORGANIZATION 1980MIRAMARE-TRIESTE
IC/60/158
International Atomic Energy Agency
and
United Nations Educational Scientific and Cultural Organization
INTERNATIOHAL CENTRE FOR THEORETICAL PHYSICS
GEOH-TYFE SOLUTIONS OF THE HOH-LIHEAR HEISEHBERG-KLEIH-GORDOH EQUATION •
Eckehard W. Mielke
International Centre for Theoretical Physics, Trieste, Italy,
and
Reinhard Scherzer
In3t i tu t fiir Beine und Angewandte Kernphysikder Chriatian-Albrechts Universitat Kiel, 2300 Kiel ,
Federal Republic of Germany,
MIRAHABE - TRIESTE
Octoter I960
• To be submitted for publication.
ABSTRACT
As a model for an "un i ta ry" f i e l d theory of extended p a r t i c l e s -Je
consider the non- l inear Klein-Gordon equation - associa ted with a "squared"
Heiseaberg-Pauli-Weyl non- l inea r spinor equat ion - coupled to s t r o n g g r a v i t y .
Using a s t a t i ona ry spher i ca l a n s a t i for the complex sca l a r f i e ld as well as
for the background metr ic generated v ia E i n s t e i n ' s f i e ld equat ion , we a re
able to 3tudy the effects of the scalar self-interaction as veil as of the
classical tensor forces. By numerical integration we obtain a continuous
spectrum of localized, gravitational solitons resembling the geons previously
constructed for the Einstein-Maxwell system by Wheeler . A self-generated
curvature potential originating froa the curved background partially confines
the Schrodlnger type wave functions within the "scalar aeon". For zero angular
momentum states and normalized scalar charge the spectrum for the total
gravitational energy of these solitons exhibits a branching vith respect to
the number of nodes appearing in the radial part of the scalar field.
Preliminary studies for higher values of the corresponding "principal quantum
nuaber" reveal that a kind of fine splitting of the energy levels occurs,k I,
which may indicate a rich, particle-like structure of these quantized geons.
A fundamental theory of matter based on the quark hypothesis has to
accommodate an in-built mechanism of (at least partial) confinement of the
constituent field in stable particles, otherwise they should be observable at
2)some detectable rate . In order to circumvent the Paul! exclusion principle,
these fundamental fermion fields are assumed to obey paraetatistics, or
equivalently, have to carry, besides, flavour, additional colour degree3 of
internal freedom. These colour models aro .! I :;t inguished by the binding
i.e. 10mechanism of quarks in hadrons.(Whether thi.; is mediated by scalar vector{see e.g. Ref.5) or tensor "gluons" 6''7'.
In "quantum chromodynamics" (QCD) , nowadays the moat prominent model
for strong interactions, the dynamics of the mediating vector gluons is
determined by an action modelled after Maxwell's theory of electromagnetism.
The resulting model is a gauge theory of the Tang-Mills type . However itQ)
is known that in such sourceless non-abelian gauge theories there are noclassical glueballs which otherwise would be an indication for the oceurr«nceof confinement in the quantized theory. (The phenomenological consequences ofthe possible existence of glueballs in QCD have been di3cused by Robson •)The reason simply being that nearfey small portions of the Yang-Hills fieldsalways point in the same direction^ internal 3 p a c e amj therefore must repeleach other as like charges. Nevertheless, vector gauge fields might be an
f-)important ingredient of any model in order to explain saturation
The confinement i tself , according to the proposals of an unconventional
scheme termed "colour feometrodynamics" {CGMD) * may be achieved by
strongly 1 ^ ' ' 1 5 ' >l(>> interacting massless tensor gluons, their dynamics
presumably being determined by Einstein-type field equations. CGKD is a
GL(2N,C) gauge model in curved space-time vhich nay be regarded as a generalization
of Einstein's gravity theory. The la t t e r corresponds to a gauging of the covering
group SL(2,C) of the Lorentz group. Since CGMD i s , in general, based on a
Riemann-Cartan space-time , Cartan's notion o f torsion is known to
induce non-linear spinor terms into the Dirac equation. This has a profound
effect on the "fundamental" spinor fields
(1.1)
-1-
-2-
distinguished by S colour (or flavour) internal degrees of freedoE. It car.12)
also be shown for this GL(2N,C) gauge invariant generalisation *" that these
quart type fields have to satisfy the Heisenbers-Pauli-Weyl
linear spinor equation
l I'
~0 (1.2)
generalized to a curved space-time (compare also with Ivanenko ). Here
L are space-time dependent generalizations of the Dirac matrices ya tensored vith
the U(n) vector operators X (generalized Gell-Mann matrices). Essentially
the modified PlancK length
jf = ti*k
of strong gravity lM,15).l
aelf-interaction in (1.2).
Old k/crf IO~'cm (1.3)
occurs also as the coupling constant of the
If we transfer the ideas of Mach and Einstein to the microcosmos, the
curving-up of the hadronic background metric should be self-consistently
produced by the stress energy eontent TU(I)J) of the spinor fields (1.1)
via the Einstein equations
J*. -v »
with "cosmological term" (we employ the sign conventions of Tolman 7- In
thiB new geometrodynamical model extended particles owninginternal symmetries
should be classically described by objects which closely resemble the geona and2k)
wormholea of Wheeler . In some sense this approach is also related to the25)
issue to vhich Einstein and Rosen addressed their 1935 paper .
"Is an utonistic theory of matter and electricity conceivable
which, while excluding singularities in the field, makes use of no
other fields than those of the gravitational field (g ) and those
of the electromagnetic field in the sense of Maxwell (vector
potentials tf )7"
-3 -
The geoit, L.t;. :i "gravitational electromagnetic entity" was originally
devised by Wheeler ' ' to be a self-cor.sjstent, non-singular solution of the
otherwise sourcefree Einstein-Maxwell equations having persistent large scale
features. Euch a geon provides a well-defined model for a classical "body"
in general relativity. If spherically symmetric geons would stay completely
stable objects they could acquire the possibility to derive their equations
of motions solely from Einstein's field equations without the need to
introduce field singularities. In a sense this approach also embodies the
goals of the so-called unitary field theory *
Geons, as we are using the term, are gravitational solutions, which
are held together by self-generated gravitational forces and are composed of
localized fundamental classical fields. The coupling of gravity to neutrino29)
fields has already been considered by Brill and Wheeler . The latter work
lays the appropriate groundwork for an extension of their analysis to non-
linear spinor geons satisfying the combined equations (1.2) and (I.U). In
this paper, however, we have avoided algebraic complications resulting from
the spinor structure as well as from the internal symmetry by considering
rather non-linear scalar fields coupled to gravity. In order to maintain a
similar dynamics we assume - as in a previous paper (Ref.30, hereafter referred
to as I) - a self-Interaction of these scalar fields which can "be formally
obtained Ijy "squaring" the fundamental spinor equation (1.2). "Linear" Klein-
Gordon geons have been previously constructed . However, we view the
additional coa-linearity of the scalar fields as an important new ingredient
for our model.
