INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/80/114.pdf · By integration we find three different special solutions and col-lect them in Fig. 2. The conventions

IC/80/llH

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

VACUUM SOLUTIONS WITH DOUBLE DUALITY PROPERTIES

OF A QUADRATIC POINCARE GAUGE FIELD THEORY

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

Peter Baekler

Friedrich W. Hehl

and

Eckehard W. Mielke

1980 MIRAMARE-TRIESTE

IC/60/lll*

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

IHTEHHATIOHAL CEHTRE Mffi THEORETICAL PKISICS

VACUUM SOLUTIONS WITH BOBBLE DUALITY PROPERTIES

OF A QUADRATIC POIHCARE GAUGE FIELD THEORY •

Peter Baekler, Friedrich W. Hehl

Institute for Theoretical Physics, University of Cologne,D-5000 Cologne kl, Federal Republic of Germany,

and

Eckehard V. Hielke

International Centre for Theoretical Physics, Trieste, Italy.

SIBAIMRE - TRIESTE

July 1980

• Submitted for publication.

ABSTRACT

We look for exact vacuum solutions of the gauge

theory of the Poincarfe group with a lagrangian

quadratic in torsion and curvature. First we find

the trivial spherically symmetric solutions with

vanishing torsion, also knwon from the Stephen-

son-Kilmister-Yang gravitational theory (see Fig.l).

Then, by means of a double duality ansatz for the

curvature, we derive in a systematic way three

different spherically symmetric vacuum solutions

with dynamic torsion (see Fig. 2) all giving rise

to a Schwarzschild-de Sitter metric with a fixed

"cosmological" constant. We rederive the Baekler

solution, find a time-reflected "anti-Baekler"

solution, and discover a new solution with non-

vanishing translational and rotational energy.

CONTESTS

Abstract

Introduction

1. Field equations of the Poincarfe gauge field theory and Bianchi

identities

2. A quadratic model lagrangian

3. The duals of the Bianchi identities

4. Rewriting the 1st field equation

5. The 2nd field equation fulfilled by a duality ansatz

6. Vanishing torsion: SKY-gravity

7. Solving the 1st field equation and the duality ansatz for spheri-

cal symmetry

8. Outlook: Confining properties of Baekler's solution?

Acknowledgment sLiterature

INTRODUCTION

Within the last two decades, gauge theories of gravity have been

developped as alternatives to the general relativity theory of

Einstein (GR). The success of the gauge idea in electroweak and

strong interactions is part of the motivation to try these concepts

in gravitational theory, too, where they historically come from,

aftenall. We have argued elsewhere (cf. [l] and refs.), why the

Poincarfe group seems to be the moat natural starting point for

setting up a gauge theory of gravity. In Sect. 1 we present the

basic tools needed to formulate the two general field equations

(1.7) and (1.8) of the Poincarfe gauge field theory (PG). They con-

trol the torsion and the curvature of spacetime, respectively. The

two Bianchi identities are also stated.

So far, the explicit form of the field lagrangian was left open.

In Sect. 2 we start from the quadratic model lagrangian (2.4),

the naturalness of which we have shown elsewhere (cf. [l]). This

leads us to a quadratic Poincarfe gauge field theory (QPG). Certain-

ly, the torsion-square part of the QPG-lagrangian yields, in the

teleparallelism limit of vanishing curvature, a viable macroscopic

gravitational theory, which is indistinguishable from GR for pre-

sently feasible experiments (cf. [2-8]).

The curvature-square part, however, representing the dynamics of

the Lorentz gauge potential and coupled to the spin of matter, des-

cribes a new interaction of a Yang-Mills type, which may be called

strong gravity and the mediating particles of which may be named

"rotons" (see [1,2]). Whether the complete lagrangian, including the

curvature-square part, has anything to do with nature, can only be

decided by exhibiting exact solutions of the QPG and by working out

their consequences.

The program of looking for exact solutions of the QPG was started by

Baekler & Yasskin [9], see also Wallner [lO,ll]. They found, in a

£j.rst step, all spherically symmetric vacuum solutions for vanishing

- 2 -

torsion. As shown by Mielke [12], see also the literature given

there, these solutions can be classified (see Fig. 1) according

to their double duality properties in the curvature tensor. We

rederive these solutions from this slightly more advanced point

of view in Sect.6.In these solutions torsion is suppressed, which

is responsible in QPG for (weak.) macroscopic gravity. Hence we are

in need of non-trivial solutions carrying also torsion.

Recently Baekler [13] found such a spherically symmetric vacuum

solution with dynamic torsion by explicitly integrating the field

equations. To our knowledge, this is the first non-trivial solu-

tion with dynamic torsion of any PGI Some of its properties were

discussed in [14].

Soon it became clear that the curvature of Baekler's solution,

apart from a piece of constant curvature, is anti-self double dual.

This, and the remarks above related to ref. [12], suggested to us,

in analogy to the procedure of Belavin et al. [15] in SU(2) gauge

field theory, to derive exact vacuum solutions of the QPG in the

following systematic way: We make a double duality ansatz for the

curvature, not forgetting the vital constant piece, and solve

thereby the 2nd field equation of the QPG by reducing it to the

2nd Bianchi identity. Left over is the 1st field equation and the

duality ansatz, a system of equations which can be solved with re-

lative ease in the case of spherical symmetry.

Our article is organized as follows: After the fundamental equa-

tions of the QPG have been set up in the first two sections, we

compute the duals of the Bianchi identities in Sect. 3 in order

to have them available in a form which is more reminiscent of the

field equations. In Sect. 4 we rewrite the symmetric part of the

1st field equation in a pseudo-einsteinian form. In Sect. 5 a

double duality ansatz is employed and thereby the 2nd field

equation solved. In Sect. 6 the degenerate case of vanishing

torsion is treated and our results for the corresponding spheri-

cally symmetric solutions are summarized in Pig. 1.

In Sect. 7 the symmetric part of the 1st field equation, by means

of the duality ansatz, is reduced to an Einstein equation with

"cosmological" constant acting in a riemannian spacetime. For

spherical symmetry, this leads to a Schwarzschild-de Sitter metric

inter alia. The anti-symmetric part of the 1st field equation and

the duality ansatz yield, in the spherical case, four differential

equations for the four unknown components of the torsion tensor.

By integration we find three different special solutions and col-

lect them in Fig. 2.

