The phase structure in minisuperspace cosmological models

Physics Letters B 272 (1991) 11-16 North-Holland

PHYSICS LETTERS B

The phase space structure in minisuperspace cosmological models S.V. S h a b a n o v Laboratory qf Theoretical Physics, Joint Institute for Nuclear Research, Head Post Office, P.O. Box 79, SU-IOI 000 Moscow, USSR Received 11 July 1991

A phase space (PS) structure in minisuperspace cosmological models with gauge fields is investigated. It is shown for the SO (n), n > 3, gauge group that the physical PS differs from an ordinary plane. Due to this phenomenon, the path integral representation gets modified for the ground-state wave function of the Universe. It is also argued that the wormhole size quantization should change due to a non-trivial physical PS structure of gauge fields.

1. Recently much attention has been devoted to the study o f the E i n s t e i n - Y a n g - M i l l s system in q u a n t u m cosmology. This system is rather difficult by itself. Following, however, ref. [1], one m a y introduce a set o f simplifying a s s u m p t i o n s and consider closed cosmologies with an ~ ® S 3 topology. In this gauge fields on a homogeneous space are described by the S O ( 4 ) - i n v a r i a n t ansatz [ 2 - 4 ] . The reduced system (the minisuperspace m o d e l ) contains only a finite n u m b e r o f degrees o f freedom corresponding to gravitational and gauge fields. Nevertheless, it is believed that the model keeps some d y n a m i c a l features o f the original field theory. In particular, it has two local s y m m e t r y groups, the r e p a r a m e t r i z a t i o n and gauge ones. It is known that models with a gauge symmetry may have a non-trivial phase space ( P S ) o f physical degrees o f freedom [ 5 - 7 ]. In the present letter we study the physical PS structure o f the minisuperspace model. In particular, the PS for one o f the physical variables in the case o f the S O ( n ) gauge group, n > 3, turns out to be a cone unfoldable into a half-plane. We also argue that this p h e n o m e n o n gives rise to a modification o f the path integral representation for the ground-state wavefunction o f the Universe and, as a consequence, the quasiclassical a p p r o x i m a t i o n changes in the minisuperspace model. Then we consider wormholes. Wormholes can be treated as grayE-mail address: [email protected].

itational instanton type solutions, i.e., as solutions of the euclideanized classical field equations corresponding to tunnelling through a classically inaccessible region [ 8 , 2 - 4 ] . We observe that the wormhole quantization rule [2,4] should be m o d i f i e d due to the physical PS reduction i f a gauge group has a rank higher than one. 2. In the minisuperspace a p p r o a c h to the E i n s t e i n Y a n g - M i l l s system, the metric has the SO ( 4 ) - i n v a r iant form. The most general form o f such a metric, i.e., a metric which is spatially homogeneous and isotropic in a space ~ ® S 3 topology, is given by the F r i e d m a n n - R o b e r t s o n - W a l k e r ansatz 2G ga"dxUdx"= ~n [-N2(t)

dt2+p2(t)°Yc°']'

(1)

where N ( t ) and p ( t ) are arbitrary non-vanishing functions o f time; G is the gravitational constant and ~o' are the left-invariant one-forms ( i = 1,2, 3 ) on the three-sphere satisfying the condition doJ~= -- (_ok09j A O)k. The ansatz for gauge fields in the metric ( 1 ) was suggested in refs. [ 4,9 ]. Gauge fields with the SO ( n ) group, n > 3, are described by a scalar X = Z ( t ) e N , a vector x = x (t) e N/, l = n - 3, and a real a n t i s y m m e t r i c l × l matrix y = y( t ), i.e., y = y , T a, yaEN, T a are generators o f SO (l), so that the effective minisuperspace action of the E i n s t e i n - Y a n g - M i l l s system reads

0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

11

Volume 272, number 1,2

PHYSICS LETTERS B

t2

find that all the secondary constraints are equivalent to the following:

