fulltext

Semiclassical time variables in a quantum cosmology

with a massless scalar field (*)

Y. OHKUWA

Department of Mathematics, Miyazaki Medical College Kiyotake, Miyazaki 889-16, Japan

(ricevuto il 24 Giugno 1996; approvato il 30 Agosto 1996)

Summary. — We calculate explicit expressions of two semiclassical time variables by the WKB approximation and the Ehrenfest principle in a quantum cosmology with a massless background scalar field. Two time variables are compared and confirmed explicitly to be identical when the scale of the Universe is large.

PACS 04.60.Ds – Canonical quantization.

PACS 98.80.Hw – Mathematical and relativistic aspects of cosmology; quantum cosmology.

PACS 03.65.Sq – Semiclassical theories and applications.

1. – Introduction

One of the most puzzling aspects of quantum cosmology is that the wave function is time independent owing to the constraints in the Dirac quantization [1]. Though the problem is still controversial, several candidates for the time variable have been advocated. One of them is the one introduced by Banks and others in terms of the WKB approximation [2-13]. Another is that introduced by Greensite and Padmanabhan in terms of the Ehrenfest principle [14, 15].

In the former formalism the assumption is used that the solution to the Wheeler-DeWitt equation has the form of the WKB approximation, namely C 4Fe(i/ˇ) S0_,

where S0is Hamilton’s principal function. Banks and others introduced a time variable

TW, using S0. They showed that, when a quantum matter field fQ is coupled to the

system, its wave function satisfies the Schrödinger equation with respect to TW in the

region where the semiclassical approximation is well justified. However, this formalism crucially depends on the assumption that the solution has the WKB form.

In the latter formalism this notion is extended. Greensite and Padmanabhan introduced a time variable TE, using the phase of an arbitrary solution to the

(*) The author of this paper has agreed to not receive the proofs for correction.

Wheeler-DeWitt equation. They obtained TE by requiring that the Ehrenfest principle

holds with respect to this time variable.

In this paper we calculate explicit expressions of these two time variables in a quantum cosmology with a massless background scalar field and compare them. In sect. 2 we investigate the WKB approximation to this model and calculate the WKB time variable TW. In sect. 3 we derive the exact solution to the Wheeler-DeWitt

equation and calculate the Ehrenfest time variable TE. In sect. 4 we show that two

time variables are identical when the scale of the Universe is large and we summarize.

2. – WKB approximation and time

We consider _{the following minisuperspace model in (n 11)-dimensional} space-time [13, 16]. Though n 43 in reality, we calculate in the more general case. The metric is assumed to be ds2_{4 2 N}2(t) dt2_{1 a}2(t) dVn2, where dVn2is the flat metric on Rn_{. We take a massless background scalar field f}

B(t) and a quantum matter field

fQ(t). The Wheeler-DeWitt equation for a wave function C(a , fB, fQ) reads

.

`

/

`

´

HC 4 ( HB1 HQ) C 40 , HB4 ˇ2 2 vnan 22

g

cn 2 apoo ¯ ¯aa poo ¯ ¯a 2 1 a2 ¯2 f2B

h

1 U(a) , cn4

o

16 pG 2 n(n 21) , U(a) 4vn 2 L 16 pGa n_. (1)

Here HQ(a , fB, fQ) is the Hamiltonian constraint for the quantum matter field fQ,

poois a parameter of operator ordering, L is a cosmological constant, vn is the spatial volume, and we assume that vn is some properly fixed finite constant.