The conventions in this article are those of ref. [l] unless sta-

ted otherwise (k = relativistic gravitational constant, fi * modi-

fied Planck constant, c = velocity of light).

1. FIELD EQUATIONS OF THE POINCARE GAUGE FIELD THEORY AND

BIAHCHI IDENTITIES

In the PG (cf. [l]) spacetime is described by a Riemann-Cartan

geometry. As independent geometrical variables we can take the

tetrad coefficients et" and the connection coefficients P ; " =

— — ["*•• . Here i,j ... = 0 ... 3 are holonomic (world)

indices and«,fj ... = O ... 3 anholonomic (Lorentz) indices. The

tetrads are chosen orthonormal, i.e. the local metric 11—a

coincides with the Minkowski metric diag (-+++). The tetrad co-

efficients e|* can be interpreted as translational and the connec-

tion coefficients P-* "as rotational (or Lorentz) gauge potentials.

The torsion

and the curvature

(12)U.2) P* * P ' *are the corresponding gauge field strengths, respectively. The

operator D. represents the covariant exterior derivative with re-

spect to a holonomic basis.

Let be given a matter field represented by a Poincare spinor-ten-

sor WC**) an

det e"* , and 2C : = aL, whereas 1?": = eV is the gauge field

lagrangian depending on some coupling constants K4 } K^ • • • } on

the local metric, and on the anholonomic components of torsion and

curvature, respectively; for a detailed discussion see ref. [1].

Let be given the field momenta by

the momentum current of the gauge fields by

and the spin current of the gauge fields by

then the field equations of the PG read

(1.7)

(1.8)

The sources on the right hand sides are the canonical momentum

current and the canonical spin current of the matter field. The

field equs. (1.7, 1.8) are Yang-Mills type field equations. It is_

specific to the PG, however, that the gauge currents ( £ ̂ } £ ̂ '» * )

have tensorial character. For simplicity we will concentrate in

this article on the vacuum, i.e. we put the material sources in

(1.7, 1.8) equal to zero.

The field equations are supplemented by the two Bianchi identities

for torsion and curvature, respectively:

(1.9)r •'V3

(l.io) J) . (T

For the model lagrangian of our choice, see the next section, the

field equations will turn out to be of fir3t differentiation or-

der in torsion and curvature, similar as the Bianchi identities.

2. A QUADRATIC MODEL LAGRANGIAN

If we insist that the field equations (1.7, 1.8) be linear in the

2nd derivatives of the tetrad and the connection, i.e. that they

are quasi-linear, then the field momenta ' * ) must be

linear in torsion and curvature. A particularly plausible choice,

motivated toy the Gordon-decomposition of the momentum and the spin

current of the Dirac field, reads (see [l])s

(2.1)

(2.2)

l e

at flIf we introduce the modified torsion tensor

(2.3)

then the corresponding gauge field lagrangian can written as

i.e. it consists of a translational part if .depending on torsion,

with the Planck length Z = -/fcfic* as coupling constant, and of a ro-

tational (or Lorentz) part ^ which depends on curvature and is

characterized by a dimensionless strong coupling constant "it .

Recently there have been 3ome objections against our model lagran-

gian (2.4) by Sezgin and van Nieuwenhuizen [16], It should be clear,

however, that these arguments, based on perturbative methods on a

given Minkowaki background, do not necessarily apply to our theory,

since Baekler's spherically symmetric vacuum solution [13], for

example, is not asymptotically minkowskian, but rather has a back-

ground of constant non-vanishing curvature.

If we substitute (2.4) into (1.4-1.8), we find for the vacuum

field equations of our quadratic PG (QPG) the expressions

(2.5) PifeT!*

(2.6)

with the momentum current of the translational field

and the momentum current of the rotational field

Observe the properties

(2.9) £ ' * = 0 ,

and

(2.10) e i. = o .

r.ROT£ r = 0.

Let us consider the symmetric and the antisymmetric part of the

1st field equation separately. For this purpose we transvect (2.5)

with :

fi

= DjCeT^f J -

The first term can be partially integrated. If we remember the de-

finition of torsion (1.1) then we find (T- : = T' '. )

(2.12)

Hence the symmetric part ot (2.11) can be written as

(2.13)

with

(2.14), ±rpt£ rn _ L.

If we express £V a in terms of the modified torsion, we have

(2.;TR

Then X .„ , defined in (2.14), turns out to be

(2.16)

' fSe ' ' i) •Because of (2.9, 2.1O), the trace of (2.13) reads

In an analogous way we can derive the antisymmetric part of (2.11)

Using (2.9), a short calculation yields

(2.18)

Because of angular momentum conservation, equ. (2.18) can also be

derived by differentiating (2.6) with D±. Because of J). J). (ef~.'?

we immediately find (2.18).

In (2.13) as well as in (2.18), there appears a divergence of the

type JLfeT**",,.) . For later purposes we transform such a diver-

gence of an arbitrary 3rd rank tensor into the corresponding holo-

noraic expression. Simple algebra yields

We substitute the equs. (2.42, 2.47) of ref. [l]:

(2.20, $ , 7 % - im*yDkCeT!^.V^ I- 'K + *\ t ' w h e r e *KHere we used the abbreviation ^

the operator of covariant differentiation (see Schouten [17]).

3. THE DUALS OF THE BIANCHI IDENTITIES

In the field equations there enter covariant divergencies, whereas

in the Bianchi identities covariant curls show up. In order to

bring the Bianchi identities into a similar form as the field

equations, we will take their dual3.

- 8-

Let £** be the Levi-Civiti tensor density with £ = - i

(cf. MTW [IB], p. 87). Then 1£ ' "* k l •• •= G,' * L/-/~ g" is an (orienta-

tion dependent) pseudo-tensor. The dual of an antisymmetric 2nd

rank covariant tensor A. . can then be defined as

u * " * * 1 j. /

U ** A I M Lr I m ™ 4 J1 1 A * * -̂ - —S" >ht d

' ~ X t kt •

We introduced the imaginary unit i here in order to guarantee that

the operation of taking the dual is involutive, see (3.4) below.

We define the reciprocal of the Levi-Civit^ tensor density by the

relation

(3.2) ' ' ' *

where o

Of course, the formula (3.1) is also valid in anholonomic coordi-

nates.