2

S=½~dtN[-(Potp).-l-p2-,~2p tl

4

+(Po,x)Z+(PD,x)2-2V] ,

(2)

where D, = 3, + y is the covariant derivative, 22= 2GA/ 9~r, A is the cosmological constant,

O~ [-/" ~

2

3/r']2+4Z2X21

-2o~J

(3)

is the potential of the Yang-Mills fields, 0¢=g2/4~is the fine structure constant. The action (2) is invariant under two local groups, the reparametrization one dl'

t--,t'(t), N(t)-.N(t') d t '

(4)

and the SO(I) gauge group, under which only the variables x and y transform as follows:

where ~2=exp(~o~T~)zSO(l), ~(2T=(2T.Q=I and ~o. = ~o~(t) are arbitrary functions of time; other variables remain unchanged. Due to these local symmetries, the variables N and y turn out to be purely unphysical because their canonical m o m e n t a vanish,

5S

pN = ~

=0,

8S

ga = --8)>a= 0 ,

(6)

where the dot means the time derivative. So, N and y play the role oflagrangian multipliers. Determining the canonical m o m e n t a pp, p and Px of the remaining variables p, x and X, respectively, we may find the canonical hamiltonian in the standard way,

N ( -p,,2_p2+)2p4+p2+p2+2V_y~aa) I4= yp N

- -- ( H u D - - y ~ a ~) , P

(7)

where a~=pT~x. However, the system has the primary constraints (6) [ 10 ], therefore it is necessary to find all the secondary constraints calculating the Poisson brackets of H with the constraints (6) and putting the results equal to zero [ 10 ]. In doing so, we 12

28 November 1991

HUD = 0 ,

(8)

aa=0.

(9)

As a consequence, the hamiltonian vanishes, which is always the case for systems with a reparametrization symmetry. It is easily to be convinced that all the constraints are first-class ones [ 10 ]. Eq. (8) is the classical Wheeler-DeWitt equation for the minisuperspace model. The constraints (9) generate the SO(I) gauge transformations of the canonically conjugated variables x andp. However, not all of them are independent. The number of independent constraints is l - 1 since any vector in ~ has a stationary subgroup SO ( l - 1 ) (a vector x~ ~ can always be directed along one of the coordinate axes by a gauge transformation). Therefore a "particle" described by a vector x has only one physical degree of freedom. Really, it is a radial motion because the quantities o ~ coincide with components of the particle angular m o m e n t u m in ~R1. 3. Consider now the physical PS structure of the gauge field degrees of freedom. Obviously, the total PS of these variables consists of points (x, p ) ~ BR2~and (X, PX) ~ 2 [WE ignore the pure unphysical degrees of freedom (Ya, ~ = 0 ) ]. The physical PS is a subspace of the total PS picked out by the constraints a a = 0 and an identification of all points connected by gauge transformations on the surface of the constraints (the latter points correspond to the same physical state of the system). Variables X and Px are gauge invariant therefore their PS is the usual plane, ~2. The general solution of eq. (9) is p = ~x where a function of time ~ is determined by dynamics (by the potential V). This solution means that only radial excitations are admissible. Further, one may always direct a vector x along one of the coordinate axes with the help a gauge transformation (the unitary gauge), for example, we put xi=d,~x. As a consequence, pi=6Hp, p=~x. However, this is not the end. There remain residual gauge transformations forming the 7/2 gauge group with the help of which one may change the sign of x: x ~ _+x (the gauge rotations through the angle n). The residual gauge group cannot decrease the

