Retrogress to the past of an expanding universe

0083-6656/93 $24.00 @ 1993PergamonPressLtd

Vistas in Aawaomy, Vol. 37, pp. 539--542, 1993 Printedin GreatBritain.All rightsreserved.

RETROGRESS TO THE PAST OF AN EXPANDING UNIVERSE Hirofumi Hayashi Department of Physics, Faculty of Education, Shizuoka University, Ohya 836, Shizuoka 422, Japan

1.

We adopt

L.Parker's

formalis~ as a scalar field

model in an expanding universe, the expanding universe. probability

amplitude

and go back to the past of

We obtain an expression for the

of finding zero particles

when we start at the present vacuum state,and

in the past

others.

L. Parker l) studied a scalar field model containing only one scalar field coupled to gravity.

The Lagranglan density

is --~ where ~ d e n o t e s

~"

~'~-~ ~)~

the scalar field with mass m. g p V

the metric tensor,

g is the determinant

tensor g~A~ ,satisfying gMJ~

is

of the metric

~.~ = ~ .

The Robertson.-Walker metric is adopted,i.e., invarlant

(i)

the

line element ds is written as follows,

where R(t) is a cosmic scale factor. Then the explicit Lagranglan density and equation of the scalar field are expressed

H. f-laya~hi

540

(4) where dots on ~ and R(t) mean the time derivatives,

and

H = R(t)/R(t) The scalar field components

is expanded

,

in terms of Fourier t

~

f

;*'

~¢ ~t~(,~lelgf[ilir~lr- ~/tk,tJdl'~.t-I,c. ft

where

a~(t)

is

an annihilation

field,

and W(k,t)

The momentum ~

Is

a real

operator

~

of the

scalar

o f k = Ilk I a n d t

is defined a s ~ =

and the equal time commutation relations them.

(5)

"

function

conjugate to

}

.

~

T are assumed among

The following ansatz is made

where

are the present time(t O ) annihilation

operators, and

dI~,~) and

~,k)are

and creation

complex c-number

function~ satlsf~ing inltia~ condition a~.)---l, ~ , % ) ~ .

2.

The tlme-dependent

with the requirement is diagonallzed

Hamiltonian H(t) is calculated

that the present time Hamiltonlan

t in terms of , .~Ig and

.~

.,i.e.,

(8)

,

Retrogres~Ou~Past with

another

~

541

relation

---Z H ~

~ ¢ ~1~) ~ .

.

(9)

From these results, we obtain a plausible solution,

-~-~I

(i0) q

3.

We also cal0ulate the probability amplitude of finding

zero particle corresponding to a~(t) at bhe p a s t i m e

t

when we start from the vacuum state at the present time t O We define the present time vacuum state [O> as follows, ~i~

1°"7 " ~ - O

for all k

as well as the past time vacuum state ~;Rt~) These

vacua

operator

U(t).

~O)C are

=

O

related

This

unitary +

some combinations of / 4 ~

IO~

for

to

each

other

operator

and

is

A|g

(ll) ,

all

k

.

(12)

by a unitary in

turn

made o f

as follows~

where

E

= N { c, ~ . * ) A , ~ . A ~ , .

c'.(~.,)A;

A&J

Cl(k,t), c2(k,t) and c3(k,t) are complex functions of k and t, and satisfy the following relations,

c , (:~,-t) . - = - c~ ( ~ ,.~ ),- __ _tz o ( ~ , , ) ~ ~, ~ ( c g )

%

cu,

In terms of these quantities, we can express the trans-

H. Hayashi

.$42

formations of A k

,

"4-

= A~,cecl, fo).A-4~.~,~ti-~r)~.,I, The p r o b a b i l i t y corresponding

amplitude

to a~t)

from the p r e s e n t

of finding

to}.

(z4)

zero p a r t i c l e

at t h e p a s t t i m e t , w h e n we s t a r t

t i m e vacuum s t a t e

is

where

C =

~ I c,(~,..~ I z - ~ .~ ~ k , , j

t---1

~-

C j,

( Z ~: ( <: ( ~ - z ) ) Clz..._,

J~'

= 9-'.30~--,)

This result is different from that which L.Parker cited from SoKamefuchi and H. Umezawa 2) . We also calculate ~t.-~Ib~+ by differentiating to

the above ~ 0 ' 0 . ~

with respect

~(R,~).

Feferences l) L.Parker, Phys.Rev.183(1969)1057. 2) S. Kamefuchi and H. UMezawa, Nuovo Cimento 31(1964)492.