So as to look for a WKB solution to eqs. (1), write C(qa_{, f}

Q) 4F(qa, fQ) e(i/ˇ) S0(q a₎

, (2)

where we have used the abbreviation qa

4 (a , fB). Substituting eq. (2) to eqs. (1) and

equating powers of ˇ, we obtain (3) ₂ cn 2 2 vnan 22

g

¯S0 ¯a

h

2 1 1 2 vnan

g

¯S0 ¯fB

h

2 1 U(a) 4 0 , (4) iˇ

y

cn 2 2 vnan 22

g

¯2S0 ¯a2 F 12 ¯S0 ¯a ¯F ¯a 1 poo a ¯S0 ¯a F

h

2 2 1 2 vnan

g

¯2S0 ¯fB2 F 12 ¯S0 ¯fB ¯F ¯fB

hl

1 HQF 40 ,

where we have considered HQ to be of the order of ˇ. Equation (3) is the

Hamilton-Jacobi equation for the gravity coupled with a background scalar field, and S0

is Hamilton’s principal function.

field fQ. That is cn2

g

¯2S0 ¯a2 F01 2 ¯S0 ¯a ¯F0 ¯a 1 poo a ¯S0 ¯a F0

h

2 1 a2

g

¯2S0 ¯fB2 F01 2 ¯S0 ¯fB ¯F0 ¯fB

h

4 0 . (5) Now we write C(qa_{, f} Q) 4F0(qa) e(i/ˇ) S0(q a₎ c(qa_{, f} Q) . (6)

Then from eqs. (4)-(6) we obtain

iˇ

y

cn 2 vnan 22 ¯S0 ¯a ¯c ¯a 2 1 vnan ¯S0 ¯fB ¯c ¯fB

z

1 HQc 40 . (7)

If we define a time variable TW as

2 cn 2 vnan 22 ¯S0 ¯a ¯TW ¯a 1 1 vnan ¯S0 ¯fB ¯TW ¯fB 4 1 , (8)

eq. (7) can be written as

iˇ ¯c

¯TW

4 HQc .

(9)

This is a Schrödinger equation, so TW is a semiclassical time variable in the WKB

approximation [2-13].

Let us calculate the explicit expression of TW. First the Hamilton-Jacobi equation

(3) can be solved by the separation of variables:

cn2a2

g

¯S0 ¯a

h

2 2 eva2 n4

g

¯S0 ¯fB

h

2 4 k2, ev4 4 vn2L 16 pG . (10)

Here k is an arbitrary constant, and ¯S0O ¯fB4 k corresponds to the classical

momentum of fB. Therefore the solution is

.

`

/

`

´

S04 ea cn IS1 kfB1 const , IS4



da a

k

eva 2 n 1 k24 1 n

y

k

eva 2 n 1 k21 k 2ln

u

k

eva 2 n 1 k22 k

k

eva2 n1 k21 k

v

z

, (11)

with ea4 6 1. We can also solve eq. (8) by the separation of variables, since

eacna

k

eva2 n1 k2 ¯TW ¯a 1 vna n 4 k¯TW ¯fB 4 j , (12)

where we have used eqs. (10), (11) and j is a separation constant. When k 40, TW is solved as TW4 2 eavn cnkev ln a 1tW(fB) , (13)

where tW(fB) is an arbitrary function of fB. On the other hand when k c 0, we obtain

.

`

`

/

`

`

´

TW4 ea cn (jIW12 vnIW2) 1 j k fB1 const , IW14



da a 1

k

eva 2 n 1 k2 4 1 2 nk ln

u

k

eva2 n1 k22 k

k

eva2 n1 k21 k

v

, IW24



da an 21

k

eva 2 n 1 k2 4 2 1 2 nkev ln

u

k

eva 2 n 1 k22kevan

k

eva 2 n 1 k21keva n

v

. (14)

Owing to eqs. (10) the variables in eq. (5) are separated, too: (15) eacn

k

eva2 n1 k2

m

[ (n 211poo) eva2 n2 ( 1 2 poo) k2] F01 2 a(eva2 n1 k2) ¯F0 ¯a

n

4 4 2 k¯F0 ¯fB 4 g . In the case of k 40, the equations can be derived from those of kc0, if we first set

g 40 and then kK0. So we have only to consider the case kc0. Then eqs. (15) can be

rewritten as

.

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/

`

`

´

dW(a) da 1 PW(a) 1 Q 4 0 , F04 W(a) 1 gfB 2 k , P 4 (n 211poo) eva 2 n 2 ( 1 2 poo) k 2 2 a(eva2 n1 k2) , Q 4 g[ (n 211poo) eva 2 n 2 ( 1 2 poo) k2] 4 ka(eva2 n1 k2) fB2 eag 2 cna

k

eva2 n1 k2 . (16) The solution is

.