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PHYSICS LETTERS B

n u m b e r of physical degrees of freedom, but it reduces their PS. Indeed, the sign of p should change simultaneously with the one o f x due to the equality p=~x. Hence the points (x, p) and ( - x , - p ) on the phase plane ~2 are gauge equivalent and should be identified. The phase plane turns into a cone unfoldable into a half-plane that is just the physical PS because the gauge arbitrariness is exhausted. The same result may be found by considering the motion in the gauge-invariant canonical variables r = Ixl, p~= (p, x)/r [71. A modification of the physical PS leads to some consequences in the corresponding quantum theory. For example, the path integral approach differs from the usual one [6,7,11] and, as a consequence, the quasiclassical approximation changes [ 7,12 ]. Quantum Green functions have unusual analytical properties [ 13 ]. Therefore we may expect the appearance of analogous features in our system. As we show below, that is really the case. 4. According to the Dirac quantization method [ 10 ] of systems with the first-class constraints, to get a quantum theory corresponding to the classical one, we change all the canonical variables q~ and p,, where q,, p~ mean the sets (x, p, Z), (P, lop, &), respectively ~', by operators with the canonical c o m m u t a tion relations

[O., ~o~] = i ~ . ~ ;

(10)

the operators of constraints (8), (9) must annihilate the physical states. The equation corresponding to eq. (8) is the Wheeler-DeWitt equation. In the coordinate representation it reads [9]

6"~ph(q)=0,

(12)

w h e r e / i = - ih ~ . The functions ~'ph are normalizable by the following condition:

(13)

i dp PP i dz f dx l ~ph ( q )12= l . 0

--oc~

621

The operator/lwD in ( 1 1 ) is hermitian with respect to this scalar product. Eq. (12) means that the physical states are invariant under SO (l) rotations of the vector x, i.e., they are the s-states/~/ph (X, Z, P) = 0 ( r , Z, P) =--0 ( z ) , where r-- Ixl. Moreover, these s-states should be even,

O(r,x,p)=O(-r,z,p)

(14)

because the potential Vis an analytical function. One may also prove the same property for the variable p,

(15)

#(r,z,p) = O ( r , z , - P ) .

Indeed, due to the parity 0 f H u D in p we may divide all solutions of eq. (11 ) into even and odd ones. However, only the even ones have a regular behavior at p = 0 (wave functions should be regular [15,1 ] ). When p ~ 0 , the first term in HwD is only essential, therefore, O~p"Jv, u= ½(p-1 ), J. being the Bessel function. We must select from all solutions of ( 1 1 ) only those which possess the asymptotics p"J,, p-~0; the odd wave functions, obviously, do not have it. To formulate our quantum theory only via physical variables, we introduce spherical coordinates. Since the 0 are independent of angular variables, we get instead of ( 1 1 ) ½(_/}2_D2..l_..~2p4

½[hZp-P(Opop p OF,)--p2+22p4

h2

-v + 2V+ 2Ven')OE(z ) 3x2 +pr

=EOa'(z),

_ h 2 3 ~ _ h 2 3~ +2V]~//ph(q ) =Egtph ( q ) ,

28 November 1991

( 11)

where E is an arbitrary constant arising from the matter-energy renormalization [1 ]. The real number p > 0 is usually introduced in order to take into account the curvilinearity of the minisuperspace (it reflects also the operator ordering problem here) [ 1 ]. The quantum version ofeq. (9) reads ~tl We omit again the purely unphysical degrees of freedom y, and N [14,7].

(16)

v e n ' = - h2p(p-2) + h 2 ( l - 1 ) ( l - 3 )

81)2

8r 2

(17)

,

/)a = - ihp -" Opop", v = 1 (p_ 1 ), 1)r= --ihr-"'O,.or ~', v ' = ½ ( l - - 2 ) are the hermitian mo-

where

m e n t u m operators. The corresponding scalar product reads

i d r d p ? dZp(Z)~'(Z)Ou,(Z)=~EE,

(18)

13

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w h e r e / t ( z ) =r(l-~pV; we include the total solid angle into the norm of Ou. Let us turn now directly to describe the q u a n t u m theory in terms of the path integral (PI). The amplitude

g~h(z,z')=(zlexp

('

