This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 202.114.6.37

This content was downloaded on 10/10/2013 at 12:19

Please note that terms and conditions apply.

The ground-state wavefunction of a radiation-dominated universe

View the table of contents for this issue, or go to the journal homepage for more

1991 Class. Quantum Grav. 8 1271

(http://iopscience.iop.org/0264-9381/8/7/005)

Home Search Collections Journals About Contact us My IOPscience

Class. Quantum Grav. 8 (1991) 1271-1282. Printed in the UK

The ground-state wavefunction of a radiation-dominated universe

0 Bertolamif and J M MourHotB t Cent10 de Fisica da Materia Condensada, AV. Prof. Gama Pinto 2, 1699 Lisbaa Codex, Portugal t Centro de Fisica Nuclear, AV. Prof. Gama Pinto 2, 1699 Lisboa Codex, Portugal 8 Departamento de Fisica, Instituto Superior ‘Ecnico, AV. Rovisco Pais, IC40 Lisboa, Portugal

Received 26 November 1990, in final form 21 March 1991

Abstract. We propose a minisuperspace construction of the ground-state (no boundary) wavefunction for a radiation-dominated universe. As in the case of a free massless confor- mally invariant scalar field discussed by H a d e and Hawking, the functional integrations over the gravitational and gauge fields can he performed independently. The gauge part of the wavefunction is semiclassically computed with the help ofthe (anti-Iselfdual solutions of the Euclideanized S0(4)-symmetric Einstein-Yang-Milla (EYM) systems. Implications of our construction are discussed.

1. Introduction

Recent developments in high energy physics and cosmology have raised pressing issues concerning the initial conditions which in the early universe gave origin to the universe we observe. However, progress in the direction of clarification of these and other issues requires a consistent understanding of the intertwining between quantum and gravita- tional phenomena. In constructing a quantum theory of gravity one finds that many of its features are already present in quantum cosmology. In the latter approach one relies on the canonical Hamiltonian formalism. Although quantum cosmology, clearly, does not solve all the difficulties that arise in a full quantum gravity theory, it has the virtue of allowing one to insulate and sometimes to circumvent the most common obstacles preventing the extrapolation of the cosmological standard model back to energies of the order of the PIanck mass, Mp= 1 . 2 ~

In the canonical approach, the dynamics of the universe is determined by its possible quantum mechanical states which are characterized by a wavefunction. This wavefunc- tiox is a Finctiona! oii the space af 3-1~-e:;’.cs. k,,(r? az-d of --at:-: Seld cozfig.;:a!iozs, generically denoted by Qo(r), and satisfies the Wheeler-DeWitt equation [1,2]:

GeV.

where G,,, is the metric on the space of all 3-geometries:

h = det(h,), ’ R is the scalar curvature of th,e spatial hypersurface, A stands for the cosmological constant, k2= M’p/ 16v and Taa(-i8f &Do, ‘Do) is the time-time com- ponent of the energy-momentum tensor promoted to an operator.

0264.9381/91/071271+ 12S03.50 @ 1991 IOP Publishing Ltd 1271

Go:; =qh-1’2(hikE,: + h,:h,, - h,h,,! (12)

1272 0 Bertoiami and J M Mourlo

A striking breakthrough in quantum cosmology was due to Hartle and Hawking [3]. The Wheeler-DeWitt equation can be seen as the gravitational analogue of the Schrodinger equation. Since the solution corresponding to the lowest energy state of the latter equation can, in quantum mechanics, be obtained by the Wick rotated path integral over the configuration space, Hartle and Hawking suggested that a solution of the Wheeler-DeWitt equation associated with the ground state could be found via the Euclidean functional integral over 4-metria 4g and matter fields on a 4-manifold M weighted by the Euclidean action:

r

+PE,, = J B4g9@ exp(-sd4g, @I) (1.3)

where the spatial 3-metric h, and O0 are the restrictions of *g and @ on Z, a boundary of M. Moreover, Hartle and Hawking proposed that the boundary conditions of (1.1) should be fixed in the most economical way, i.e. by considering that M in (1.3) is a compact Euclidean manifold with no other boundary than Z.