The precise set-up of this theory is given in Sec.II. For the intended
construction of localized geons, the stationary, spherical ansatze of I are
employed for the scalar fields, whereas the metric is taken in its general
spherically symmetric canonical form. As in the case of a prescribed
Schwarzschild background - analysed in I - the curved space-time effects the
resulting Schrodinger equation for the radial function essentially via an
external gravitational potential.
The stress energy content of these scalar solutions determines the
currature via Einstein's field equations. In Sec.II we review the spherical
symmetric case and include also a method which enables us to incorporate non-
zero angular momentum states into this framework hy averaging the stress energy
of these scalar fields over a spherical shell.
Our geons contain a fixed (quantized) scalar charge. By imposing this
restriction in Sec.V we not only fix an otherwise undetermined scale of our
geons but may also increase their stability. The main concern of Sec.V i s ,
- I t -
however, to contrast two notions of energy for our gravitational solitons,l) tb-e field ener^ of the general relativistie scalar waves and 2) the total
32)gravitational energy of such an isolated system . In order to probe ourconcepts ve construct in Sec.VI & simplified geon by considering radially
constant scalar solutions owning the particular constants admitted by the non-
linear self-interaction. Outside a ball of radius C_ the scalar fields are••/ is
diseontinuously set to zero. Although this procedure is rather artificial,
we thereby obtain a ""bag"-like object having inside a portion of an
Einstein microcosmos and outside a Schwarzschild manifold as background
ap&ce-t ine.
In general, the resulting system of three coupled non-linear equations
for the radial parts of the scalar and the (strong gravitational) tensor
fielda has to be solved numerically. In order to specify the starting values
for the ensuing numerical analysis we derive in Sec.VII asymptotic solutions
at the origin and at spatial infinity. Sec.VII is then devoted to a discussion
of the numerical results. Preliminary speculations are offered with the aim
to interpret particles as "quantum geons".
Sec.VIII concludes the paper with a prospective overview of other
developments concerning "gravitational solitons".
(8.3)
ia chosen to be similar to that used in I. Such a model has recently been
treated in 1+1 dimensions according to quantum field theoretical methods
Variation for £JCK,,/jf(» yields the non-linear KLein-Gordon equation
[a u /Jew*)] <r =where
(2.U.)
(2-5)
denotes the generally covariant Laplace-Beltrami operator. When (2.1*) laout
explicitly written/for thechoiee (2.3) of the self-coupling i t will be referredto as the non-linear Helgenberg-Klein-Gordon equation
I I . THE MODEL
Following the outline given in the Introduction we may consider as a
simplified model a theory consisting of H complex scalar fields
(2.1)
Their dynamics i9 governed by the Lagrangian density
(2.2)
defined on curved pseudo-Riemannian spaee-tiae with metric tensor f . In
to obtain a similar dynamical probl™ 13 in the non-linear spinor theory
given by (1.2) the self-interaction potential
IS 6O.6)
In I i t has been shown that (2,6) is formally similar to that obtained by
"squaring" the fundamental spinor equation (1.2). This is part of the
motivation for considering a If I -term in the corresponding Lagrangian
density (2.2).
Although the resulting quantum field theory, contrary to the \<f\
model, would not be renormallzable according to standard criteria of perturbation
theory we include in this paper the additional |*pj self-interaction
term. For a semiclaaaical approach it may be instrumental for the construction
of quasiatable, spherically symmetric and localized solutions. At least In
flat apace-time,Anderson has shown by means of a phase-space analysis thatfor
8 s -f (2 .T )
-6-
particle-like (stable) solutions can exist. CO is the ratio "between dynamical
mass and the "bare" mass **. of this model (see Sec.III). Intuitively, ve
auspect that the stability of these solutions (and possibly also their degree
of "confinement") is enhanced by the attractive forces exerted on them via the
coupling to strong tensor gluona ' . Geometrically speaking, this would
correspond to a curved background manifold.
the badcgroun
(HW, p.5OU)
In our model this curving up of
the background Is self-consistently produced from the stress energy tensor
Cf) * - £JLHHS
* a., . - • ' « , (2.8)
of the scalar fields tf via Einstein's field equations ( l . i t ) . In effect,our geometrodynamical model is then completely determined by the Lagrangiandensity
UK 6
since (2.6) and (2.9) can be derived from it by a variation for
and for
III.
(2.9)
SPHERICAL SCAIAK WAVES IH A CURVED BACKGROUND
As a semiclassical model for a particle we are considering sptericalgeon type ' solutions which minimize (2.9). More precisely,we are looking forspherical wave configurations which solve tbe HKfi equation (2.6) in a s t a t i c ,spherically symmetric background space-tiise which in turn is determined by ( l . b ) .
vei l 3 'As is veil known 3 ' ( see , e.g. MTW, p.59k and box 23.3) a canonical
form of the general (flimensionless) line element for this background reads:
2trIF* it*
- e(3.1)
if the sign conventions of Tolman 3' are adopted. Here V = V (̂ )
X S /l(?) are functions which, depend solely on the dimensionless
"Schwarzschiia type" (MTW, p.T2l) radial co-ordinate
and
M*cr * Ul (3.2)
The determinant of this metric Is given by
5lh<? . (3.3)
For the construction of spherical scalar waves, we take up the veil-knownfact that solutions of the free, linear^ Klein-Gordon equation can be expandedIn terms of spherical harmonics Y^(^.f) which are elgenfunctions
t Ax * LCC + OT, Y"(e,0
* fife % Vof the Laplace operator 4 g on 2-sphere S2. Although a non-linear theory
in general does not respect such an expansion, the non-linear terms of the
field equation (2.6) admit the two dist inct separation anaStss of I ;
X - l - 1ce>
elsewhere
(3.5)
(3.6)
Due to the familiar addition theorem (Landau and Lifschitz,
-8-
L
I
a 3elf-i.nteracti.0n given by a polynomial in |c#| =q = 1remains spherically symmetric as required by separability.
(3.7)
P 1 flp^
In order to see how space-time curvature affects the vave equation it
is instructive to define
and #.respect to the new co-ordinate $« / Mie Schrodinger type vfave function
th.e structure of the Laplace-Beltrami operator is similar to that of a
coaformally flat Klein-Gordon operator with an external potential .
Consequently, the stationary ansatia (3.5) and (3.7) together with
the property (3.11) cast the HKG equation (2.6) into the Schrodinser type
equation
(3.8)
(in Borne equations belov abbreviated by H or F, respectively) and tointroduce Wheeler'a tortoise co-ordinate" a* (MTW, p,663) via thedifferential form 3 "
* CJL-v)/SL ,
5 s e 6
Then the line element (3.1) takes the form
(3.9)
(3.10)
It resembles the metrical ground form of a space-time vith two conformally39)flat portions. Then a kind of "conformal change" of the
Laplace-Bertrami operator (2.!*) aay be calculated with the formal result
CY-X)/Z\e ) (3.11)
-9-
;F - -=-|- F + 1 - e co
with an effective curvature potential (compare with MTW , p.868)
implicitly given by
(3.12)
(3.13)
Here and in the following the factor
denotes the dimensionless ratio between the Planck mass M» and the "bare
mass" it of the a "constituent" fields. So far for the formal
aspects of the theory. For the ensuing numerical calculations, however, it
is more convenient to use the equivalent radial equation
e\z) R(3-15)
written explicitly in terms of 6 . (The dash denotes differentiation vith
respect to ^ .) It may he obtained from (3.12) and (3.13) by resubstitutiona,
or more directly from (2.6) and the original ansatze (3.5) and (3.6) founded
on the background (3.1). It generalizes Eq..(3.9) of Kaup;
linear Klein-Gordon equation. -10-
31) derived there from a
IV. THE EINSTEIN FIELD EQUATIONS
By applying Hachian ideas to the microcosmos, the strong gravitational
background vill be determined from the stress energy content of the scalar
waves via the Einstein equations (1.1*).