The duals of the torsion and the modified torsion read respective-

ly:

(3.8) h

Their traces turn out to be

(3.9) F := F . V « = / •= ' . C it*and the reciprocal reads

(3.1O) C~2

By squaring we find the useful formula

The expression r* =• I plays a particular .role in our con-

siderations, since it vanishes identically for spherical symmetry.

For the right, the left, and the double dual of the curvature we

have, respectively:

(3.12)

(3.13)

(3-14) r # • u. u '

The double dual (3.14), in contrast to the single duals (3.12,

3.13), is distinguished by the property of being a pure (orienta-

tion independent) tensor (W. Kopczynski, priv. comm.).

The self double dual and the anti-self double dual of the curva-

ture are defined according to

(3.15)

We have

(3.16)

+Ff? '•-i(

= • F *F* =-Obviously, the curvature splits into a sum as follows:

(3.17) f = F + IF .

These notions, in the context of the curvature tensor, were de-

veloped by Lanczos [19, 2O], Debever ^ J ^ a ] , Nomizu J 23] ,

- 10 -

others, see also Gu et al. [24] and Penrose & Rindler [25]. A

finer splitting of the curvature tensor than in (3.17) can be

achieved by using also the left and the right (anti-) self duals,

aee Xin [26].

Often one needs the explicit form of (3.14). If we define the trace-

free Ricci tensor by

(3.18) /"„,. :*= F .

then the application of (3.2), after some algebra, yields

The first contraction of (3.19) reads

(3.2O) F ' **"

and the second contraction

(3.21) f. yf

Debever [21] has shown that in a riemannian 3pacetime V. the

curvature tensor ft P . *£ c a n ^e decomposed into its irreducible

parts under the action of the Lorentz group by means of [R and

IR . If we define the Weyl conformal curvature tensor as the

traceless part of jj by

(3.22) CV,

and note

(3.23)

- - 2 M ' c-

then the curvature tensor of a V4 can be written as

If we define the unit tensor in the space of bi-vectors "fl

then we can express (3.24) symbolically asX I -

.»> B -This is the irreducible decomposition looked for.

Let us now turn to the Bianchi identities. Transvect (1.9) with

i- € '*kl and finda

Dk(e s eor, by multiplication with e^ and partial integration,

(3.27)

Decomposed into symmetric and antisymmetric part, (3.27) reads:

C3.28

The equs. (3.28) should be compared with (2.13) and (2.18),

respectively.

In an analogous way we treat the 2nd Bianchi identity (1.10). We

transvect it first with j £'*** .

^ = 0 .Then we multiply by -j ^ **P >* ' ;

(3.30)

Equs. (3.29) and (3.30) should be compared with (2.6)!

4. REWRITING THE FIRST FIELD EQUATION

Before one makes use of the analogous form of the field equations

and of the corresponding Bianchi identities, ohne should rewrite

the 1st field equation in two respects. To this end we first put

it into a quasi-einsteinian form by decomposing the curvature ten-

sor R,«,» C*) of the Riemann-Cartan spacetime U. into its rie-

mannian part R a » * : = ^A* (* */ a n d into its torsion depen-

-dent terms, ̂ "he'respective Einstein tensors read:

- J l - t -

(4.1)

Then a straightforward but tedious calculation, the result of which

we may take from equ. (3.78) of ref. [27], leads to

(4.3) G -

Symmetrization of (4.11) yields

_ i

(4.13)

The explicit form of the last term we find by substituting the

definition (3.11) twice and by applying contractions of (3.2):

(4.14)

Collecting our results, we have for a U.

ROT

(4.15)

3e c' H-»*~~ -Jt (Cot

f ror more specifically for a

(4 16) V m^ If '

The translational energy can be put into a similar form,

(4.17) T £ A = £-%

but this formula will not turn out to be as useful as (4.15).

On substitution of (4.15) into (4,6), the symmetric part of the

1st field equation finally reads

(4.18)

5. THE SECOND FIELD EQUATION FULFILLED BY A DUALITY ANSATZ

A strategy for finding exact solutions in a Yang-Mills theory

has been developped by Belavin et al. [15]. It consists in sol-

ving the field equations by means of a duality ansatz, see also

Actor [28], Certainly our 2nd field equation (2,6), i.e.

(5.1)

being very similar to a pure Yang-Mills equation, should be most

appropiate for such an undertaking.

The modified torsion tensor T '. M a enters (5.1). We recall (cf.[l](

equ. (2.43))

(5.2) e I - / s

The unit tensor if —(

self double dual:

in the space of bivectors is

(3.3) or

The antisymmetric tensor'JO s (if',* **) • perhaps surprisingly, has

the same property:

Because of (5.1)-(5.4), we may take the 2nd Bianchi identity in

it3 equivalent form (3.3O), i.e.

(5 si TV (e F" J «) — 0.

We compare (5.1) and (5.5) and remember (5.2). We try to solve

(5.1) by the double duality ansatz

(5.6)

with the unknown constants (_ ^ if) ^''

We take the double dual of (5.6) and substract the result from (5.6):

(5.7) • •

For ̂ £ - 1 , the curvature of the solutions ia self double dual:

p = 0 . If we add or subtract (5.1) and (5.5), we have instead

of (5.1) the two equations

(5.8) ?:*i.e. any solution with self double dual or anti-self double dual

curvature leads to the trivial case of vanishing torsion:

T] r -a 3=fr ==t> F;'i* ** * 0. A look at the 1st field equation

(2.5) cum (4.15) then shows, since |F = 0 or F 3 &t that it is fulfil-

led identically. Consequently we find the

PROPOSITION 1: The subclass of solutions of the QPG with self

or anti-self double dual curvature tensors resides in a rie-

mannian spacetime V^, - We will discuss these solutions in Sect. 6.

Comming back to (5.7), we conclude that the only non-trivial case

with dynamic torsion, which we can find by means of the ansatz

(5.6), i3 represented by

(5.9) £ = - 1 .

Then we can write (5.6) alternatively as

? : ? « - £ S *e:ra,e:•fil

For the duality ansatz (5.6) with £ = -1 we calculate the left hand

side of (5.1) ,

(5.12)= - 5.

or, because of (5.5), and after some algebra:

T). ft* F'* ^(5.13)

In this article we will put 0 = 0; 6 ^ 0 will be studied elsewhere.

Of course, the field equation (5.1) can be solved for HP, A ' = 0

a=* F"J1 ' s ff . Then the right hand side of (5.13) vanishes. This" I *

is again the trivial case of a V̂ ,, see Sect. 6.