--~r//twD

) Iz'),

(19)

where r/is the euclidean conformal time (dr/=d~/p, r = - i t ) , is the standard object in this approach because in the limit r / ~ this kernel gives the groundstate wave function of the Universe [ 1 ]. When deriving the PI for the amplitude (19) in the usual way based on iterations of infinitesimal kernels U~h, e-,0, we meet certain difficulties. The first is the existence of the non-trivial measure ~t(z) in the scalar product (18). The reduction of the integration region in (18) represents the second, more serious problem for calculating the iterations. Indeed, according to ( 18 ) a convolution of two infinitesimal amplitudes (18) contains integrals over a semiaxis ( r > 0 , p > 0 ) . So, in the limit e--+0 we get a PI on a semiaxis and its calculation is indefinite even for the simplest systems such as a free particle and an oscillator (we cannot calculate an infinite dimensional gaussian integral in semi-infinite limits). The latter, in fact, reflects the physical PS structure in the theory. Note that in contrast with the variables r, the PS of which is reduced due to the gauge symmetry ~2, the PS ofp is reduced at the very beginning [ see ( 13 ) ] because only p > 0 has physical meaning. Nevertheless, the properties (14), ( 15 ) allow us to avoid these difficulties and to get a PI with a standard measure, where integration is carried out over all variables in infinite limits. We do not give here its derivation because it may be easily done by the method suggested in ref. [ 6 ]. The final formulas read

U~"( z, z')

= i

dz" ~ [lt(Z)lt(z,,)]l/2 U;r"(z,z")Q(z",z'),

(20)

~2 One should not, however, think that the PS of the radial variable in the corresponding non-gauge model [y---0 in (2) ] is a half-plane. A careful analysis shows that the PS of each variable in the spherical coordinate system is a complete plane [7]. 14

28 November 1991

O~(z, z') =O(Z-z')Q(p, p')Q(r, r') ,

(21)

Q(r, r') =&(r-r') +a(r+r') ,

(22)

where d z - d r dp dz, the kernel Q (p, p') coincides with the one in (22) and

.f

U, ( z , z )=

f Dzexp

-

Serf[z] ,

(23)

sefr[zl = ½ I dr/ [ -/02-p2+~.2p4q-22q-/'2 rt '

+ 2 ( V + Veer) ] .

(24)

Here, Dz = Dp Dr DZ means the standard measure of the lagrangian PI and the initial conditions are defined as z(q') =z', z(q") = z " ; the dot in the effective action S e~r denotes the derivative with respect to the conformal time r/. The operator Q entering into (20) shows that together with a direct trajectory connecting the points p, p', or r, r' one should take into account contributions into the transition amplitude of trajectories going from - p ' to p and from - r ' to r. It resembles the motion on a semiaxis restricted by an impenetrable barrier at zero when the trajectory reflected from the "wall" contributes to the transition amplitude as does the direct trajectory [ 17 ]. The difference, however, is that in the latter case there exists the boundary condition qJl r=0 (or ~hp=o) and because of it the contribution o f the reflected trajectory is taken with the opposite sign. Therefore the operator Q(r, r') =O(r-r')-6(r+r') is antisymmetric in r and r'. There is no "wall" at zero in our theory, but the existence of the residual gauge group [ see (14), ( 15 ) ] gives effectively the boundary condition 0pO[p=o = 0,~l r=0 = 0, which leads to the operator Q (22), symmetrical in both arguments. It resembles also a reflection of a quantum particle without any change in its phase. Thus, we have found that the PI for the minisuperspace model in a quantum cosmology is modified due to the physical phase (configurational) space reduction. The amplitude (20) gives the ground-state wave function o f the Universe in the limit r/--+oo. Therefore, we may expect that its quasiclassical calculation turns out to be modified also.

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PHYSICS LETTERS B

5. When calculating the integral (23) quasiclassically, one should find a stationary trajectory Zs~ satisfying the equation

8sen-[ z ] = 0 ,

z(q)=~,(q).

(25)

28 November 1991

Thus, the measure/~ as well as Venare essential only for calculating quantum corrections to the leading term of a quasiclassical series, but the 7/2 symmetrization in r should be always done as it is ordered by eq. (20).