With respect to the diagonal metric of (3.1) these equations reduce to
(see Tolman 2 3 , p.
e V- 1 U. it.v'-x'
and
Here
gravitationa.1 fields from the field equation (2-9)- Cue to the contracted
Bianchl identity V G = 0 there exists the same relation between the
diagonal elements of the Einstein tensor G . Therefore it is enough to
consider the two remaining equations (It.l) and (H.3) only. (Bef.^l, p.1*88).
For the supposed spherically symaetric background (3.1) we notice that
the non-trivial ansatz (3-5) would lead to an inconsistency in the
gravitational field equations. The reason being that the corresponding
stress energy tensor (2.8)#in the scalar case given by
would also depend on the angular distribution of the solutions, contrary to
the Einstein tensor.
For localized solutions the spherical, asymmetry of the scalar waves
(3.5) ia expected to be negligible sufficiently far away from the centre
of the geon. Therefore it ia physically Justifiable not to discard ansatz
(3.5) but rather consider the Einstein equations (lt.l)-(lt.3).with respect to
an averaged stress energy tensor
p.1*88). Suitable is an average
property
a s proposed by Pover and Wheeler (see
over a spherical 3hell defined by the
o e
(U.6)
denotes a dimensionless "cosmological" constant and the dash again means
differentiation with respect to £ .
Although this set of equations may look like an overdetermined
system, a simple argument sbovs that this is not the case.
According to second-order variations! principles which can be
itO) >generalized to a curved space-time there exist the conservation lawvt VYv T ^ = 0 for the stress energy tensor (2.B) provided that the matter fieldequatlolia(2.i*) hold.. In our case, this law relates the two tangential
0tensionsc
knovledge of
= T.*to the radial tensions 1 T , T n and d T /dr. Thisf , * r 0 r= T," Is not instrumental for the determination of the
The evaluation of ^ T u . V ^ w i l 1 'be facilitated by employing the identity
=* 0-11-
which results from the application of Stoke'5 theorem.
With respect to the ansEtze (3.5) and (3.6) the averaged radial tensions
come out as
(it.8)
and
vhere the spherical average of the Lagrange function is explicitly given by
x ~
("4.10)
In order to bring the radial Einstein equations in close analogy withthose known for a perfect fluid (Ref.23, p.2lfM we may formally introduce In(U.8) the dimensiooless proper density ? 0 0 of the scalar field. It turnsout to be explicitly given by
Furthermore, the dimensionlesa proper "nydrostatic pressure" p- of a
scalar field can (implicitly! be defined by (U.9).
By subtracting (l(.l) from (1».3) and evaluating the combination
f- =
ve obtain
(fc.ia)
Ct-13)
whereas in this notation (U.3) is equivalent to
-fU.Ik)
These equations are generalizations of those considered by Kaup
Ua)and e.g. by Kodama et aX.
31) for a l inear
for a non-l inear Klein-Gordon f i e l d .—4.
It should be noted that f t . l i t ) Is a l inear d i f f e r e n t i a l equation In • .
As la w e l l known the general formal so lut ion can be written as
& — I ~ * ~ 3 ? C * i i 5 )
where
(it.
denotes a mafla function. The meaning of the latter terminology will lieilluminated If ve consider the Einstein equations outide the region where"natter" fields vanish, i . e . R = 0 in our case. Then tbe non-linearequations (U.13) become linearized and the vacuum Einstein equations admit the.(with respect to the canonical metric (3.1)) unique set of exact solutions
- 1 3 - -Ht-
- A.- e. U.IT)
exteriorThey descritie the/Schvarzschild-de Sit ter geometry for a. mass distributionlocated at the Origin. Here CiS 0L {to) ia the parameter measuring thegravitational mass ot M* at spatial infinity.
V. GRAVITATIOHAL ENEBCT OP GEOMS WITH QUANTIZED CHAHGE
In order to associate some quantum meaning to the time-dependent
localized solutions of the HKG equation the Bohr-Sommerfeld quantization rules
may be imposed. For a field theory with infinitely many degrees of freedom
this semicl&seical quantization condition reada
j1s1
<d0 a,(5.1)
the time integration being performed over the 3emlperiod ffji of the solution.
In • curved space-time the canonical conjugate field momenta are defined {see,
e.g. Fulling UU)) by
•jr „
In a. s ta t ic background and for the stationary anBatze {3.2) or (3.3) ovnlng
the semlperiocl If A • - eo u 0 /ft i t is not difficult to see that the Bohr-
Sommerfeld oondition (5.1) is equivalent (see also Ref.l45, Sec.3.6) to tbe
charge quantization
( 5 .3 )
vhere
-15-
in a curved space-time ia- the conserved tota l charge of the complex scalarfields. The condition (5-3) may also increaae the stabi l i ty of theae
26)"quantum geons" provided that this stabilizing device for non-linear
*̂5)seiolclassical field theories applies also in curved space-time.
By insertion of the ansatae (3.2) or (3.3) we obtain the expression
11
(5 .5)
For fixed A and pre-aasigned «u this condition normaliiea the a prioriarbitrary Planck length £* with regard to the coupling constant Jt of ournon-linear model. On the other hand, i f we fix this ra t io to be e.g. Jlmljtm 1as we will assume in our numerical calculations, the condition (5.6) determinesthe physically immaterial i n i t i a l constant C^ appearing in the asymptoticsolutions (7.1) discussed in Sec.VII.
Our normalization ) i s the same as that used by Kaupdeviates from the condition suggested by Feinblum and HcKinley
3D but
In a curved space-time the energy concept is known to be rather subtle^
Let us recall that for matter fields coupled to gravitation the locally
conserved it-momentum is given by
the integration being performed over a space-like hyperaurface. Differently as
in the case of flat Minkovski space, in (5.6) tht stress-energy pBeudotensor
t » y of Landau and Lifshltz (WW, p.!*66) must be included in order to
account for the contribution from the gravitational f ield.
-16-
For a quasistatic isolated systen and .A. = ° Tolrr.an "' Uee alsoRef.23, p.235) has derived the following equivalent expression for its totalenergy
(5 .7)
is operationally more useful
This result which is exact in the static casef, in particular in numerical
calculations, since the volume integral has to be extended only over the region
actually occupied by the "matter" fields.