`

/

`

´

W(a) 4e2sdaP

g

2



da QesdaP 1 const

h

, esdaP 4 a(poo2 1 ) /2_(e va2 n1 k2)1 /4,



_{da Qe}sdaP 4 (n 211poo) g 4 k fBIF

k

1 4

l

2 ngk 4 fBIF

k

2 3 4

l

2 eag 2 cn IF

k

2 1 4

l

, IF[x] 4



da a(poo2 3 ) /2 )(eva2 n1 k2)x. (17)

special case of g 40, we have Q40 and

F04 cfa( 1 2poo) /2(eva2 n1 k2)21 /4, (18)

with an arbitrary constant cf. When we ignore the quantum matter field fQ, choose

poo4 1 and take the limit a K Q, we obtain from eqs. (11), (18)

C Kcfev21 /4a2n/2exp

y

ieakev ˇ_ncn a n 1 ik ˇ fB

z

. (19)

3. – Ehrenfest principle and time

When poo4 1 and there is not the quantum matter field fQ, the Wheeler-DeWitt

equation (1) becomes

.

`

`

`

/

`

`

`

´

HBC(qa) 4

y

2 ˇ2 2 1 k2 G ¯ ¯qa k2 G G ab ¯ ¯qb 1 U(q a₎

z

_C(qa ) 40 , a fB Gab 4 a fB

u

2 cn 2 vnan 22 0 0 1 vnan

v

, G 4det Gab4 2 vn2 cn2 a2 n 22_. (20)

According to ref. [14, 15] we write an exact solution of the Wheeler-DeWitt equation as

C(qa

) 4e(i/ˇ) u(qa₎

r(qa_{) ,} (21)

where u and r are real quantities. Note that this is not the WKB approximation

(

cf. eq. (2)

)

, but this is only a separation of phase and absolute value. Let us define a time variable TE as Gab ¯u ¯qa ¯TE ¯qb 4 2 cn2 vnan 22 ¯u ¯a ¯TE ¯a 1 1 vnan ¯u ¯fB ¯TE ¯fB 4 1 . (22)

As proved in ref. [14, 15], the Ehrenfest principle holds with this time variable TE:

.

/

´

iˇ ¯ ¯TE aOb 4 a[O, HB]b , aOb 4



DXk_{2 G C* (T}E, X ) OC(TE, X ) , (23)

where O is an arbitrary observable and ¯X

Since the Wheeler-DeWitt equation (20) is the type of separation of variables, it can be solved exactly as follows. If we assume a solution as

C(qa

) 4up(qa) 4Xp(a) e(i/ˇ) pfB, (24)

then Xp must satisfy

y

ˇ2_cn2

g

d 2 da2 1 1 a d da

h

1 p2 a2 1 eva 2 n 22

z

_X p(a) 40 . (25)

Here p can be regarded as the momentum of fB. This equation can be rewritten as

y

d2 dz2 1 1 z d dz 1 1 2 n2 z2

z

Xp(z) 40 , (26) where z 4 kev ˇncn an and n 4 iNpN_ˇnc n

. Therefore, from eqs. (24) and (26) we have

up4 N Zn(z) e(i/ˇ) pfB. (27)

Here N is a normalization constant and Zn is a Bessel function. We may write the perfect solution of the Wheeler-DeWitt equation as

C(qa

) 4



dp[c1(p) Hn( 1 )(z) 1c2(p) Hn( 2 )(z) ] e(i/ˇ) pfB, (28)

where c1, c2 are arbitrary functions of p and Hn( 1 )(z), Hn( 2 )(z) are Hankel functions.

Suppose we take the boundary condition of Vilenkin [17]; we choose C(qa ) 4 NHn( 2 )(z) e(i/ˇ) pfB, (29) since C(qa_{) Kconst exp}

y

₂n 2 ln a 1 i ˇ

g

pfB2 kev ncn an

h

z

, (30)

when a KQ [18]. Note that this corresponds to eq. (19), when k4p and ea421 . Using eqs. (21), (29) and [Hn( 2 )(z) ]* 4H2n

( 1 ) (z) 4enpi_H n( 1 )(z), we obtain

.