Quantum cosmology becomes especially useful in the study of the early universe when one considers minisuperspace models. These are obtained by freezing all but a finite number of modes of the 3-metric h,, and matter fields and using the canonical quantization methods to treat the unfrozen modes. In this approximation, the Wheeler- DeWitt equation is obtained by imposing the Hamiltonian constraint, resulting from the ADM formalism [4], to the wavefunction and by promoting the canonical conjugate momenta into operators. The simplest model considers a single gravitational mode a ( t ) corresponding to the radius of a Friedmann-Robertson-Walker (FRW) universe and a homogeneous scalar field + ( t ) , which sustains the cosmic expansion-t being the cosmic time. This framework allows one to find, among other desirable features, an inflationary phase for the various scalar field potentials that have been analysed in the literature [5-111.

Hartle and Hawking have also advanced a concrete proposal for the ground-state wavefunction of the universe 131. They considered the sum over minisuperspace variables in a system with a positive cosmological constant and a single free massless conformally coupled scalar field. This choice implies that the field dynamics is in a sense trivial since a free massless conformally invariant scalar field lacks any particle physics scale. Moreover, as the Hamiltonians of gravity and of the scalar field split up, the boundary conditions can be implemented for each field separately and the wavefunction corresponding to the scalar field can be easily obtained. The Hamiltonian for the scalar field turns out to be that of a harmonic oscillator, implying that the corresponding minimally excitation wavefunction is the Gaussian associated with the ground state of the harmonic oscillator.

In this paper we construct the minisuperspace no-boundary ,wavefunction of a radiation-dominated universe. This proposal arises in the context of Einstein-Yang- Mills (EYM) systems, which share with the free massless conformally invariant scalar field case the important property that the Hamiltonians for gravity and for the Yang- Mills fields separate (see below, equations (2.12) and (2.17)). The latter property is due to the conformal invariance of the Yang-Mills fields. One also finds that the path integral (1.3) over the gauge variables is semiclassically dominated by (anti-)self-dual solutions. These solutions have vanishing energy-momentum tensor and features similar to those encountered for the free massless conformally coupled scalar field by Hartle and Hawking. Furthermore, we believe that a setting in which the evolution of the very early universe-earlier than the inflationary period-is

Ground-state wavefunction of a radiation-dominated universe 1273

dominated by gravity and non-Abelian gauge fields is more realistic from the physical viewpoint than the one of [3, 121, where the scalar field plays, together with gravity, the prominent role. Naturally, as pointed out in [12], the mathematical simplicity of a conformally invariant theory has the disadvantage that the gravitational part of the wavefunction is very close to that obtained in the absence of matter. Thus, in agreement with the Hartle and Hawking proposal that the quantum state of the universe is determined entirely by the ground-state wavefunction, one has necessarily to introduce other types of matter in order to achieve a more realistic description of our universe. For this purpose a massive scalar field has been considered in [ 121 and in [6] quadratic curvature terms have been included. Within our framework an interesting development would consist in including scalar fields. This could allow a quantum description of the transition from a radiation-dominated universe to an inflationary period.

Recently solutions of Euclideanized EYM systems have been studied with interest.

to emphasize that they correspond to a tunnel connecting disjoint regions of spacetime. Although wormhole solutions are usually seen as solutions of the Euclideanized classical field equations, it has been argued that wormholes can be obtained from the quantum mechanical Wheeler-DeWitt equation with appropriate boundary conditions [13]. Wormhole-type solutions of EYM systems with SU(2) gauge group have been found by various authors [14-161. The generalization to arbitrary gauge groups was discussed in [17] and also in [18]. In these references, a key ingredient to the generaliz- ation of the EYM wormhole solutions is the use of the theory of symmetric fields on homogeneous spaces to obtain S0(4)-symmetric ansatze for the fields in a R x S3 topology. This method has been extensively used in the dimensional reduction of multidimensional models (see e.g. [19]) and in the theory of spontaneous compacti- fication of extra dimensions (see [20] and references therein). The main virtue of the theory of symmetric fieids is that it aliows the ciassification of ali gauge ana matter field configurations which give rise to homogeneous and isotropic observables as, for instance, the energy-momentum tensor. Furthermore, the formalism of symmetric fields turns out to be useful in studying closed spatially homogeneous and isotropic cos- mologies in the presence of fields with gauge degrees of freedom [21].