For 0 = 0, non-vanishing torsion can only be achieved for / - T.

Hence, as can be seen from (5.1) and (5.13), the 2nd field equa-

tion is non-trivially solved for

(5.14) ={ >

(5.15)

(5.15a,b)

It is remarkable that the constant curvature term in (5.1) yields

a non-trivial solution only for a specific choice of the constant

/.namely for / = 1/2.

Let us study some properties of the duality ansatz (5.15). By

means of (3.15), (3.19), and (5.15), we have

F r f , «or, by contraction,

1 r 3X

A ,decomposition of (5.17) into its anti-symmetric and its sym-

metric parts yields, respectively,

(5-18> F c < t / n = 0 ,

(5.19) p - -

The property (5.18) looks very riemannian-like, of course. To-

gether with the contraction of the first Bianchi identity, it

yields

(5.2O) -ff

-il-

or, together with (2.IB),

(5.2D Dk (e I ) =

logical" constant and equ. (6.11) a HordstrgJm type vacuum theory

(cf. MTW, p. 429). However.it should toe stressed that (6,10), say,

does not really correspond to Einstein's theory with cosmological

constant since, afterjall, the source of (6.2) is the spin of matter

and the arbitrary "cosmologieal" constant is of a strong gravityorigin.

If we relate (6.1O, 6.11) to (6.9), then we recognize that every

spherically symmetric solution of (6.1, 6.2) is either a solution

of Einstein's vacuum theory with arbitrary "cosmological" constant

or of Nordstrfrfm's vacuum theory. Consequently, in order to classify

all spherically symmetric solutions of SKY-gravity, we have to

look for the corresponding solutions of Einstein's and Nordstromps

theories, respectively.

In Einstein 's theory with coamological constant .A.,one has to be a4

bit careful in ennunciating Birkhoff's theorem . It states {Kra-

sinski & Plebanski [40], cf. Bonnor [41]) that all spherically •

symmetric solutions can be reduced to either the Schwarzschild-de

Sitter (SdS) metric

{cf. MTW, p. 843 for a discussion,

IR:V-

IR =fR =

00

vanishing energySKY-equation

A

0

7

IR = 0 or fR = 0self I anti-self

Einstein with A

SdS (6.12)Nariai(6.13) Ni (6.20)

Fig. 1: SKY-gravity and solutions with double duality

properties (the symbol ̂ ) means spherical

symmetry).

-II'

(7.

where the tensor t,,. = tftt»j , depends solely on the modified

torsion, as can be seen by substituting (4.8):

Let us now use the duality ansatz (5.15) and its consequences

(5.18, 5.19, 5.23). From (7.2) we find:

(7.4) '*/»

The energy tensor ~t « vanishes together with the totally anti-

symmetric part F t i , r t = T^v*.*l of the torsion. Hence, because

of (5.15a, b ) , we have

(7.5) *., = " (ffl 9*,i.e. torsion drops out altogether and we are left with an Einstein

equation with the "cosmological" constant -A.: — ~ A*ffi acting in a

riemannian spacetime V*.

Accordingly, the symmetric part of the 1st field equation (2.5) is

reduced to (7.5), whereas its antisymmetric part stays the same,

see (2.18):

(7.6) Dk (e T . Cot(3

The 2nd field equation (2.6) is solved by the duality ansatz (5.15,

5.15a, 5.15b):

Consequently (7.5, 7.6, 7.7) are the remnants of the two field

equations (2.5, 2.6). The equations (7.5, 7.6, 7.7) need to be

solved.

Because of (7.7), the totally antisymmetric part of the torsion is

required to vanish identically. Therefore we would not be able to

find axially symmetric solutions with the duality ansatz (7.7),

since in this case torsion has certainly a leading F ^ j y ] -part.

Perhaps & =£ (T is a way out in this case. Let us concentrate in

the present article on the simplest case compatible with (7.7),

namely on spherical symmetry, where £•[,*a»1 ~ ̂ • S o far we have net

completed the proof that the duality ansatz (7.7) encompasses all

spherically symmetric solutions, but we know already that it does

include most cases.

We start off from the most general spherically symmetric metric

(cf. ',40j, equ. (2.13),and MTW, p. 361)

-xx(7.8)

with the unknown functions U =• yM CR,TJ , X - -1 ̂ ff,~O,and r = r(R,T).

Differentiation with respect to R and T will be denoted by a prime

and a dot, respectively. A tetrad field which is naturally asso-

ciated with this metric reads

i \'x X «(7.9) (no summation!).

Now we impose O(3) symmetry and space reflection symmetry on tor-

sion and curvature. We find 4 nonvanishing anholonomic components

of the torsion and 6 components of the curvature. We display them

in two matrices, where the antisymmetric index pairs (Ol, O2, O3,

23, 31, 12) are numbered by (1,2 .

(7.1O)

361):

(7.11) (F.7") =

/A . . . . .\

. C . . . G

. . C . -G .

D . -H .

\. -D . . . -Sj

We have f = f(R,T) etc. and A = A(R,T) etc. Simple calculations

yield for the modified torsion

-15-

(7.12)

and for the (anti-)self dual parts of the curvature

f-2q

•

V •

2k

•

h+k

•

f - g

h+X

-f+g

(7.13)

and

(7.14)

(?;.'% iA-L ^

C-H . . . -D+G

C-H . D-G

A-L

D-G . C-H

t . -D+G . . . C-H

•A+L

C+H . . . D+G

C+H . -D-G

. -A-L

D+G . -C-H

-D-G . . . -C-H

, i.e. IF is then a diagonal matrix.

Incidentally, the (6x6)-unit matrix reads ^ o ^ op ) —

= diag (1,1,1,1,1,1).

In a V4 we have always D=G

For writing out (7.11) and (7.6) explicitly, we will need the

anholonomic components of the connection. We express the connec-

tion in terms of the metric (7.8) and the torsion (7.10) [see [l]

equ. (2.51)]:

(7.15)

/

T

Then, by means of its definition {1.2), we can compute the curva-

ture in terms of metric and torsion:

fB- • 6 —

« Vi W= -H - - ̂ f,r - r'^y- ( h + 1

(7.16)

Now everything is prepared for solving the equations (7.5, 7.6,

7.7). Using the metric (7.8) and Schwarzschild coordinates r » Rt

t=T, equ. (7.5), via Birkhoff's theorem, leads to the SdS-metric

(6.12) with _/\- - -

Here we have e — S — fl. JJ . The case of the Nariai metric,

which emerges also in this context, will be left to a separate

study.

