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IL NUOVO CIMENTO VOL. 112 B, N. 11 Novembre 1997

Wormhole wave function in the Einstein-Yang-Mills theory (*)

Z.-Q. TAN(1) (2) and Y.-G. SHEN(1) (**)

(1_{) CCAST (World Laboratory) - P.O. Box 8730, Beijing 100080, PRC}

(2_{) Department of Physics, Guangxi University - Nanning, Guangxi 530004, PRC}

(ricevuto il 30 Ottobre 1996; approvato il 25 Giugno 1997)

Summary. — In this paper, the wormhole wave function in the

Einstein-Yang-Mills theory is presented by using the method proposed by Hawking.

PACS 98.80.Hw – Mathematical and relativistic aspects of cosmology; quantum cosmology. PACS 04.20.Cv – Fundamental problems and general formalism.

PACS 04.60 – Quantum gravity.

In recent years, quantum cosmology, baby universe and wormhole have been widely studied [1]. It is known that for the Wheeler-De Witt equation [2] two kinds of solutions exist [ 3 , 4 ]. One kind of solution is the solution of quantum cosmology. This kind of solution gives a wave function of the universe and coincides with the evolution of the universe. In the classical forbidden zone (a EH21_{, here a is a scale factor of the}

Robertson-Walker metric, and H 4k_{LO3 with the cosmological constant L) the wave}

function decreases exponentially. In the classical permissible zone (a DH21_{), the wave}

function is of oscillatory type. Another kind of solution is the solution of quantum wormhole. In the classical permissible zone (corresponding to a from 0 to r0A LP, here

LPis the Planckian scale), this solution is of oscillatory type. In the classical forbidden

zone (corresponding to a from r0to Q), the solution decreases exponentially. It is just a

reasonable picture of the baby universe and the corresponding wormhole.

In general, classical wormholes are instantons, solutions of the Euclidean Einstein field equations [5-9], which consist of two large regions of space-time connected by a throat. And the quantum wormholes are solutions of the Wheeler-De Witt equation with definite boundary conditions [4, 10-13]. Hawking, Page et al. have pointed out [ 4 , 12 ] that these kinds of boundary conditions make the present wave functions to exponentially decrease in a large three-dimensional space, and to remain canonical in some proper way when the three dimensions tend to zero.

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) Permanent address: Shanghai Observatory, Academia Sinica, Shanghai 200030, PRC.

Z.-Q.TANandY.-G.SHEN

1516

In this paper, by using the Hawking method [10] in the Einstein-Yang-Mills theory, we give the corresponding Wheeler-De Witt equation and calculate the wave function of the wormhole.

It is known that the Einstein gravity coupled to SU( 2 ) Yang-Mills gauge field is described by the action [14]

S 4



M d4_xk9

g

_L 02 1 16 pGR 1 1 16 pag Fa mnFamn1 iu 16 p2F a mnFAamn1 (1) 1 iu 8 48 p2RmnabR Amnab 1 gRAmnabRAmnab

h

2



¯M dSkh 1 8 pG(K 2K0) . Here agfg2O4 p denotes the fine-structure constant of the SU( 2 ) gauge field. L0

denotes the cosmological constant. The CP-violating parameters in the Yang-Mills and the Einstein gravity theories are u and u8, respectively. The induced metric on the boundary is denoted by hmnfgmn2 nmnn, where nm is the unit normal vector on the boundary ¯M . The extrinsic curvature at the boundary M is denoted by K.