In the next section, we shall review the main ideas contained in [17,21] and derive

universe. In section 3, we present our construction for the ground-state wavefunction of the universe and discuss the role played by the (anti-)self-dual solutions in such construction. Section 4 contains our conclusions and an outlook of possible implications of our proposal.

Thnrn m--.,:+o+:--ol :--+n-+n- r-l..t:--o mFe-r.rl +- nlr- .l- r,r--h-ln +..--0-1..6:-+.~ L U r D C a,c 6 ' a " ' L a L ' Y " a L L I I O I ~ L I ~ Y ' I J V L U L I Y I ' D , lrlrllr" L V a,uu an "V"L"'"'c-Lyyc J V l U L L V l l U ,

the EinisCperspcc EAae!er-nc\Titt pquation corrpsponding to a radiatien-do~~inatpd

2. EYM systems with SO(n) gauge groups (n>3)

We are interested in EYM systems, whose dynamics is defined by the action:

1 8e

S = k2 d4x 6 (R - 2A) - 2k2 c13x f i K +T d4x 6 Tr(F,,F") (2.1)

where g,, is the Lorentzian metric, with signature (-, +, +, +),in the four-dimensional manifold M, R, g and K are the scalar curvature corresponding to gKu, g = det(g,,) and the trace K = K*, of the extrinsic curvature K', of M respectively. Constant e denotes the gauge coupling constant and F,. the usual field strength tensor.

1274 0 Bertolami and J M MourEo

The most general form of an S0(4)-invariant metric, i.e. a metric which is spatially homogeneous and isotropic in a M = W x S' topology, is given by the Friedmann- Robertson-Walker ansatz:

3

-N2(t)dt2+a2(r) w w h = l

where N ( f ) and a ( t ) are arbitrary non-vanishing functions of time t ; the constant u2=2G/3?r is introduced for later convenience and w b are the left-invariant 1-forms

in S U ( 2 ) = S3 satisfying diff

3

o .b=l dwC= - 1 ccabwLI A wb.

The metric (2.21 gives rise to an Einstein tensor Gpp which is also S0(4)-invariant rnrl in r h m non an^ AI = 1 i c m i w a n hu

"i -..-, 1.1 .I._ 61"6" .. ., .~ . . _Y

3

b = l G = Gco)(t) dt2+ G,,)( t )a ( z w bu (2.3a)

where

( 2 . 3 )

( 2 . 3 ~ )

and a dot denotes a time derivative.

of the Einstein tensor (2.3a); that is, it must be of the form For consistency, the energy-momentum tensor (EMT) must share the symmetries

3

b= l T = T,,,(t) dt'+ T, , , ( t )a ( t )2 z wbwb. (2.4)

This i s the EMT of a comoving perfect fluid with energy density p ( f ) = T o , ( ( ) and pressure p ( f ) = Tclr(t). There is a large class of fields AV and generically of all matter fie!& 4, which possess the reqcired property of gegera!ing SQ(4)-icvar;.ant CMT. This class, or in other words this truncation procedure, is also consistent with the correspond- ing equations of motion. These fields are the so-called S0(4)-symmetric fields A, and a, meaning that they are S0(4)-inuariant up to a gauge transformation [17,21].

For the simplest embedding of the isotropy group SO(3) in the gauge group SO(n), the SO(.l)-symmetric ansatz for the gauge fields is given by [ 17,211:

-,.,-,.,.,I A l e \ = b f r \&Yl t$ ,.,., :z.sa:

where

is a gauge field in R with gauge group SO( n -3).