Equ. (7.6) has only one independent component. By means of (7,8,

7.9, 7.12, 7.15), it can be put quite generally into the form

e * - / • •(7.18)

If we restrict ourselves to static solutions in the torsion, i.e.

g = f = h = k = O , and use the SdS-metric (7.17), then (7.18) re-

duces to

Let us now turn to the duality ansatz (7.7): The relation K-L

is fulfilled identically for spherical symmetry, as can be seen

from (7.1O). The set (7,7) corresponds to 3 different equations.

This can be read off from (7.13) in a most direct way. If we

again assume staticness in torsion and use the SdS-metric (7.17),

eqs. (7.7, 7.13. 7.16) yield:

(7.20) A~L- 2JL3-

(7.21) C-H =

(7.22) D_i

- hfr

= 0 = )

The metric is already determined in (7.17). We should remember

that in (7.19) to (7.22) we always have e'* = e'^sfl"? The four

unknown torsion functions (g, f, h, k) are governed by the four

ordinary first order differential equations (7.19, 7.20, 7.21, 7.22).

Addition and subtraction of (7.19) and (7.22) yield, respectively,

if we remember (7.17):

±L? + k) = Or ,(7.23)

(7.24)

Equ. (7.23) leads to two different cases: Either to

h = -k (case I),(7.25)

or to

(7.26)»" ~,

-X(case II).

Let us first follow up case I. For the specific choice g = f,

equ. (7.24) results in

(7.27) = - k

with tft» iHLgyiation constant to. On substituting (7.27) into (7.2O,

7.21), we find

Sit(7.26) = -

The other components look similarly. Clearly, the tetrad field al-

ways carries 3pin. The spin current (1.6) cum (2.1) turns out to

be

(7.37a,

,--f/st

= ± • •tall other components = o.

Whereas the mass dipol moment density £^*° stays the same for

both solutions (7.31a,b), the corresponding flux in r-direction

reverses it3 sign.

The case II of (7.26) has different characteristics. We eliminate

from (7.2O) and from (7.22) multiplied by e ̂ the term OFe'O.Thia

leads to a second order differential equation for k(r) alone:

(7.34) [ ~ax (t* k)']'A specific solution of (7.34) is

(7.35) k* = - | ^ f ; OC = with

de Sitter metric with the associated torsion and curvature struc-

tures could lead to a classical model for confinement. Some of

our ideas presented in this section ate frankly speculative, but

we hope that they have at least some plausibility. - In any re-

lativistic theory of particles possessing quarks as building

blocks, a mechanism of (at least partial) confinement of the basic

constituents has to be inherent in the model.

In quantum chroroodynamics (QCD, see [47] for a review), nowadays

the moat prominent model, such a confining phase is suspected to

occur by taking non-pertubative effects properly into account.

Despite much efforts, this has not yet been demonstrated. A classi-

cal analysis does not yield encouraging hints either. On the con-

trary: It is known [48 j that pure Yang-Mills fields cannot form

classical glue-balls. The reason being that nearby small portions

of the vector gauge fields must always point in the some direc-

tion in internal space and therefore must repel each other similar

as charges of the same sign. Therefore, in phenomenological devi-

ces, such as the MIT-bag model [49], see also [50], a color sen-

sitive "bag constant" - B has been included ad hoc into the la-

grangian in order to compensate for the pressure of the quark gas.

This, and appropriate conditions for the Dirac spinor at the

boundary of the bag, yield a reasonable description for hadrons

with confined quarks.

On the other hand a mechanism of confinement, which is intrinsic

and in accordance with gauge ideas, emerges if we adopt the

hypothesis that strong interactions are (at least partially)

mediated by tensor gauge fields. In the f-g theory of gravity,

modeled after Einstein's theory in a V4, Salam and Strathdee [51]

have shown that there exist a completely confining configuration:

the anti-de Sitter space. Usually this space is obtained as a so-

lution of the Einstein equation with (scaled) negative"cosmologi-

cal" term. In the f-g theory such a term can be simulated by

adding a mixing term Jf ? to the Einstein-Hilbert lagrangian

for the f and ̂ ^v tensor fields. For a X -^ -term of the

Pauli-Fierz form, however, f̂ ,y represents a massive spin-2 par-

ticle. This is inconsistent with the aim to achieve confinement

by (tensor) gluons, which have to stay massless.

With respect to this issue, the QPG is claimed to be conceptual

superior as compared to other tensor gauge theories. Without having

included it from the beginning, we obtain for our modified anti-

self double dual solutions an effective bag constant

(8.1) B - X £+ )which is of microscopic origin. Furthermore, as a promissing new

feature, a Schwarzschild-de Sitter solution (7.31a,b) with dynamic

torsion is the result.(Other choices of the QPG lagrangian would

yield also massive spin-2 mesons (see [16]) without the need to

introduce a Pauli-Fierz mass term if«r.)

In order to exhibit the confining property of this solution, we

may adopt here the crude physical picture that the hadron is re-

presented by a Schwarzschild-de Sitter microuniverse, with a -=-=•

singularity at the center originating from a "source" quark of

mass1 OC . Into this world one can think of putting an additional

quark-treated as a test particle, in so far as its contribution

to the source is ignored - in order to complete the hadronic

structure.

For the case of scalar test (and source) "quarks", we will solve

the Klein-Gordon equation

(8.2) C a -in the curved background given by (7.17) with 0£= O. Here

is the generally covariant Laplace-Beltramic operator (for the

following discussion See also Mielke [52 ]). Stationary, radial

solutions of (8.2) may be obtained by using the familiar separa-

tion ansatz

(8.4)

In this ansatz

(8.5) S • = ~^£

denotes a dimensionless radial coordinate. After introducing the

"tortoise coordinate" (MTW, p. 663)0

{8.6) § ' " J ~T~L—"5 - ardanQ {=? Avtn $) for+

equ. (8.2) reduces to the Schrodinger type equation

(8.7, C *) +f*] C -afor the radial function R U M

formulae the quantity

. In this and the following

(8-8) I* =has been used. It denotes the (dimensionless} ratio of the re-

duced Compton wave length Xu. of the test particle and the

Planck length JL, modified by our strong constant 3C .