We take the metric as [14] dS2 4

g

3 p 2 G

h

21 [N2_{(t) dt}2 1 a2(t) sa 7 sa] 4

g

3 p 2 G

h

21 [em 7 em] (2)

and the ansatz of the gauge field, in A04 0 gauge,

Aamdxm4 ( 1 1 H(t) ] sa. (3)

The one-form sa_{satisfies the SU( 2 ) Maurer-Cartan structure equation} dsm

1 eabcsb_{R s}c 4 0 . (4)

In the above gauge the field strength is

Fa 4 H n N dt Ls a 1 (H22 1 ) 1 2e abc_sb_{R s}c 4 (5) 4 dai

y

H n Nae 0_{R e}i 1

g

H 2 2 1 a2

h

1 2 eijke j_{R e}k

z

_. Let L040. Using the ansatz of eq. (3) and the field strength eq. (5), the action becomes

S 4 1 2



_{dt Na}3

y

2

gg

a n Na

h

2 1 1 a2

h

1 r02 a4

u

(H 2 2 1 )21

u

H n a N

v

2

v

z

; (6)

here the rescaled fine-structure constant is

r22 0 4

2 ag

3 p . (7)

As a matter of convenience, we choose a conformal gauge

N(t) 4a(t) .

WORMHOLE WAVE FUNCTION IN THE EINSTEIN-YANG-MILLS THEORY 1517

Hereafter, a dot above the symbols denotes the derivative with respect to conformal time. The equation of motion of the Yang-Mills field follows from the variation of the action with respect to H(t):

H

n n

22 H(H22 1 ) 4 0 . (9)

Using eq. (5), it is easy to prove the Bianchi identity

D R Fa4 0 (10)

to be satisfied. The first integral of motion of eq. (9) is the conserved energy density of the gauge field:

[H

n

2(H22 1 )2] 42E . (11)

From eqs. (11), (8) and (6), we have

S 4 1 2



_dt[2(an₂ 1 a2) 1r2 0( 2 H n 2 1 E) ] . (12) Let 2 r02H n 2 4 Wn2, (13)

then eq. (12) becomes

S 4 1

2



_dt[2an₂

2 a21 Wn21 r02E] .

(14)

By canonical quantization, we have the Wheeler-De Witt equation as follows:

y

1 ap ¯ ¯aa p ¯ ¯a 2 ¯2 ¯W2 2 2 a 2 1 2 r02E

z

C(a , W) 40 , (15)

where p denotes the operator-ordering ambiguity in quantum gravity, and we take p40. Let

C_{(a , W) 4c(a) c(W) .} (16)

By separating eq. (15), we get 1 c(a) d2_c(a) da2 2 2 a 2 4 2l , (17) 1 c(W) d2_c(W) dW2 2 2 r0 2 E 42l , (18) where l is a constant. From eq. (17) we obtain

d2c(a)

da2 1 (l 2 2 a 2

) c(a) 40 (19)

and its solutions are

c(a) 4N1e2a

2

Hn(k2 a) , (20)

Z.-Q.TANandY.-G.SHEN 1518 with l 42n11 , n 40, 1, 2, R . (21) Equation (18) becomes d2c(W) dW2 1 (l 2 2 r 2 0E) c(W) 40 . (22)

The analogous model to eq. (11) is that of a particle moving in the inverted double-well potential V(H) 42(H2

2 1 )2with a total energy 2 E. The classical motion is possible only if the total energy is in the range 21 G2EG0. So, from eq. (20) we can choose properly l to make l D2r2

0E . Thus, the solution of eq. (22) is

c(W) 4N2eikl 22r 2 0EW_. (23) Then C_{(a , W) 4N}1N2e2a 2 Hn(k2 a) exp ikl 22r2 0EW_; (24)

here N1and N2are normalized constants.

From the conclusion obtained by the ref. [14] we have

k2 r0

k

1 2kE GW(t) Gk2 r0

k

1 1kE,

(25)

0 Ga(t) E1Q . (26)

From eq. (21), it is known [13] that the wave function will oscillate in a for a2

ElO2, and decrease exponentially for a2

D lO2 . So the wave function obtained is the wave function described for the quantum wormhole.

* * *

The authors would like to thank Prof. YUAN-ZHONG ZHANG for helpful discussions. The work is partly granted by the National Science Foundation of China.

R E F E R E N C E S

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