In our construction of localized aeons we can satisfy the criteria for
the applicability of (5,7) if we require the radial part of the scalar field
to be exponentially decreasing in the asymptotic region & -¥ °* (see Sec.VI).
Thereby the gravitational background field (3.1) tends sufficiently
fa»t to that of a Schwarischild geometry given by (U.1T) with .A • 0 . By
construction the total mass of our geon is then known to be ot H* (compare
with Eq.(6.2) of Ref.Ul) and the Einstein relation
M'c1 (5.8)
holds in a rest frame.
Using the static background (3-1) and the formal relations (It.8) andthe Tolman energy (5-7) can be cast into the form
(5.9)
23)(Tolman , p.248). With respect to a non-linear scalar field theory defined
by (2 .2) , the to ta l energy is equivalent to
the integration over angular variables formally being absorbed in the averages
defined earl ier by (h.6). After inserting the ansatse (3-5) or (3.6) we obtain
tfec mam «a»iici t result
(5.U)
The model dependent rat io J[*IJL may be eliminated by the previously derived
normalization condition (5-5)- Then in view of (3.1U) the formula
JL 1fin '
finally determines the total gravitational mass ctM* of a "scalar neon withquantized charge". It is interesting to note that gravity alone modifiesthe "bare" mass p. by 2l(» , vhereaa a "mass renormalUatic-n" on this aemi-classlcal level is due to the (non-linear) self-interaction. The relation(5-12) may be contrasted with the curved space-time definition (see, e.g.BefAU)
£1 e (5.13)
of the field energy of the N scalar "constituent" fields <P' of the geonwithout the contributions from the self-consistently generated gravitationalfield.
Inserting (U.9) together with (4.10) and then substituting thenormalisation condition (5.4) yields the "mass formula"
-aft-
J£ (JV,««•)" *
0. As a precautionary measure for the case that this
(5-lU)
expression diverges at the origin •, we have introduced a "cut-off" length p > 0itt (5.1U), enabling us to study the "regularized"instead. After subtracting the boundary term
field energy E,
VI. GEON WITH A CONSTANT "BAC-LIKE" CORE
Before we turn to a numerical analysis of the spherical Einstein-Klein-Gordon system it is instructive to study an exact solvable geon containing aconstant scalar "core" of radius ( . . In order to obtain this highly idealizedconfiguration one has to note the fact that the non-linear equation (3.1ci) forthe radial distribution of the scalar field <q admits for L = 0, besides zero, theconstant solution
(6.1)
5.15)
fron (5-13) and then using the normalization condition (5.5) we may alternatively
consider vhst we call the normalized energy
£ - F, =• k { co
+ ST T**^ S 2 ^ (5.16)
which should be compared vith (U.9) of I. Furthermore, it could be phyBiaally
interesting to study the binding energy
e*- 4 /I/f «
(5.17)
also in a curved space-time (3.1), provided that the f.- component of themetric tensor is also a constant, i .e.
e - (6.2)
35)In the case that this condition holds Anderson's analysis of the classicalphase apace (R.E1) of the HKG equation can be applied to a certain extent.The main result is that stable (particle-like) solutions can exist ifthe parameter B^ . employed already in (2.7) satisfies the condition
n -«" (6.3)
Then the stable equilibrium points in the phase space of the asymptotic
version of (3-15) correspond to the constant solution (6.1) with the negative
sign of the square root. This solution which the geon may adopt in its
interior gives rise to a constant scalar density
of a scalar particle within a geon as a function of CO and A . FollowingRef.31 we «ay define it as the energy of a "free" scalar particle from whichits energy contribution to the total geon mass at rest is subtracted. A"free" teat particle means free of gravitational and self-interaction. Inview of (5.16) the corresponding normalized energy is 6) Me which is thefactor appearing in the phase of the ansatze (3.5) or (3.6).
8ISO
-20-
aa well as to a. constant "pressure" term
If we abaort (6.U) into an effective "cosmologicftl"
(6.5)
(6.6)
solutions of {It.lb) which are regular at the origin read
si * (6.7)
From e » const, I.e. Eq.(6.2), we can infer that the constant radial
solutions (6.1) exist only In (a portion of) an Einstein "microcosmos" (compare
with Tolaaa 2 , $135 and fl39). (Note that (a.6) admits also non-trivial1«7)
radial solutions in an Einstein Universe. These exact solutions however
are notgeon-type solutions, i.e. they do not satisfy the Einstein equations
(1.1*) at the same tine.)
The remaining radial Einstein equation (U.13) yields the "equation of
state"
h * (6.8)
for the "density" in terms of the "hydrostatic pressure" pQ . After
insertion of (6.U) and (6.5) into (6.8) we obtain
{ 6. 9 )
which, in view of (6.1), determines • V (W) as a function of W and TV. .
Our geon construction may now follow closely those by which
Schvargschild (see Hef.23, $96) obtained the exterior and interior solutions
for & spherical star consiting of an incompressible perfect fluid of constant
property density £ Q ( r To this end we may use (6.l) together with the condition
(6.2) and the resulting metric function (6-7) as solutions for the Interior
OS <} * gg of aballJTaen the curvature potential (3.13) associated Vith this
metrical background is also a constant, i .e . acre precisely
In order to interpret the interior solution (6.1) as a kind of "bound state"
within a negative potential produced hy a gluonlc "bag" of tensor forces we
have to require -A.eff > 0.
In view of (6.6) and (6.8) this condition can be satisfied for
ba&^ 2 ̂ 0 * ° o n l T - S u c h a non-sero "bag constant"-A^ in necessary for
the interior of the geon in order to compensate for the "vacuum" pressure (6.5).
of the "<viark-type" scalar fields. A similar mechanism haa been proposed in
the phenomenological MIT "bag" model of hadronB ' (If -A. would be
zero, the condition (6.8) is the same as that for a random distribution of23)electromagnetic radiation (Tolman , p.217) except for the Bign.)
Outside the constant bag-like core of radius y_ we may simply continuewith ft » 0 ( i f In this Idealized construction we are contented vith solutionsvhich are only piece-wise dlfferentiable and continuous) and obtain for
/V.,^. =• 0 a Schwarzschild solution (lt.lt) for
(6.U)
ThiB crude geon construction allows us to evaluate the total charge (5. K) in
closed fora:
Insertion of (6.1), (6.2) sad (6.7) into (5.5) yields
(6.12)
The remaining integration can be performed vith the aid of an integral
representation (see Eq,. (30) of Bef.ItT) of the hypergeometric function. The
charge quantization (5-3)then leads to the normalization condition
-21- -22-
If •= l/3V X z
for simplicityFor-A = 0 and assuming herej/also at = 3 ^
'8
(6.13)
the above result can be used' to
determine the scalar density (6.U) respecting the condition (6.8) as
VII. LOCALIZED GEON SOLUTIOHS FROM NUMERICAL INTEGRATION
According to our . introductory remarks we will reserve the term "geon-
type solution" (compare Sec.VIII for other notions) for configurations resulting
from a self-consistent coupling of fields to gravity in which both the "matter"
waves andJmetric tensor are (non-singular) and sufficiently localized. A
precise criterion for localization depends crucially on the circumstance of
whether or not a cosmologlcal constant -A. is included. For the present we
put .A, = 0 and may then require for localized, spherically symmetric scalar
geons the following set of
eo 3
In a similar way we may calculate the already normalized expression (5.12)
for the gravitational mass otM*.