Formally, the curved background affects the wave equation only via

an effective curvature potential (compare with MTW, p. 868) which

in the de Sitter case is implicitly given by

for ie > o Cte * eO.It is instructive to consider the particular simple case of zero

angular momentum states (L = 0) for yt> 0 . Then the curvature

potential of the anti-de Sitter space reads explicitly

This potential belongs to a family of examples studied by Poschl

and Teller [53] for the oscillations of diatomic molecules (see

also [54], p. 89). For ft9"< 0. . it gives rise to a smooth "wall"M It-infinite at Q - 3 , which the test "quark" can never penetrate.

Equ. (8.7) can be exactly solved [55,51] in terms of hypergeo-

metric functions. The regular, square-integrable solutions form

a discrete set having quantized energy values

(8.11, C2N | + VIApart from the zero-point term, these are exactly the energy

levels of the 3-dimensional non-relativistie harmonic oszillator

([54], p.70). The result (8.1O) may be interpreted as a prototype

mass spectrum for an excited hadron of integer spin. This spec-

trum has discrete eigenstates, no continuum, and thus no disso-

ciation of the "quarks".

In a sense, the confinement in an anti-de Sitter microuniverse

incorporates the features of both, the bag and the potential

model in QCD (see Sect. 1.3.5 of [47]). It should be noted that

the Schwarzschxld-type source term -=£- in the solution (7.31a,b)

is necessary for the observed asymptotic freedom of the consti-

tuents at short distances. By treating this term as a pertuba-

tion, it can be shown [51 in the limit Ur* 0 of the scalar test

mass that the degeneracy of the energy levels of (8.1O) in the

parameter (2N 4- L) is removed.

For a more realistic analysis of the implications of the solution

(7.31a,b) for the confinement problem, we may consider its effect

on a spin-1/2 test particle. In an l?4 not only the Schwarzschild-

de Sitter metrical background occurs in the generally covariant

Dirac equation, but also the dynamic torsion of the vacuum solu-

tion is expected to yield essential contributions. Morover, in a

QPG theory coupled to Dirac fields, the completely antisymmetric

part of the torsion induces a nonlinar term of the axial vector

type into the spinor equation (see [56]). Therefore, the analysis

of the spinor solutions in Baekler's background presents a high-

ly interesting but formidable task in itself, which we will defer

to a later publication.

The main problem for a realistic description of hadrons within the

QPG paradigm, which must be also solved, concerns the incorpora-

tion of color and, needless to say, of flavor. We need to make

a distinction between quarks and antiquarks and also introduce

a color selectivity in order to obtain only the physical hadronic

states.

In the vierbein description of the tensor fields, the (covering)

symmetry of the PG includes a group of local SL(2,C) transfor-

mations. This local group can easily be enlarged to incorporate

a color symmetry via the subgroup chain SL( 2 ,C)C SL(2 ,C) ®SU C ( 3)

to obtain only one anti-de Sitter "microuniverse" with absolute

confinement for all kind of test particles and no outside space-

time, a color sensitivity has to be built into the theory. Similar

as in the colored extension [51] of the f-g theory, this has to be

achieved in such a way that for color singlet states the tele-

parallelism limit ?C —* 0 will be obtained. '

Expressed differently, the observed hadronic states should not

feel the strong Yang-Mills type gravity ("rotona") and, unlike

their constituents, should not participate in the geometrodynamical

confinement mechanism.

ACKNOWLEDGMENTS

One of the authors (FWH) is most grateful to Professor Remo

Ruffini for- the invitation to present some aspects of the QPG

at the 2nd Marcel Grossmann Meeting in Trieste. We all are very

much obliged to Professor Hubert Goenner for his advice on the

Birkhoff theorem and on related questions, and to Dr. Phil Yasskin

for most interesting critical remarks and for sharing with us

hia knowledge in solving the field equations of QPG. One of us

(EWM) would like to thank Professor Abdus Salam, the International

Atomic Energy Agency, and UNESCO for hospitality at the ICTP in

Trieste. This work was supported by the Deutsche Forschungsge-

meinschaft, Bonn.

found implicitly in Baekler & Yasskin [9]: Take the discussion

following their equ. (64). The 6 cases are reduced to the cases

I and II, i.e. to Nordstrom's theory comprising anti-self double

dual solutions, cf. our equ. (6.11), and to case VI, i.e. to

Einstein's theory with "cosmological" constant comprising self

double dual solutions, cf. our equ. (6.10).

Goenner [36] gave a careful exposition of the Birkhoff theorem

with references to the original literature. Birkhoff type theo-

rems in theories with torsin were investigated by Yasskin [37],

Ramaswamy & Yasskin [38], and Neville [39].

This proposition subsumes in a unified way and in a hopefully

final form results of different authors on SKY-gravity, see

Thomson [45], Pavelle [46], Ni [33], Olesen [34], Gu et al. [24],

Mielke [12], and Baekler & Yasskin [9]. In the past several

spherically symmetric solutions of the SKY-equation (6.2) have

been given. We turn first to the solution of Thomson [45] and

Pavelle [46]:

"

)~

For (F.I) we find R = 0, i.e. it also fulfills (6.1) and is a

special case of (6.20). The examples of Ni [33], see also

Pavelle [46],

(F.2) Js*- - dix +and of Thomson [45]

( F . 3 ) d $a = - r f < " *

both have no double duality properties, i.e. they do not fulfill

(6.1).6 It ia always understood that (1 -~- + TfcT1}'* -/ f'^ etc.

FOOTNOTES

We have changed our conventions in the meantime. The holonomic

torsion now reads • p; - — y? 'Ci^l •

An additional term t(*F '.^JA + ~̂ .7*/*) does not seem to work,

even being a self dual combination, 3ince there is no Bianchi

identity for the right dual, see also [24,26].

Compare in this context Ni [33], Olesen [34], Mielke [12J, andthe literatur cited there. The proof of our assertion can be

LITERATURE

[l] F.W. Hehl, "Four lectures on Poincare gauge field theory",given at the 6th Course of the International School ofCosmology and Gravitation on "Spin. Torsion, Rotation, andSupergravity", Erice, May 1979, P.G. Bergmann und V. DeSabbata, eds., Plenum Press, New York (to be published 198O),

[2] F.W. Hehl, Y. Ne'eman, j. Nitsch, and P. von der Heyde,Phys. Lett. 78B, 1O2 (l?78).

38

[3] M. Schweizer and N. Straumann, Phys. Lett. 71A, 493 (1979).