(2.5b)

Ground-state wavefunction of a radiation-dominated universe 1275

Thcso(n)=Lie(SO(n)) are the generators of SO(n), ,yo([), , y k ( t ) - k = 1,. . . , n - 3 - a r e arbitrary functions and (Ak'"(t))LZ=, is for any 1 an arbitrary anti- symmetric matrix. In ( 2 . 5 ~ ) we have introduced, for convenience, the fine structure constant n = e 2 / 4 r .

It is shown in [17,21] that the energy-momentum tensor generated by the gauge field (2.5) is that of a perfect fluid (2.4) satisfying the equation of state for a radiation fluid:

P Y M ( f ) = f P Y M ( t ) (2.6)

and that, inferms of the arbitrary functions ,yo([) and , yk ( t ) , the energy density is given by

In (2.7) x=j,yk};::, 6, is the covariant derivative of x with respect to the gauge connection A(t) in R

d G , x = - x + i i x (2 .8a) d t

I \

= (Ak'"( t))t;3= I and ,

(2.86)

is the effective potential of self-interaction for ,yo and x (see (2.9) below). Substituting (2.2) and (2.5) into action (2.1) we obtain a Lagrangian density

independent of the space coordinates. The integration over these coordinates, gives just the volume p = 2 r 2 of S3 (for U = 1) as a multiplicative factor. We are then led to an effective action for an S0(4)-symmetric degrees of freedom

S e d ~ ~ o . ~ ; ~ , ~ ~ " ' l

where H 2 = 2GA/9m. Notice that,in (2.9), N ( I ) and A'"'( f ) are not dynamical variables. These functions play the role of Lagrange multipliers, being the latter associated with local SO(n -3 ) symmetry of the minisuperspace model remnant from the original SO(n) gauge symmetry.

It is worth mentioning the striking similarity, due to conformal invariance, between our effective action (2 .9) for the EYM system and that of Hartle and Hawking for the free massless conformally coupled scalar field (equation ( 5 . 5 ) in [3]). The only differences being that we have more variables (,yo and x ) describing the matter sector and a self-interacting potential which is quartic rather than quadratic.

The absolute minima of the potential (2.28b) correspond to pure gauge configur-

vacua which survive the condition of SO(/I)-symmetry, among the infinitely degenerate vacua of the full Yang-Mills theory.

..a:-"" L.*,-"-:..,. *^ +¶.--a *,...-I :"-,...:..-,A..* ..^^..^ r 1 7 1 I%.*"* - - ~ *I... allullJ vs'u"gLL1~ L U LlllCiC Lup"'u~1ca"y "Lcqu,Y'a's,,r Y ' l L U ' a L , ,,. L 1 1 T J S a,= L 1 1 S U ' L L Y

1276 0 Bertolami and J M Mourio

Action (2.9). with its local SO(n - 3 ) symmetry is an example of an action for the so-called Yang-Mills mechanical systems [22] . It has the additional property of being invariant under arbitrary time reparametrizations. Quantization of Yang-Mills mechanical systems has been a subject of interest, as it is thought that it allows for a better understanding of the quantization of four-dimensional Yang-Mills theories [ 2 2 ] . Similarly, it is believed that quantum cosmology could lead to a better understanding of the full quantum theory of gravity.

We stress that the above results for SO(n) gauge groups can be generalized to any arbitraly compact simple Liegroups-SU(n), S p ( n ) , G2, F4, E*, E, or E8. In particular, the S U ( n ) case turns out to be quite similar to that of an SO(n) group (see [17,21] for details).