/

´

u 4 ˇ 2 iln

y

Hn( 2 )(z) Hn ( 1 )_(z)

z

2 npˇ 2 1 pfB, r 4 Nenpi2

k

H_n( 1 )(z) H_n( 2 )(z) . (31)

Substituting eq. (31) into eq. (22), we find that the variables in the equation of TE are

also separated: 22 ncn 2_ˇ p a Hn( 1 )(z) Hn( 2 )(z) ¯TE ¯a 1 vna n 4 p¯TE ¯fB 4 z , (32)

where z is a separation constant and we have used the formula of Lommel, Hn( 1 )(z) dHn( 2 )(z) dz 2 Hn ( 2 )_(z) dHn( 1 )(z) dz 4 2 4 i pz .

We can write down the solution of eq. (32) as

.

`

`

/

`

`

´

TE4 pvn 2 ncnkev IE11 tE(fB) TE4 pvn 2 ncnkev IE12 pz 2 ˇn2_c n2 IE21 z pfB1 const IE14



dz Hn( 1 )(z) Hn( 2 )(z) 4



dz [Jn(z)21 Nn(z)2] , IE24



dz z Hn ( 1 )_{(z) H} n( 2 )(z) 4



dz z [Jn(z) 2 1 Nn(z)2] , (p 40) , (p c 0 ) , (33)

where tE(fB) is an arbitrary function of fB, and Jnand Nnare the Bessel functions of the first and second class, respectively. In the case of p c 0 we can calculate the integrals IE1, IE2 as

.

`

`

`

`

`

`

/

`

`

`

`

`

`

´

IE14 1 sin2_np[IE1 ( 1 ) 1 IE1( 2 )2 2 cos npIE1( 3 )] , IE1( 1 )4



dz Jn(z)24 z2 n 11 22 n ( 2 n 11)[G(n11) ]2 3 33F4

g

2 n 11 2 , n 11, 2 n11 2 ; n 11, n11, 2 n11, 2 n 13 2 ; 2z 2

h

_, IE1( 2 )4



dz J2n(z) 2 4 z 22 n 1 1 222 n_{(22n11)[G(2n11) ]}2 3 33F4

g

22 n11 2 , 2n11, 22 n11 2 ; 2n11, 2n11, 22n11, 22 n13 2 ; 2z 2

h

_, IE1( 3 )4



dz Jn(z) J2n(z) 4 4 z G_{(n 11)G(2n11)}3F4

g

1 2 , 1 , 1 2 ; n 11, 2n11, 1, 3 2 ; 2z 2

h

, IE24 1 2 n [Jn(z) 2 1 Nn(z)2] 1 z 2 n

k

Jn 11(z) ¯ ¯nJn(z) 2Jn(z) ¯ ¯nJn 11(z) 1 1Nn 11(z) ¯ ¯nNn(z) 2Nn(z) ¯ ¯nNn 11(z)

l

. (34)

Here 3F4 is a generalized hypergeometric function and we have used the formulae Nn(z) 4 1 sin np[ cos npJn(z) 2J2n(z) ] ,



_{dz Jm(z) Jn(z) 4} zm 1n11 2m 1n_{(m 1n11) G(m11) G(n11)} 3 33F4

g

m 1n11 2 , m 1n12 2 , m 1n11 2 ; m 11, n11, m1n11, m 1n13 2 ; 2z 2

h

[ Re (m 1n) D21] ,



dz z Zn(z) 2 4 1 2 n Zn(z) 2 1 z 2 n

k

Zn 11(z) ¯ ¯nZn(z) 2Zn(z) ¯ ¯nZn 11(z)

l

(Zn4 Jn, Nn) [ 19 ] . In the case of p 40, the integral IE1 can be derived from that of p c 0 by taking the

limit n K0.