Let us consider the case 6 = 1 . The insertion of (6.1) and further-
Sore {6.9) into (5.12) yields
(6.15)
a result which could also be inferred from the comparison of condition (6.8)
with the equivalent expression (5-9) of the Tolmao energy.
Since for y > J f i the space-time geometry Is determined by the
Sehvarzsehild solution (U.17) with_A. _,. " 0 and ct given by (6.15), an
"observer" placed outside the core wi l l regard this gravitational hound state
of scalar fields as an object having the mass ot. M*. For_/t, = 0 thebag
hadronic environment, i.e. the strong curvature generated by the "tensor gluons"
f-.j inside the "constant core" gives rise to such a strong "Archimedes' »T)effect on the scalar constituent fields, that their binding energy (5.lit)
2freeones equal to the rest mass UUc of a self-interacting"quark".
This can be summarized in a Wheeler-type phrase: A constant'*bag-like"
geon may have "mass without mass" (Ref.3T, P-25).
a) Asymptotic solutions at spatial infinity
We proceed similarly as in Sec.III.c of paper I
radial solutions which behave asymptotically as
30)and consider
(T.I)
If \u\<-i , such boundary conditions for the numerical integration would
necessary lead to exponentially decreasing Yukava-type solutions (Bee also
Bef.US) irrespective of the parameter 6" • Since the>sealar waves would
already be sufficiently localized, the equations (U.13) and Ct.lU) pass into
the Einstein vacuum equations. Given the canonical form (3.1) these yield
uniquely the Schvarzschild solutions (U.9) with A • 0.
Therefore
y ^ (7.2)
will hold. The asymptotic forms (7.1) and (7.2) are then inserted Into- (3.15)
in order to determine tf
coefficients of the 1/9 - expansion we obtain
onVgrounds of self—consistency.
6- - - 1 ffi
By equating the
(7.2)
This result being independent of the quantum number L of angular momentum
agrees for the plus sign with that obtained by Kaup
- 2 3 -
Let us turn to the
b) Asymptotic solutions at the origin
Guided by the "constant core" case analysed in the preceding
section ve found it reasonable to assume
y 2s. 0 '** C ^ C «=• con if.
in the vicinity of the origin.
0 ) Suppose ve find
- c0 (7.5)
Then from (it.lU) and (U.13) i t follows that the radial metric function behaves
(T.6)
at the origin. This corresponds to the exact result (5-7). Then the radial
equation (3.15) takes the asymptotic form
c V*(I R+ - Xcf?z + 1 - e - V c ^ ^ (7.7)
(compare with tq,.(3.11) of t). A familiar argument expanded in I yields
~ C' 5 (7.8)
as asymptotic solutions regular at the origin. The system of approximate
solutions (7.M, (7.6) and (7.8) turns out to be consistent.
OJJ ) Another set of asymptotic solutions can be obtained by proposing
instead of (7.5) the trial function
(T.9)
for i » 0.
Assuming that e ft 0 for £ -» 0 such that
(7.10)
ve obtain from (U.lU) the result
e — (7.11)
This together with (7.M satifies also (U.13). Furthernore, the insertion of
(7.11) into the radial equation (3-15) yields
R'±5
(T.ia)
Its integration results in (7.9), the integration constant already being
determined by (7.10). The sets (7.<0, (7.9) and (7.10) of asymptotic solutionsU6)
have earlier been discussed by Feinblum and HcKinley
In. spite of the fact that in the latter case the radial part of thescalar waves i s logarithmically divergent at origin, ve should not disregardthese solutions. For a more precise reasoning ve have to also take thestrongly deformed Bpace-tine manifold at the origin into account. Thiseffect l>econies more transparent i f we consider the function Fo^$#) definedvia (3.8) in terms of the "tortoise"co-ordinate (3.9). In view of (T.M and(7.12) the latter acquires the asymptotic behaviour
near the origin. By applying 1'Hospital' a rule ve find that
(7.13)
IT.HO
-26-
tends to zero for^-*>0. Therefore the charge integral (5.5) should be
bounded even at the origin. Since the subsidiary conditions of bounded square-
integrability and fast decrease at spatial infinity turn out to be fulfilled t
solutions with the asymptotics 0 in curved space-time can also be regarded
a3 "eigensolutions", according to the criteria of quantum "echanics (see § 16
of Ref.38). With respect to the formal SchrSdinger-type equation (3-12) the
dynamics corresponds to the motion of a "particle" in a centrally symmetric
field characterized by a centripetal potential
(7.15)
near the origin (compare with Ref.38, p.109).
With th is information at hand we have performed the numerical cal-culations on a DEC-PDP-10-computer using single precision BAG and IMSL Librarysubroutines. The evaluation of the functions R^($), e * and e*1 '^ hasbeen done in double precision mode. For a l l calculations the free parametersof the model have been fixed according to :
Then for each given at and n.^ the system of ordinary differential equations(3.15), (U.13) and (U.II4) has been numerically solved by Hunge-Kutta formulasof order 5 and 6 as developed by 3.H. Verner and coded by Hull and co-workersin the IMSL siibroutine DVERK. The global error of the solutions has beenestimated to be less than 1.10 . Using the asymptotic conditions (7.1) and(7.2) as start ing values the integration has been performed going from f^a= 30backwards to zero. Self-consistent solutions are constrained by two additionalconditions. F i r s t , they have to ful f i l l the charge normalization (5,5) andsecond, they have to reproduce the parameters OL chosen for the i n i t i a lconditions consistent with the Tolaan integral (5.12) for « . This has beenachieved by an i terat ive least squares f i t using the HAG subroutine E0UFAFwhich is based on a method due to Peckham . Thus the parameters at and C^have been determined by minimizing the sura of squares
V(7.16)
where J denotes the j t h iteration step, ct,_1 is the result of the (j-l) t h
atep which fixed the initial conditions (7.1) and (7.2) and Q., * are cal-
culated from (5.5 ) and (5.12), respectively.
Using appropriate starting values for C ^ and at , a convergence of
(7-16) better than l.io" has been obtained by the method resulting in a
relative error in Ot and Q of less than 1.10 . The numerical integration
of Q, 0C and E has been performed by using the HAG Library procedure
D01GAF which estimates the value of a definite integral(when the functional
is specified numerically) using the method described by Gill and Miller
The maximal relative error should in no case be larger than 1.10
As can be deduced from the asymptotic solutions (7.10) and (7-12) of
the second set 0 , the energy expression (5-ll») contains a term I n £ Q
which diverges for - Therefore in Fig.I* we have computed only the
finite part of E, i.e. E , corresponding to the "cut-off" parameter f_ = 0.001.