The condition

S~ff[z~ ] << h

(26)

is assumed to be valid for this trajectory. Since the effective q u a n t u m correction V err to the potential is proportional to h 2, we may approximate (25) by the classical equations of motion 5 S = 0 . However, the contribution of the effective quantum correction taken on the classical trajectory may be infinite due to the singularity at p = r = 0 . Hence, the condition (26) is broken if the classical trajectory goes too close to the points p = 0, r = 0. This singularity cannot be eliminated by using trajectories satisfying (25) because they have a gap at r = 0 and p = 0 . Therefore, the quasiclassical approach in the neighbourhoods of these points is forbidden. It is necessary to solve the exact quantum problem. An exceptional case arises at p = 2 a n d / = 3 when V ~ff- 0 and the quasiclassical approach is correct in the neighbourhoods o f p = 0 and r = 0. The singularity of the multiplier [/~(z)/z(z') ] -~/2= (pp'rr') -~ in the kernel (20) cancels because of the operator (~ action. If we are interested in the wave function behavior far from the points p = 0 , r = 0 (when the effective quantum correction to the classical action is much smaller than the action itself), then eq. (20) orders only the symmetrization o f the quasiclassical kernel (23) with respect to the group Z2®~-2 (p~+-p, r--. + r) ~3. We may also neglect the contribution of the measure/t in (20). Since the scalar product contains/~, we may consider the kernel/tU,~ h eliminating then ~t from the scalar product. The multiplier [~ (z') / /~(z") ]~/2 is equivalent to an additional term ( ~ h ) in the system action because

where z(r/') = z ' and z(q") =z".

~3 The symmetrization in p is trivial here because the quasiclassical amplitude (23) depends analytically on p2 [ 1].

6. Another example where the physical PS structure of gauge fields plays an b.aportant role, is wormhole quantization, first encountered in ref. [2] for the gauge group SU (2), but in this minisuperspace model the PS o f gauge fields is a plane because S U ( 2 ) ~ S O ( 3 ) , i.e., / = 0 . The case of an arbitrary gauge group was considered in ref. [4] where, however, the non-trivial structure of the physical PS was not taken into account. Consider the euclidean version of the equation of motion for the system (2) ( - it = r, y--, iy) and introduce the euclidean conformal time r/. Due to the SO(l) gauge invariance o f the equation of motion we may always put xi(t/) = 3 n x ( q ) , i = 1,2, ..., l. It means that the physical state changes are described by x varying along the first axis (the unitary gauge). In other words, one may always choose arbitrary functions y,(q) (choose a gauge) so that xi(fl) = 0 , i = 2 , 3, ..., l. It was shown in ref. [4] that there exist periodic solutions x(r/), p(q) and Z(q) with periods Tx, Tp and Tz, respectively. If we interpret the solution p (q) as a wormhole connecting two points in the same space, the gauge fields should be the same at both sides. Since Z(q) and x(q) are periodic, the period T~, (the time between two p-maxima) should be an integer multiple of their periods [ 2 ], i.e.,

Tp=nTx=mT, ,

(28)

where n and m are numbers. The relation (28) leads to the exponential quantization of the wormhole size

[2]. The relation (28) is valid if the physical PS of gauge fields is assumed to be a plane. However, as we have shown above, that is not the case for the variable x. Its PS is a cone unfoldable into a half-plane. Since x oscillates around x = 0 [4] with period 7:~, we find for the physical period T~ h =½Tx (points x < 0 are gauge equivalent to points x > 0; T p~ is the time during which the system returns to an initial physical state). Therefore the quantization rule of wormholes (28) should be changed, 15

Volume 272, number 1,2

Tp=nTz=mT~h=½mT~.