Let us now turn to the derivation of the Wheeler-DeWitt equation corresponding to the minisuperspace with dynamics described by the effective action (2.9). We start by computing the canonical momenta conjugate to a ( f ) , x a ( f ) and x ( t ) :

The Hamiltonian constraint is obtained by considering the variation of Se, with respect to the lapse function N ( f ) , that is:

(2.11)

which yields, after using (2.10) 1 N - -{-vi - a 2 + H'a' + "io+ ai+ ZV,,(xO, x ) } = 0 . 2 a

(2.12)

It is relevant to remark that, in obtaining (2.12), we have allowed v. and a to c o n x ~ t e , ;vvhich, a!thoagh perkc!!;? !egitim~te I! the r!sssic~! !eve!, leads to ambiguities at quantum level due to the ordering of operators. Quantization proceeds by promoting the canonical conjugate momenta in (2.12) into operators, in the following way:

a r,,+-i-

Ja a nxo+-i-

axa a

ax nx + -i -.

Due to the operator ordering ambiguity, it is usual to parametrize d as [ 3 ] :

,+,-."(a."\ aa\ aaJ

(2.13a)

(2.136)

where p is an arbitrary real constant. The Wheeler-DeWitt equation is then found to be:

= E 8 ( a , x o . x ) (2.14)

where E is an arbitrary constant arising from matter-energy renormalization.

Ground-state wavefunction of 4 mdiution-dominated universe 1277

An additional constraint emerging from the effective action (2.9) is the vanishing of the components of the orbital angular momentum corresponding to the motion in the subspace R"-' of the variables x. This arises from the variation of Se, with respect to A:

(2.154)

--.,.:.,. _.. 1. wnicn rraus

Xk"x, -XmTxk =O. (2.15 b )

Thus, the wavefunction +(a, ,yo,x) has to satisfy besides (2.14), the condition:

In (2.12) we can see that the Hamiltonians corresponding to the gravitational and gauge degrees of freedom effectively decouple:

U 4 -R=- (xgra"+-Tgs"ge) . (2.17) N N

Therefore, solutions of the Wheeler-DeWtt equation can be obtained by separating variables in the following way:

$(a, x o , x ) = E Cn(4)Un(x0, x ) (2.18)

where C.(4) and U.(x0 ,x ) are solutions of:

and

(2.20)

Notice that equation (2.19) is the very one obtained by Hartle and Hawking (equation (5.11) in [3]). Furthermore, the functions U,,(x0,x) must satisfy constraint (2.16):

Ah U" ( X " , x ) = 0. (2.21)

We shall restrict ourselves to the case of an SO(3) gauge group for which the vector x does not exist and therefore:

$=+(a ,xo)=C C.(4)Yn(xJ (2.22) n

where C,,(4) satisfy (2.19) and Y.(xo) are solutions of:

(2.23)

This equation coincides with the Schrodinger equation for an anharmonic oscillator of the double-well type. On general grounds, the quantum anharmonic oscillator has been studied in [23]. More concretely, numerical solutions have been known for a long time [24] and perturbation theory at very high order has been analysed in depth

1278

1251. Moreover, variational methods involving more than one variational parameter prove to be fairly accurate in the description of the energy levels [26 ] . In the next section, we shall see that, as the ground-state solution of (2.23) can be computed with the help of the path integral, in the semiclassical approximation, this solution can be related to (anti-)self-dual solutions of the Euclideanized YM equations.

0 Bertolami and J M Mourio

In what follows, we shall use the Euclidean path integral (1.3) to obtain the ground-state solution of (2.14) for an S O ( 3 ) gauge group.

As in the case considered by Hartle and Hawking 131, conformal invariance implies ihai ihe energy-momentum iensot of ihe exiremizing YM co&guraiions, consisieni with the Hartle and Hawking boundary conditions vanishes. Also, the evaluation of the ground-state wavefunction is greatly simplified as the gravitational and gauge functional integrals in (1.3) may be computed separately:

~ ~ a , x a l = C ( a ) W x o ) = ~ % exp(-sfr.J4?)l) 1 ~x"exp(-Sf*.,.[x0(1))1) (3 .10)

where

and 1) is the conformal Euclidean time defined as dq = d ~ / a it!.

boundary conditions, appropriate for the state of minimal excitation: The functional integrals in the RHS of ( 3 . 1 ~ ~ ) are evaluated with the following

a(-oo)=O a(0) = a ( 3 . 2 ~ )

and

xo(--30)= F(t7rjcr)l ' ' X d O ! = xo. (3.26)