So far our method of integrating backward has produced solutions
belonging solely to the set O-j. of asymptotic solutions at the origin. In
this case we found solutions with and without nodes. The number of node*
of the radial part R*($) of the wave function for finite values of £
(excluding the point ^ = 0) may be used to define aa a radial quantum number
n as in the non-relativistic Schro'dinger theory (Ref.38, p.109). Prom the
theory of the hydrogen atom we suspect the relation
*- L- A (7.17)
to hold (Kef.38,p.123) where n would denote the "principal quantum number of
the geon"
Figs.l and 2 reveal that our numerical results interpolate rather well
between the asymptotic solutions at infinity and at the origin (set 0._). In
the case without nodes (Fig.la) the radial scalar function B^tt) Joins smoothly
the asympotic solution (T-9 ) with the localized solution (7.1). Both metric
functions show a Schwarzachild-tyjie behaviour for large e . For small 5 ,
e becomes constant (Figs.lb, 2b) similar as in the constant core case,
whereas e tends to zero in accordance with (7-11). The latter function
develops inbetween a noticeable peak (Figs.lc, 2c) which corresponds to the
confining barrier in the effective curvature potential (3.13) Figs.le, 2e).
An interesting phenomenon can be observed on the level of the
Schrodinger-type vave function *'0{^)t) being defined vith respect to the
"pseudo-flat" space-time (3.10). F*(j*) is concentrated (see Figs.Id, 2d)
-28-
-27-
within the negative veil of V ff(**) with its maximum close to the zero of
the potential. For Bmaller values of o» this maiimum is shifted by the
barrier of the curvature potential closer to the origin. This seems to
indicate a self-generating effect of the geoaetro-dynamical confinement
mechanism (being here only partial)! This confinement scheme and its
13) extension including colour may be compared with, e.g. the MIT "hag"proposed•odel (see also Hasenfratz and Kuti for a reviev). There an ad-hocIntroduction of a "vacuum" pressure term is needed to compensate forthe outside directed pressure of the "quark gas". In contrast to thisphenomenological device our approach ratlwr resembles Creutz's 5 3 ' re-construction of a hag model from local non-linear field theory. Similar to hia,the "core" of our "bag" i s produced by employing the stable(quantun-mechanicallyIieastable solution)of the KKG equation for an extended part of the space.Surrounding this "core" i s a transition region,the "skin" of the "bag" consistingof an exponentially decreasing lukawa-type radial solution for the scalar
field and a Coulomb type potential for the "tensor gluona".
The total gravitational mass (5.12) as measured at infinity exhibitsa branching for the zero and one node solutions with respect to i ts (^-dependence(Fig.3). For n • 0 and low A) we may understand the qualitative behaviourof U(U) by comparing i t with (see Ref.51")
(T.18)
hut for higher values of to Eq.(5.13) tends to
* ~ /J* (7.19)
The resulting predictions for the zero and the maximum of the Tolman energy,
i.e.
W. f ' 0. 7\and
(T.20)
(7.21}
(7.22)
respectively, agree quite well with our results (Fig,3).
Already for n • 1 and higher node solutions we found parts of several
sub-branches in tf(ft>), i.e. the Tolman energy of these aeons Satiates in a noticeable
way (not shown in Fig.3). In a preliminary study we could distinguish the
corresponding solutions among others by the number of nodes in H".
Further analysis is in progress in order to understand these highly
interesting instances of a possible "fine structure" in the energy levels
of the aeons. In view of these rich and prospective structures does there261
exist the speculative alternative to interpret quantum geons as extended
particles capable of internal excitations?
VIII. OTHER GRAVITATIOHAL SOLITOHS
To some extent Wheeler's ' geon concept has anticipated the, (non-1*5)
integrable )Boliton solutions of classical non-linear field theories.
As mentioned in the Introduction a geon or gravitational soliton originally
was meant to consist of a spherical shell of electromagnetic radiation held
together by its own gravitational attraction. In the idealised case of a thin
spherical geon the corresponding metric functions have the values e c • —
well inside ande*e = 1 - jr well outside the active region. The trapping
area for the electromagnetic wave trains haa a radius of
This result has been confirmed by applying Ritz variations! principles
Constructions with toroidal or linear electromagnetic wares have been
given by Ernst
Vheeler .
55)vhereas neutrino geona have been analysed by Brill and
56)Brill and Kartle > u ' could even demonstrate the existence of puregravitational solitonS. By expanding the occurring gravitational waves in terasof tensor spherical harmonics i t can be shown that the radial functionexperiences the same effective potential (3.13) except that an additionalfactor =• appears in front of the contribution from the background metric.
In a relevant paper the coupling of linear Klein-Gordon fields togravity has been numerically studied by Kaup . Moreover, the problea ofthe stability of the resulting scalar geons with respect to radial perturbation*
-29- - 3 0 -
is treated. I t i s shown that such objects are resistant to gravitationalcollapse (related works include Kefs.58, k6 and U6). These considerationsare on a semielassieal level. However, using a Hartree-Foek approximationfor the second quantized two-body problem i t can be shown that the same
coupled Einstein-Klein-Gordon equations apply.
As an important new ingredient, Kodano. et a l . U2)have considered a
real scalar field with a. Cf self-interaction as a source for the gravitationalfield. In this preliminary study the Klein-Gordon operator corresponding toflat Space-time is assumed.
A general re la t iv i s t ie Klein-Gordon field with an effective <g self-interaction for an interior ba l l has also been analysed by this group .In order to avoid a singular conitguration at origin a repulsive (or "ghost-like")scalar field has been chosen as a source e'f Einstein's equations.In a further step Kodana constructed a spherically symmetric kink-type
solution for a repulsive scalar field with Ct self-coupling (compare alsowith Ellis ' ) - As is common for kinks, the radial function at spatialInfinity i s chosen to be a constant characterizing th i s non-linear model.If we want to transfer the method to our case, we may use instead of (2.3)the symmetric self-interaction
The additional constant in (8.1) necessarily eliminates the gravitational
source which otherwise would occur for the constant aolution
;f(8.2)
characterizing the kink solution asymptotically. In flat Minkowski space
It is a conjecture that (8.2) constitutes a first approximation to the vacuum
expectation value ^o} j | 0 ^ of the corresponding quantun field.
For the general re la t iv is t ic kink of Kodama the radial solution
becomes ierc at a certain radius rB at which the background geometry develops
a Schwarzachild-type horizon. (Geon-type solutions exhibiting an event
horizon may be termed "black solitons" .) The boundary condition at r ,
however, allows an extension of these solutions into a 3-manifold consistingof two- asymptotically Euclidean spaces connected by an Einstein-Rosen bridgeArguments are given that this extended, non-singular configuration isstable with respect to radial osci l lat ions.
I t should be noted that such solutions cannot be constructed for the1 2"worMhole" topology R x S x S which would be obtainable by identifying the
asymptotically flat regions. The reason simply being that the radial functions
of the kink has an opposite sign in the other sheet of the Universe. Since thep
quantum-mechanical probability density l^fWj i s single valued and completelyregular also for the wonnhole topology, such scalar kinks provide interestingobjects for further studies.