4. – Discussion and summary

Let us consider the limit when the scale of the Universe is large, i.e. a KQ. In this region the Universe is expected to become classical. Then eq. (19) corresponds to eq. (30) with p 4k, ea4 21, and eq. (12) corresponds to eq. (32) with j 4 z, where we have used the fact that Hn( 1 )(z) Hn( 2 )(z) K

2

pz (z KQ) [18]. So TWand TEare expected

to be equal. We can confirm this also by explicit calculation, as follows. In the case of

p 4k40 the first equation of eqs. (33) becomes TEK vn cnkev

ln a 1tE(fB), and this

corresponds to eq. (13). In the case of p 4kc0 eq. (14) and the second equation of eqs. (33) become TWK 2 eavn cnkev ln a 1 j_kfB1 const and TEK vn cnkev ln a 1 z_pfB1 const.

Hence these time variables TW and TE are identical when the scale of the Universe is

large.

To summarize, we have calculated explicitly two time variables, TW by the WKB

approximation

(

eqs. (13), (14)

)

and TE by the Ehrenfest principle

(

eqs. (33), (34)

)

in a

quantum cosmology with a massless background scalar field. We have confirmed that these two are identical when the scale of the Universe is large. We think that these results are useful for further investigation on semiclassical time variables in a quantum cosmology.

R E F E R E N C E S

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Theories, edited by L. A. IBORTand M. A. RODRIGUEZ(Kluwer Academic Publishers) 1993; KUCHARˇ K. V., in Proceedings of the IV Canadian Conference on General Relativity and

Relativistic Astrophysics, edited by G. KUNSTATTER, D. E. VINCENTand J. G. WILLIAMS (World Scientific, Singapore) 1992, and references therein.

[2] BANKS T., Nucl. Phys. B, 249 (1985) 332. [3] HALLIWELL J. J., Phys. Rev. D, 36 (1987) 3626.

[4] HALLIWELLJ. J., in Quantum Cosmology and Baby Universes, edited by S. COLEMAN, J. B. HARTLE, T. PIRAN and S. WEINBERG (World Scientific, Singapore) 1991.

[5] HALLIWELL J. J. and HAWKING S. W., Phys. Rev. D, 31 (1985) 1777. [6] WADA S., Nucl. Phys. B, 276 (1986) 729; 284 (1987) 747(E).

[7] HARTLE J. B., in Gravitation in Astrophysics, edited by B. CARTER and J. B. HARTLE (Plenum, New York) 1986.

[8] VILENKIN A., Phys. Rev. D, 39 (1989) 1116.

[9] SINGH T. P. and PADMANABHAN T., Ann. Phys. (N.Y.), 196 (1989) 296. [10] PADMANABHAN T., Class. Quantum Grav., 6 (1989) 533.

[11] KIEFERC. and SINGHT. P., Phys. Rev. D, 44 (1991) 1067; KIEFERC., in Canonical Gravity

-from Classical to Quantum, edited by J. EHLERS and H. FRIEDRICH (Springer, Berlin) 1994.

[12] OHKUWAY., in Evolution of the Universe and its Observational Quest, edited by K. SATO (Universal Academy Press, Tokyo) 1994; Nuovo Cimento B, 110 (1995) 53.

[13] OHKUWA Y., Int. J. Mod. Phys. A, 10 (1995) 1905. [14] GREENSITE J., Nucl. Phys. B, 351 (1991) 749.

[15] PADMANABHAN T., Pramana J. Phys., 35 (1990) L199. [16] HOSOYA A. and MORIKAWA M., Phys. Rev. D, 39 (1989) 1123. [17] VILENKIN A., Phys. Rev. D, 37 (1988) 888.

[18] ABRAMOWITZM. and STEGUNI., Handbook of Mathematical Functions (Dover, New York) 1972.

[19] OHTSUKIY. and MUROTANIY., New Handbook of Mathematical Formulae, Vol. 2, Special

Functions (Maruzen, Tokyo) 1992, in Japanese, translated from Russian; English

translation: PRUDNIKOVA. P., BRYCHKOVYU. A. and MARICHEV O. I., Integrals and Series, Vol. 2, Special Functions (Gordon and Breach, New York) 1986.