The function C ( a ) in ( 3 . 1 ~ ) is given, in the semiclassical approximation, by [ 3 ] :

C ( a ) = exp(;n'- 1/3H') for Ha K 1

(3.3)

Let us now compute semiclassically the function U(xa). For that we must find finite - - I . . + : - - ~ -n+:rr..:-r A -... .-,.-A:+:--- I I 91.1 -r +hn c . , A : A ~ - ~ : - ~ _..

equations obtained from (3.1~). Action ( 3 . 1 ~ ) can be seen as the action ora unit mass particle subject to the potential V,,(xo). With the help of this analogy, it is clear that

LICLIW,, D U l U L L W L l J r'lrlrlJ,,,g "UU"U~L'J C V I I U I I t V I I ~ \-'.'U, U, L1.L L"CLI"..aLI,&b.U , 1 1 1

Ground-state wavefunction of a radiation-dominated universe 1279

the solutions satisfying (3 .26) are those with vanishing mechanical energy and therefore are given by the solutions of the following first-order differential equations:

(3.4)

These equations coincide with the equations for (anti-)self-dual Y M fields. Indeed, by substituting the ansa& (23) into the (anti-)self-duali!y conditions;

Feu = &*F,+. (3.5)

where

*Fe, = + & E , , _ ~ F ~ ~ (3.6)

one obtains precisely (3.4). Notice that conditions (3.5) on their own ensure that t e EMT vanishes.

Self-dual and anti-self-dual solutions for the SU(2) and S O ( 3 ) Yang-Mills theories on various spacetime backgrounds [27] are already known. For a discussion of the S O ( n ) case, in an S0(4)-invariant background, the reader is referred to [17 ] . In the present case, the solutions of (3.4) satisfying (3.26) read:

X O ( V ) = ( ~ / ~ ) ~ / ~ coth(?-?o) for ,yo< -(37r/a)'/' (3 .7a)

xu(?) = f ( % / ~ ) 1 / 2 tanh(7 - v0) (3 .76)

P

for !,yo/ < ( 3 ~ / a ) ' / ~

~ 0 ( ? ) = - ( 9 7 r / n ) ' / ~ C O t h ( ? - ? o ) for ,yo> ( 3 a / a ) ' / ' . (3.7c)

It is important to remark that for !xQ! < ( $ ~ / a ) ' / ~ (see (3.76)): there are two solutions with the same boundary value ~ ~ ( 0 ) =,yo, the relevance of each being determined by the relative value of their corresponding actions. In this region, the wavefunction is approximately given by the even combination of the semiclassical contributions from these two solutions [ % I . Therefore, semiclassically, the wavefunction is the following (see figure 1 ) :

7 r/.,~\ A avni-r.,~ i I L . / ~ ) I / W + W ~ i - ) W r . , ~ i 12- i - \1 /2131 - \ A V , ' - - ' .Yt L & U ' \ A ' ' I - I .1 3\3-,**, L A U ' \ I ' . i " I J I

for ,yo< - ( $ ~ / a ) ' / ~ (3 .8a)

U(,yo) = E exp{-[~o+(~7r/~)'~2]2+~(~a/7r)'~2[~o+(~a/a)''2~3}

+ B exp {-[,yo - ( f ~ / L Y ) ' / ~ ] ~ - ;($a/ a)"2[,yo - ($a/ a) ' /2]3)

for lxol s ($n/a)"2 (3.86)

U(,yo) = A exp{-[Xu- ($7r/a)'/232 -$(fa/n)'/2[,yo- ( $ ~ / a ) ' / ~ ] ~ }

for ,yo> ($n/a)"2 ( 3 . 8 ~ )

where A =[1 +exp( -Z~ /a ) ]B and E is a normalization factor. We see that near the two vacua, ,yo= * ( : n / ~ ) ~ ~ ' , the wavefunction has essentially a Gaussian behaviour, as one should expect from the double-well form of potential (2 .86) (for ,y =O). This is because (3.8) is an approximate solution of the Schrodinger equation (2.23) with