Although we have no intention to give a complete review, we would liketo mention that other studies on geons involve massleas scalar fieldscoupled Einstein-Maxwell-Klein-Gordon systems ' or even combinedDirac-Einstein-Maxtfell field equations T 1 ' .
As a final observation we remark tha t , according to a result of Bri l l
25)
72)
a massless scalar field can be geometrized in the sense of the "already unified
field theory" or "geometrodynamies" of Rainieh, Misaer and Wheeler . Loosely,
speaking, this means that the scalar field can be completely read off from
the "footprints" it leaves on the geometry.
ACKNOWLEDGMEMTS
We would like to thank Professor H. Rufflni and Professor Abdus Salsafor useful h ints . One of us (E.W.M.) would like to express his sinceregratitude to Professors F.W. Hehl, Abdus Salam and J.A. Wheeler forencouragement and support, and would also like to'thank Professor Abdus Salam,the International Atomic Energy Agency and UNESCO for hospitality at theInternational Centre for Theoretical Physios, Trieste. TTCiiB work wassupported by the Deutsche Forschungsgemelnschaft. The computations wereperformed at the Reehenzentrum of the Christian-Albrechts-Universltat, Kiel.
-31- - 3 2 -
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Fig. 2
FIGURE CAPriONS
Scalar geon without node. The results of the numerical integration
are shown for the case t*IJL ' *. fi s 0-2,-A-= 0, £= £ = 1, n = 3,k * 1, L = 0. The solutions which depend on to and J or **,
respectively, are presented as relief.
a) Radial solutions
b) Time-like metric function e *' .
c) Space-like metric function e • .
d) Schrodinger-type wave function FQ(<") given in terms of the
"tortoise" co-ordinate • • .
e) Effective curvature potential V f f(^*) exhibiting a deep well
together with a confining barr ier .
(In Figs.la and Id only a few solutions are shown in order to
avoid a too strong screening in the 3D plot which would otherwise
occur. In Figs.Id and le the curve3 are plotted up to $ • .values
corresponding to ^ = 5-)
Scalar geon with one node- Same case and presentation as in F ig . l ,except for a braoder CO range.
Radial solutions exhibiting one node.b) Time-like metric function ec) Space-like metric function e * .d) Schrodinger-type wave function ^ ( ^ * ) having also one node
outside the origin (see the magnification (2x in F and 50x in
\*) of part of the r e l i e f ) .
e) Effective curvature potential V ,,_(ft*) with a deep well in a
confining barr ier .
(Again in Figs.2d and 2e the curves are plotted up to $* -values
correspondong to ^= 5.)
Fig.3 Tolman energy or Schwarzschild mass atM* as a function of c*J for
"quantized" scalar geons with "principal auantum number" n = 1,2.
Flg.U "Regularized" field energyof the scalar waves within scalar geons
as a function of CO , corresponding to a "cut-off" length f- - 0.001.
- 3 5 --36-
- 3 7 - - 3 8 -
- 3 9 -- U O -
" • • • ! ; :
-1*1--1(2-
Fia.2c
-1.3-
fip
-1+5-.1.6-
WL'Q BL'0 t*O O'O
-1*7- UJ
- u a -
IC/80/2
IC/80/3
IC/80/U
IC/eo/15
IC/80/21
IC/80/22
IC/80/29
ic/eo/32
ic/eo/33IBT.REP.1
IC/80/38
IC/80/39
IC/80/li3
IC/8OA7
IC/8O/U8
IC/80A9
IC/80/50
IC/80/52
IC/8o/5!«
CURRENT ICTP FREPRINTS AND INTERNAL REPORTS
A.P. BUKHVOSTOV and L.U. LIPATOV: Ins tanton-ant i - tns tanton in te rac t ion inthe 0(3) non-l inew a-model and an exactly soluble fermlon theory.
RIAZUDDIN; K •* 2i decays In non-relat iv ist ic quark-gluon model.
G. SENATORE and M.P. TOSI: Theory of the surface dlpole layer and ofsurface tension in l iquids of charged part ic les .
S. RAI CHOUDHURY: QCD effects in a model of non-leptonic hyperon decays.
SAKIX K. PAK: Electric charge as the source of CP violat ion.
T. KATTERMAHB and J. PRZYSTAWA: Locking-in and incommensurability ofthe structural transition in BaMnFi,.
E.W. MIELKE: On pseudoparticle solutions in Yang's theory of gravity.
ABDUS SALAM and VICTOR ELIAS: Induced Higga couplings and spontaneoussymmetry breaking.
A. AMUSA: Point-triton analysis of exchange free direct two-nucleon transferreaction cross-sections.
WITHDRAWN
ABDUS SALAM: The nature o f t h e " u l t i m a t e " e x p l a n a t i o n i n p h y s i c s(Herbert Spencer Lecture - 12 November 1 9 7 9 ) .
V. de ALFARO, S . FUBIHI and Q. FUELAN: C l a s s i c a l s o l u t i o n s and ex tended•upergraTity.
J.C. PATI: Unity behind diversity in Nature.
J . LUKIERSKI: Supersymmetric <J models and composite Yang-Mills theory.
W. KROLIKOWSKI: Are lepton and quark families quantized dynamical systems?
V. DEPPERT: The role of Interaction in neutrino s t a t i s t i c s .
D. KUSNO and M.J. MORAVCSIK: On the problem of the deuteron shearingcorrections - I: The conventional approach.
E.W. WELKE: The eight-fold vsy to colour geometrodynaffllcs.
ic/80/55 A.R. HASSAH: PSvonon- as slated transitions in crossed e lectr ic and magnetic
fields.
IC/8O/56 A.B. HASSAS: Tiro-photon indirect transitions in crossed e lec tr ic andmagnetic f i e lds .
IC/Bo/57 S. HAFISOS: Techniques of dimensional renormalization and application tothe two-point functions of QCD and QED.
IC/80/62 H.S. C1WIGIE: Qua.ntum chromodynamics - A theory of the nuclear force.
IC/80/67 WITHDRAWN
IC/80/68 A. AYEKSU: pynamic dislocations in high quarts.
IC/6O/69 T. PERSI: Time dispersion relations and small-time behaviour In the decay1ST.HEP.• of an unstable system.
IC/80/70 M.F. KOTKATA, A.A. EL-ELA, E.A. MAHMOUD and M.K. El-MOUSLY: Electrical"transport and structural properties of Se-Te semiconductors.
IC/8O/7I A.Yu. OROSEERO and A.R. HiOKHLOV: Some problems of the statistical theory ofINT.REP.* polymeric lyotropic liquid crystal3-
IC/80/72 J.C. PATI and ABDUS SALAM: Quark-lepton unification and proton decay.
IC/80/73 ASHOK DAS: Are massless aupersyimiietric gauge theories really snassless?
IC/8O/7U G. HAIELLA: Chiral rotations and the fermion-boson equivalence In theIBT.REP.* Schwinger model.