PHYSICS LETTERS B (29)

As a c o n s e q u e n c e , the q u a n t i z a t i o n o f the w o r m h o l e s size is also m o d i f i e d . I f the t h e o r y c o n t a i n s o t h e r fields r e a l i z i n g a certain gauge g r o u p r e p r e s e n t a t i o n , the p e r i o d s o f t h e i r physical oscillations w o u l d be d e f i n e d by p o w e r s o f the i n d e p e n d e n t C a s i m i r o p e r a t o r s for a g i v e n repr e s e n t a t i o n [ 7 ]. 7. We h a v e c o n s i d e r e d the simplest case o f the gauge group S O ( I + 3). In principle, the case o f an a r b i t r a r y gauge group changes n o t h i n g in o u r investigation, only the t e c h n i c a l details are c o m p l i c a t e d . We h a v e also not i n c l u d e d f e r m i o n fields into the m i n i s u p e r s p a c e m o d e l . F e r m i o n degrees o f f r e e d o m m a y also h a v e a n o n - t r i v i a l PS s t r u c t u r e due to a gauge s y m m e t r y [ 7 ] and, m o r e o v e r , the c o r r e s p o n d i n g q u a n t u m description has specific features as c o m p a r e d with the bosonic case [ 18 ]. We will study such m o d e l s elsewhere.

References [1]J.B. Hattie and S.W. Hawking, Phys. Rev. D 28 (1983) 2960. [2] Y. Verbin and A. Davidson, Phys. Lett. B 229 (1989) 364. [3] A. Hosoya and W. Ogura, Wormhole instanton solutions in the Einstein-Yang-Mills system, preprint RRK 89-7/INSRep.-733 (1989). [4] O. Bertolami et al., Dynamics of euclideanized EinsteinYang-Mills systems with arbitrary gauge groups, preprint IFM-9/90 (Lisbon, 1990).

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[ 5 ] L.V. Prokhorov, Sov. J. Nucl. Phys. 35 (1982) 229; L.V. Prokhorov and S.V. Shabanov, Phase space structure of the Yang-Mills fields, JINR preprint E2-90-207 (Dubna, 1990). [6] L.V. Prokhorov and S.V. Shabanov, Phys. Len. B 216 (1989) 341; S.V. Shabanov, Phase space structure in gauge theories, JINR lectures P2-89-533 (Dubna, 1989) [in Russian]. [7] L.V. Prokhorov and S.V. Shabanov, Sov. Phys. Usp. 161 (1991) 13. [8] S.W. Hawking, Phys. Len. B 195 (1987) 337; Phys. Rev. D 37 (1988) 904; G.V. Lavrelashvili, E. Rubakov and P.G. Tinyakov, JETP Lett. 46 (1987) 167; S. Coleman, Nucl. Phys. B 307 (1988) 867; B 310 (1988) 643; S.B. Giddings and A. Strominger, Nucl. Phys. B 307 (1988 ) 867. [9]0. Bertolami and J.M. Mourao, The ground-state wavefunction of a radiation dominated Universe, preprint 1FM-19/90 (Lisbon, 1990). [ 10] P.A.M. Dirac, Lectures on quantum mechanics (Yeshiva University, New York, 1964). [ 11 ] S.V. Shabanov, Phys. Lett. B 255 ( 1991 ) 398. [ 12] L.V. Prokhorov and S.V. Shabanov, in: Topological phases in quantum theory (World Scientific, Singapore, 1989) p. 354. [ 13] S.V. Shabanov, Mod. Phys. Lett. A 6 ( 1991 ) 909. [ 14] R. Jackiw, Rev. Mod. Phys. 52 (1982) 661. [15]P.A.M. Dirac, The principles of quantum mechanics (Clarendon, Oxford, 1958 ). [16] S.V. Shabanov, Teor. Mat. Fiz. 78 (1989) 4 l l ; Intern. J. Mod. Phys. A 6 (1991) 845. [ 17 ] W. Pauli, Pauli lectures on physics (Massachusetts, 1973 ); W. Janke and H. Kleinert, Lett. Nuovo Cimento 25 (1979) 297; L.V. Prokhorov, Vestn. Leningr. Univ. Fiz. Khim., Ser. 4, No. 4 (1983) 14 [in Russian]. [ 18] S.V. Shabanov, J. Phys. A 24 ( 1991 ) 1199,