1280 0 Bertolami and J M Mourio

c: < r,^..LI^ ... "I, "="".:,,- < , * A , sa_ .__" ,*_^LO" " .,-, -, " " A ,I.- "",,"- "%U'S I . """"IC-WCII L I I Z L . L I . L pY="L1"1 ,'.YVI ,", -" L","., ,,.. 6ay6..

wavefunction ofthe minimally excited state (3.8) (full curve) for: ( a ) A = 50 and a/ li = 0.01 (as expected from the ROE analysis in GUT); ( b ) A = 10 and n / l i =0.05; ( e ) A = 5 and a / ~ = 0 . 1 ; ( d ) A = 1 and o / l i=0 .5 (close to the breaking of the semiclassical approxi- mation).

n = 0 and around the minima the potential is approximately quadratic. For large ,yo. solutions (3.8) reproduce the expected exp[ -f($a/rr)"'\,yol3] behaviour of the wavefunction [24].

4. Conclusions and outlook

We have studied the minisuperspace Wheeler-DeWitt equation for Einstein-Yang- Mills systems. Due to the conformal invariance of the Yang-Mills action we find that,

Ground-state wauefunction of a radiation-dominated universe 1281

similarly to the case of a free massless conformally invariant scalar field studied in [3,12], the Wheeler-DeWitt equation can be solved by separating the gravitational and gauge degrees of freedom. We build the ground-state (no boundary) wavefunction for the case of an SO(3) gauge group in the semiclassical approximation using (anti-)self-dual solutions of the Euclideanized Yang-Mills equations. The Wheeler- DeWitt equation (2.14) for general SO(n) gauge groups will be the subject of a future analysis.

The wavefunction constructed in this way accommodates naturally the expectation that the very early universeearlier than a conjectured inflationary stage-was domi- nated by radiation. Here an interesting development would be to include scalar fields in the setting we have described in the present work. One could then, in principle, achieve a quantum description of the transition from a radiation-dominated universe to an inflationary phase.

values of the constants of nature [29,30]. Moreover, the wormhole influence can be seen as resulting from the interaction of our universe with a bath of harmonic oscillators in an unknown state. Nevertheless, we have seen that the degenerate structure of the gauge fields vacua leads to a wavefunction which is peaked for the different values of the dynamical variables which correspond to these vacua. Thus, a possible implication of our construction would consist in an indeterminacy in the process of fixing the constants of nature [29]. This may have consequences on the various phenomenological questions that are related with the breaking of global symmetries.

T+ i o k e l i a . r e . 4 +h-t +ha a ( T m r + nf thn ---sa-.-- nC..zn-h-S-- ir +ha fr:-- - F + h - ', ,., " C l l r l r " L.L*L L L l L .Ilfllll C Y b . C L ", ,"C y L c " b , , L ' c " I w"L,.III",rD 1n L l l L A L A l L L ~ "1 ,.,=

Acknowledgments

It is a pleasure to thank F Freire and J C Zambrini for discussions. We also thank P Si for the help in constructing figure 1.

References

[I] DeWitt B S 1967 Phys. Rev. 160 1113 [ 2 ] Wheeler J A 1968 Ballelle Renconrres ed C DeWitt and J A Wheeler (New York Benjamin) [3] H a d e J B and Hawking S W 1983 Phys. Rev. D 28 2960 [4] Amowitt R, Deser S and Misner C W 1962 Gravirorlon ed L Witten (New York Wiley) [5] Hawking S U' 1984 N u d Phys. 8 244 I35 [6 ] Hawking S U' and Luttrell P C 1984 Nuel. Phys. B 247 250 [71 Moss 1 G and Wright W A 1984 Phys. Res. D 29 1069 [8] Hawking S W a n d Wu 2 C 1985 Php. L l t . 1518 I S [9] Gonzales-Dias P F 1985 Phys. Lei!. 1598 19