IC/BO/75 J. SPA1EK and K.A. CHAO: Kinetic exchange interaction in a, doubly degenerate
narrow band and its applications to Fe^Cx^Sj and ^i-^x3?*
IC/80/76 L.K. SHAYO: The generalized pressures on oscillating cantilever pipesIHT.REP.* conveying inviscid fluid.
IC/80/77 E. GAVA, R. JENGO and C. QMERO: Finite temperature approach to confinement.
IC/80/78 A. AYENSU: The dependence of mechanical, electrical, thermal and acousticIHT.REP.* properties of tropical hardwood on moisture content.
IC/80/79 J.S. NKOMA: Effect of impurities on the two-dinensioneO. electron gaspolarliability.
IC/8O/8O A.M. JAYAHNAWAR and N. KUMAR: Orbital diamagnetlsm of a charged Brownianparticle undergoing birth-death process.
IC/80/81 L.K, SHAYO; On the solution of the Laplace equation In the presence of aIHT.REP.* semi-infinite boundary - The case of an oscillating cantilever plate in
uniform incompressible flow.
IC/BO/82 E.R. HISSIMOV: Higher quantum conserved current in a nev completelyXNT.REP.* integrable model.
• Internal Reports: Limited distribution.
THESE PREPRISTS ARE AVAILABLE FROM THE PUBLICATIONS OFFICE, ICTP, P.O. BOX 586,
I-3I1IOO TRIESTE, ITALY.
IC/8O/85 Tfco RUIBAO and Pu FUCHO: A proof of the absence of modulated phase transitionand' spin density wave phase transition in one- and two-dimensional systems.
M/80/66 PuFOCHO: Density of s t a t e s , Polason's formula of summation and Walfisz'sIBT.HEP.* ronnula.
IC/80/87 A.K. BASDYOPADHYAY, S.K. CHATTERJEE, S.V., SUBRAMANYAM and B.R. BULKA:IBT.REP.* Electrical r e s i s t i v i t y atudy of some organic charge transfer complexes under
pressure.
IC/8O/88 In-Gyu KOH and Tongdui. KIM: Global quantity for dyons with various chargeHT.HEP.* distr ibutions.
IC/8O/B9 K.G. AKDEHIZ, M. GOODMAH and R. PERCACCI: Honopoles and twisted sigma madela.
IC/80/90 A.R. HASSAN and F. Abu-ALALLA: Exciton-polarlzations by two-photon absorptionHT.REP.* in eemiconductors.
IC/80/91 A.R. HASSAfi: Phonon-polaritons by two-photon absorption in so l ids .IKT.REP,*
IC/80/92 L. LOHCA and J. KOHIOR: Exact results for some 2D-Ialng models v l t hIIT.REP,* periodically distributed impurities.
IC/80/109IBT.EEP."
IC/80 / l l lIMT.REP.*
IC/80/U2
1C/8O/X13IHT.REP.*
P. FURLAH and R. RACZKA: On higher dynamical symmetries in models ofrelatlvistie field theories.
A. BREZIHI and G. OLIVIER: Localisation on veakly disordered Cayley tree.
S.K. OH: The BJorken-Paschos relation In the unified gauge theory.
A.A. FADLALLA: On a boundary value problem In a strongly pseudo-convexdomain.
IC/8O/9A
ic/80/95
ic/80/96
IC/60/97
IC/80/98HT.REP.*
IC/80/99HT.HEP.»
IC/80/100
IC.80/101HT.REP.*
IC/80/102
ic/80/103DTT-HEP.*
IC/30/105
IC/So/106HT.RKP.*
ABDUS SALAH and J. STRATKDEE: On Witten's charge formula.
ABDUS SALAM and J. STHATHDEE: Dynamical mass generation in (UT(l) x U (l))2.
I* n
I . S . CRAIGIE, f. BALDRACCHIHI, V. ROBERTO and H. SOCOLOVSKY: Study offactorisation in QCD vi th polarized beams snd A production at large p_.
K.K. SINGH: Renormallzation group and the ideal Bose gas.
HABESH DADH1CH; The Penrose process of energy extraction in electrodynamics.
0 . HUKHOFADHYAT and S. LUHBQV1ST: Dynamical polar l iabi l l ty of atoms- '
A. QADIR and A.A. MUPTI: Do neutron stars disprove multiplicative creationIn Dlrac's large ruutber hypothesis 1
V.V. MOLOTKOV and I.T. TODOROV: Gauge dependence of vorld l ines and invarianceof the S-matrix in r e l a t i v i s t i c c lass ica l mechanics.
Kb.I. PUSHKAHOV: Solitary excitations In one-dimensional ferromagnets atT * 0 .
Kb.I. PUSKHAROV and D.I. PUSHKAROV: Solitary clusters in one-ctimenstonalferromagnet.
T.V. HOLOTCOY: Equivalence between representations of conformal superalgebrafrom different subgroups. Invariant subspaces.
MEHBEL SACHS: On proton mass doublet from general r e la t iv i ty .
L.C. PAFALOUCAS: Uncertainty relations and semi-groups In B-algebras.
IC/BO/117 P, KAZEKAS: Laser Induced switching phenomena in amorphous GeSejt A
-phase transit ion model.
IC/80/118 P. BUDIHIs On splnor geometry: A genesis of extended supersymaetry.
IC/&0/119 G. CAMPAGHOLI and E. TOSATTT: ABF -intercalated graphite: Self-consistentband structure, optical properties and structural energy.
IC/80/120 Soe Iln and E. TOSATTI:. Core level shi f ts in group IV semiconductors and•emimetal*.
IC/80/121 D.K. CBATORVECI, 0. SEBATORS and M.P. TOSIi Structure of l iquid alkal i .1ST.REP.* metals aa electron-ion plasma.
IC/80/126 T*e Seventh Trieste Conferenea on Particle Ptaysles, 30 June - •» July 1980IHT.RKP.* (Contributiona) - Part I .
IC/80/13T The Seventh Trieste Conference on Particle Physics, 30 June - *• July 1980IHT.REP.* (Contributions)- Part I I .
IC/60/128 ?ABID A. KHVMJA: Temperature dependence of the short-range order parameterand the concentration dependence of the order-disorder temperature for I l -Ptand Bi-Fe systems In the Improved s t a t i s t i c a l pseudopotential approximation.
IC/SO/130 A. «IADIR: Dlrac's large number hypothesis and the red shif ts of distantIJIT.REF.* galaxies.
IC/80/131 S. ROBASZKIEWICZ, R. MICHAS and K.A. CHAO: Ground-state phase diagrain of
IBT.REP.* extended attract ive Hubbard model.
IC/80/133 K.K. 8IDGH: Landau theory and Giniburg criterion for Interacting bosons.
IC/8O/13U R.P. HAZOUME: Reconstruction of the molecular distribution function* from theIHT.REP.* s i t e - s i t e distribution functions In c lass ica l molecular fluids at equilibrium.IC/80/135 R.P. HAZOUME: A theory of the nematic liquid crysta ls .IRT.REP.*
IC/80/136 K.G. AKDEHIZ and H. HORTACSU: Functional determinant for the Thlrrlng modelIHT.REP.* with instanton _ —, _