[4] L.L. Smalley, Phys. Rev. £21, 328 (1980).

[5] J. Nitsch and F.W. Hehl, Phys. Lett. 90B, 98 (19BO).

[6] M. Schweizer, N. Straumann, and A. wipf, GRG Journal, tobe published.

[7] J. Nitsch, "The Macroscopic Limit of the Poincare Gauge

Field Theory of Gravitation", Erice, May 1979, see ref. 1.

[8] P. Baekler, Phys. Lett. B (in press, 1980).

[9] P. Baekler and P.B. Yasskin, GRG Journal, to be published.

[10] R.P. Wallner, GRG Journal, to be published.

[11] R.P. Wallner, Preprint University of Vienna (1979).[12] E.W. Mielke, Preprint ICTP (Trieste) IC/8O/29 (1980),

GRG Journal, to be published.

[13] P. Baekler, Preprint University of Cologne (1980).

[14] F.w. Hehl, "Fermions and Gravity", Colloque du Centenairede la Naissance d'Albert Einstein au College de France,June 1979, to be published.

[15] A.A. Belavin, A.M. Polyakov, A.S. Schwartz, and Yu. S.Tyupkin, Phys. Lett. 59B, 85 (1975).

[16] E. Sezgin and P. van Nieuwenhuizen, Preprint State Univ.of New York, Stony Brook, ITP-SB-79-97, Phys. Rev. D, tobe published.

[17] J.A. Schouten, Ricci Calculus, 2nd ed., Springer, Berlin1954.

[18] C.W. Misner, K.S. Thorne, and J.A. wheeler, Gravitation,Freeman, San Francisco 197 3. Cited as MTW.

[19] C. Lanczos, Ann. Math .39, 842 (19 38).

[20] C. Lanczos, Rev. Mod. Phys. 34. 379 (1962).

[21] R. Debever, Bull. Acad. Roy. Belgique, Cl. Sc. 42, 313, 608,1033 (1956).

[22] R. Debever, Cahiers Physique .18, 303 (1964).

[23] K. Nomizu, Differential Geometry (in honour of K. Yano),p. 335, Kinokuniya, Tokyo 1972.

[24] C.H. Gu, H.S. Hu, D.Q.Li, C.L. Shen, Y.L. Xin, and C.N. Yang,Scientia Sinica ,2_1, 475 (1978).

[251 R. Penrose and W. Rindler, unpublished manuscript on spinors.

[26] Y.L. Xin, J. Math. Phys. 21., 343 (198O).

[27] F.W. Hehl. Habilitation Thesis (mimeographed), University ofClausthal (W. Germany) 1970; GRG Journal 4, 333, 5., 491 (1973/4).

[28] A. Actor, Rev. Mod. Phys. .51, 461 (1979).

[29] G.W. Barret, L.J. Rose, and A.E.G. Stuart, Phya. Lett. 6OA,278 (1977).

[3O] G. Stephenson, Nuovo Cimento 9., 263 (1958).

[31] C.W, Kilmister and D.J. Newman, Proc. Cambridge Phil. Soc.(Math. Phys. Sc. ) 52, 851 (1961).

[32] C.N. Yang, Phys. Rev. Lett. 33- 4 4 5 (1974).

[33] W.-T. Hi , Phys. Rev. Lett. .35, 319, 1748(E) (1975).

[34] P. Olesen, Phys. Lett. 711- 1 8 9 (1977).

[35] W, Szczyrba, in preparation.

[36] H. Goenner, Comm. Math. Phys. .16, 34 (1970).

[37] P.B. Yasskin, Ph. D. Thesis, Univ. of Maryland (1979).

[38] S, Ramaswamy and P.B. Yasskin, Phya. Rev. D19, 2264 (1979).

[39] D.E. Neville, Phys. Rev. D21,, 277O (1980).

[40] A. Kra3inski and J. Plebanski, Rep. on Math. Physics (Torun),to be published.

[41] W.B. Bonnor, in: Recent Developments in General Relativity,p. 167. Pergamon, Oxford 1962.

[42] H. Nariai, Sci. Rep. T&hoku University (Japan) 3_4, 160 (195O):25, 62 (1951).

[43] E.W. Mielke, GRG Journal 8, 321 (1977).

[44] H. Goenner and P. Havas, J. Math. Phy3. 21, 1159 (198o).

[45] A.H. Thomson. Phya. Rev. Lett. 34., 5O7 (1975): 3J5, 320 (1975).

[46] R. Paveile, Phys. Rev. Lett. 34, 1114, 1484(E) (1975).

[47] W. Marciano and H. Pagels, Phys. Rep. 36C, 137 (1978).

[48] S. Coleman, and L. Smarr, Comm. Math. Phys. 56_, 1 (1977).

[49] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorne. and V.F.Weiss-kopf, Phys. Rev. D9, 3471 (1974).

[5O] P. Hasenfratz, and J. Kuti, Phys. Rep. 4OC, 75 (1978).

[51] Abdus Salam and J. Strathdee, Phys. Rev. Dl8, 4596 (1978).

1*0

[ 5 2 , E.W. Mielke, "On de S i t t e r - t y p e Bag Models of Confinement",in preparat ion .

[53] G. Pdschl, and E. T e l l e r , Z. Phys. 83 . 1439 ( 1 9 3 3 ) .

[54] S. Flligge, P r a c t i c a l Quantum Mechanics. V o l . 1 , SpringerB e r l i n 1971.

[55j R. Prasad, Nuovo Ciraento 3Q, 1921 ( 1 9 6 5 ) .

[56 ] F.W. Hehl, and B.K. Datta, J . Math. Phys. _1_2, 1334 ( 1 9 7 1 ) .

[57 ] E.W. MielXe, Preprint ICTP ( T r i e s t e ) IC/8O/54 (198O), Int JTheor. Phys . , t o be p u b l i s h e d .

CURREJiT ICTP PREPRINTS AMD INTERNAL REPORTS

IC/79/163 0. GELMINI: The a-wave QcQc atateB.

IC/T9/165 S. MARCULESCU and L. MEZISCESCU: Phase factoru and p o i n t - a p l l t t i n g InauperBymmetry.

IC/79/166 A. FASOLIHO, 0. SANTORO and E. TOSATTI: I n s t a b i l i t y , d i s tort W . anddynamics of the W(100) surface.