[IO] Page D N 1986 Quanrum Concepts in Spaceand Kme ed R Penrose and C J lsham (Oxford: Clarendon) [ I l l Gibbons G W a n d Grishchuk L P 1989 Nuel. Phys. B 313 736 1121 Hawking S W 1984 Nucl Phys. B 239 257 [13] Hawking S W 1990 Mod. Phys. Len. SA 453 [I41 Hosoya A and Ogura W 1989 Phys. L e x 2258 117 [!5] Verbin Y a d Davifisoc A log9 Phyr k l ! . ??!!I! 364 [16] Rey S J 1990 Nucl. Phys. B 336 146 [17] Bertolami 0, Mourio J M, Picken R F and Volobujev I P Dynamics of Euclideanized Einstein-Yang-

[18] Yashida K, Hirenraki S and Shiraishi K 1990 Phys. Rea D 42 1973 Mills systems with arbitrary gauge groups Inr. J. Mad. Phys. A to be published

1282

[19] Manton N S 1979 Nucl. Phys B 158 141

0 Bertolami and J M Mourcio

Forgacs P and Manton N S 1980 Commun. Marh. Phys. 72 I5 Bais F A , Barnes K I, Forgacs P and Zoupanos G 1986 Nucl. Phys. B 236 557 Coquereaux R a n d ladcyzk A 1988 Riemannian Geomerry. Fiber Bundles. Kalum-Klein Theories and

Kubyshin Yu A, Mourio J M and Volobujev I P 1989 Inr. J. Mod. P h p A 4 151; 1989 Theor. Math.

Rudolph G and Volobujev I P 1989 Nucl. Phys. B 313 95

all rhot . . . (Lecture Notes in Physics 16) (Singapore: World Scientific)

Phys. 78 41, 191

[20] Kubyshin Yu A, Mourio J M, Rudolph G and Volobujev I P 1989 Dimensional Reduction of Gauge Theories, Sponraneons Compactifieation and Model Building (Lecture N o t a in Physics 349) (Berlin: Springer)

[21] Moniz P V and Mourio J M Homogeneous and isotropic closed cosmologies with a gauge senor Lisbon Preprint lFM.11/90

[22] Prokhorov L V 1982 Sou. 1. N u d Phys. 35 285 Sawidy G K 1983 Phys Lett. I308 303 Shabanov S V Phase space reduction and the choice of physical variables in gauge theory JINR Preprinr

EZ-90.23 [23] Reed M and Simon B 1975 Methods in Modern Mathemarim1 Physics val I1 (London: Academic) [?4] McWeeny R and Coulson C A 1948 h o e Camb. Phil, Soc. 44 413 [25] Bender C M and Wu T T 1971 Phys. Rei. Lett. 27 461; 1973 Phys. Rev, D ~ 7 1620 [26] Dias de Deus J 1982 Phys. Reu D 26 2782 [27] Pope C N and Yuille A L 1978 Phyr. Lett. 788 424

Duff M J and Madore J 1979 Phys. Reo. D 18 2788 Boutaleb Joutei H and Chakrabarti A 1979 Phys. Reu D 19 457 Picken R F 1981 I. Phys. A: Math. Gen. 14 2960

1281 Maslov V P 1984 Proc. Int. Conar. ofMurhematics (Warsaw) vol 1 (Amsterdam: Nonh-Holland1 D 139 _ . . . b j Coleman s 1988 Nucl. Phw. B 310 643 1301 Hawkina S W 1987 Phys. Lerr. 195B 337; 1988 Phvs. Rev. D 37 907

Lavrelashvili G V, Rubakov V A and Tinyakov P G 1987 JETPPLerr. 46 167; 1988 Nuel. Phy.$. B 299 757 Giddings S and Strominger A 1988 N u d Phys. B 306 890; 1988 Nurl. Phys B 307 854 Coleman S 1988 Nucl. Phys. B 307 867 Hawking S W and LaRamme R 1989 Phys. Lett. 2098 39 Klebanov 1, Surskind L a n d Banks T 1989 N u d Phyx B 317 665