IC/79/168 T.D. PALEEV: The trace formulae for rea l i za t ion to f Lie algebras withA-operators. Analysis of the Fock representat ions .

IC/SO/1 * . JBsUIKOtfSKI: Towards a dynamical preon model.

IC/80/5INT.REP.•

IC/80/61ST.REP.*

IC/80/7INT.REP.*

IC/80/8INT.REP.•

IC/80/9

IC/8O/IO

IC/8O/II

IC/80/12INT.REP.*

IC/80/13

IC/80/lliINT.REP.•

RIAZUDDIN; Two-body D-meson decays in n o n - r e l a t i v i s t l c quark model.

G. ALBERT, M. BLESZYNSKI, T. JAROSZEWICZ and S. SANTOS: Deuteron D-vaveand the non-eikonal e f f ec t s in tensor asymmetries ID e l a s t i c proton-dauteron scat ter ing .

A.H. KURBATOV and-P.P. SAMKOVICH: On the one v a r i a t i o n s ! pr inc ip le inquantum s t a t i s t i c a l mechanics.

G. STRATAN: On the alpha decay branching ra t ios of nuclei around A * 110.

H.Y. ATOUB: Effect of high ly ing s t a t e s on the ground afld few low ly ingexc i t ed 0* energy leve la of same c l o s e d - s h e l l n u c l e i .

L. FONDA, N. MANKOtf-BORSTNIK and H. ROSIKA: The decay of coherent rotat ionals t a t e s subject to random quantum measurements.

6.C. GHIRARDI, V. GORI|II and G. PARRAYICIKI: Spat ial l o c a l i s a t i o n ofquantum s ta tes and physical meaning of the msxix elementa of the resolventoperator*'

M.V. MIHAILOVIC and H.A. NAGARAJAM: A proposal for ca lcu la t ing theImportance of exchange e f f e c t s In rearrangement c o l l i o i c n a .

A. BULCAC, F. CAEST01U and 0. DUMITRESCU: Double folded Yukava. interact ionpotent ia l between two heavy i o c s .

W. KROLIKOWSKI: Leptton and quark famil ies as quantum-dynamical systems.

- u\ -* Internal Reports 1 Limited distributionTHESE PREPRINTS ARE AVAILABLE FROM THE PUBLICATIONS OFFICE. ICTP, P.O. BOX 586,

ic/8o/i6

Ic/60/17INT.REP.*

ic /80/18INT.HEP.•

ic/ao/i9

IC/80/201ST.REP.•

AJOCD OSMAN: Four-body problem for four bound alpha particles in 0.

NAMIK K. PAK: Introduction to instantons in Yang-Mills theory (Part I ) .

RIAZUDDIN: Neutral current weak, interaction vithout electro-weak,unification.

H.S. CRAIGIE, S. NARI SON and RIAZUDDIN: An apparent inconsistencybetween tbe Dyson and renormallzation group equations in QCD.

W.I. FURMAN and G, STBATAH: On alpha decay of some isomeric states inPo-Bi region.

1C/80/37

IC/80A0IHT.REP.*

IC/80AIINT.REP.*

A.H. ANTONOV, V.A. HIKOLABV and I.Zh. PETKOV: Nucleon momentum anddensity distributions of nuclei.

W. KROLIKOWSKI; An integral transform of the Salpeter equation.

Z.A. KHAH: Elastic scattering of intermediate energy protons on Heand 12C.

IC/80/23 P. ROZMEJ, J. DUDEK and W. NAZAREWICZ: Possible interplay between non-axial and hetadecapole degrees of freedom - An explanation for"enormously" large & 1

IC/80/2U REVISED.

IC/BO/25 G. MAIELLA: Path-integral measure and the Eermion-Bose eguivalence inINT.REP.• the Schvinger model.

IC/Bo/26 N.3. CRAIGIE, S. HARISOK and RIAZUDDM: A cr i t ical , analysis of theelectromagnetic mass shift problem in QCD.

IC/80/27 D.K. CHATUEVEDI, U. MAKINI BETTOLO MARCONI and M.P. TOSI: Mode-couplingINT.REP.* theory of charge fluctuation spectrum in a binary ionic l iquid.

IC/80/28 ASDUS SALAM: Gauge unification of fundamental forces {Nobel lecture) .

IC/80AU

IC/8OA5INT.REP.*

IC/8OA6INT.REP.•

A. OSHAH: Two-nucleon transfer reactions with form factor models.

E.W. MIELKE: Towards exact solutions of the non-linear Helsenberg-Paull-Weyl spinor equation.

H.S. CRAIGIE: Catastrophe theory and disorder* of th»; l inl ly

IC/80/30 B. NOJAROV, E. NADJAKOV and V. ANTONOVA: High spin structure in acoupled bands model.

rc/6o/31 AHMED OSMAH: Rearrangement collisions between four identical part iclesas a four-body problem.

IC/80/53 R. BECK, H.V. MIHAILOVIC and M. POWSAK: Calculation of nuclear reactionparameters with tk« generator co-ordinate method »nd their i t t t i

IC/60/3U I.Zh. PETKOV and M.V. STOITSOV: On a generalisation of the Thomas-FermiINT.REP.• method to finite Fermi systems.

IC/80/58INT.HEP.•

IC/80/59INT.REP.•

IC/80/fiOIDT.REP.*

3. YOKSAN: Spatial variations of order parameter wahiM fondo impurityfor T 4 Tc.

J.K.A. AMUZU: Sliding friction of some metallic glaseas.

Zi-Zhao GAB and Guo-Zhen YANG: A theory of coherent propagation ofl ight Have in semiconductors.

-11--ili-

iWTTf.f •'* 1

IC/8O/61INT.REP.•

K.G. AKDENIZ and 0. OGUZ: A nev class of meronic solutions.

IC/80/63 H.B. SIHQH and K. HAUO: Optical absorption spectrum of highly excitedINT.REP.* direct gap samiconductora.

IC/SO/fiU W. KLON0WSKI: Living protein iwcromolecule as a non-equilibriumINT.HEP.• metdatable system,

IC/8O/65 H.Q. HEIK: An approximation to the ground state of E ® E and fg ©INT.HEP.* Jahn-Teller system* based on Judd's isolated exact solutions.

IC/80/66 A.H. ERMLOV, A.S. KIREEV and A.M. KURBATOV: Spio glaaees with ncm-INT.REP.* Gaussian distributions. Frustration model.

- i v -