I

AH INTRODUCTION TO QUANTUM GRAVITY

C . J . Isham*

0. PREFACK

The purpose of my t a l k a t t he Oxford Conference was t o prov ide a

genera l i n t r o d u c t i o n t o some of t h e ideas and methods of quantum g rav i ty

as a p r e c u r s o r t o t h e r a t h e r t e c h n i c a l l c c t u r e s which fo l lowed . This i s

r e f l e c t e d in these l e c t u r e no tes which are concerned mainly with broud

a t t i t u d e s r a t h e r than with s p e c i f i c , up t o d a t e , t e c h n i c a l t o o l s . The

scheme of t h e pape r i s as f o l l o w s . The f i r s t s e c t i o n i s a s h o r t

i n t r o d u c t i o n which emphasises t h e dua l p a r t i c l e / f i e l d i n t e r p r e t a t i o n of

convent iona l quantum f i e l d t heo ry . The l a t t e r i n t e r p r e t a t i o n i s used

e x t e n s i v e l y in quantum g r a v i t y and , because of i t s r e l a t i v e u n f a m i l i a r i t y ,

i s t he s u b j e c t of r e p e a t e d d i s c u s s i o n throughout t h e s e n o t e s . The next two s e c t i o n s deal wi th the problem of d e f i n i n g a quan t i s ed f i e l d on an

unquant i sed g r a v i t a t i o n a l background. There has r e c e n t l y been

c o n s i d e r a b l e i n v e s t i g a t i o n on t h i s t o p i c (which i s a p r e l im ina ry t o

quantum g r a v i t y p roper ) and i t promises t o be of some re levance t o

a s t r o p h y s i c a l problems invo lv ing g r a v i t a t i o n a l c o l l a p s e (see t h e c h a p t e r

by S . Hawking). The f o u r t h s e c t i o n i s concerned wi th c o v a r i a n t

q u a n t i s a t i o n (see t h e chap t e r by M. Duf f ) whi le in t h e next two s e c t i o n s

canon ica l q u a n t i s a t i o n i s d i s cus sed in some t e c h n i c a l d e t a i l s i n c e t h i s

was not t he s u b j e c t of any o t h e r s p e c i f i c l e c t u r e a t t h e c o n f e r e n c e . The f i n a l s e c t i o n cons ide r s t he c u r r e n t l y popu la r quantum model/quantum

cosmology approach t o q u a n t i s i n g t h e g r a v i t a t i o n a l f i e l d , a l though

again s ince a l e c t u r e was devoted t o t h i s t o p i c ( s ee the chap t e r by

* I am g r a t e f u l t o NATO f o r t h e i r suppor t by NATO Research Grant No.815.

M.MacCollum) the t r e a t m e n t here i s concerned wi th t h e gene ra l ideas

r a t h e r than wi th s p e c i f i c d e t a i l s .

1. INTRODUCTION

The problem of q u a n t i s i n g t h e g r a v i t a t i o n a l f i e l d has e x e r c i s e d

t h e minds of a number of people over t h e l a s t f o r t y y e a r s and w i l l

d o u b t l e s s con t inue t o do so f o r t h e nex t f o r t y ' 1 ^ 3 ^ ^ 5 ^. The

importance and i n t e r e s t of t h i s s u b j e c t of s t u d y , which i s r e f l e c t e d in t h e very c o n s i d e r a b l e i n c r e a s e in a t t e n t i o n which i t has r e c e i v e d du r ing

t h e l a s t decade , d e r i v e from a number of d i f f e r e n t s o u r c e s . General

r e l a t i v i t y and quantum theo ry a r e wi thou t doubt two o f t h e g r e a t e s t

i n t e l l e c t u a l achievements o f t h i s c e n t u r y . This i s in i t s e l f s u f f i c i e n t

t o guaran tee a cont inued i n t e r e s t in t h e problem of u n i f y i n g then.; an

i n t e r e s t which i s he igh tened by c o n s i d e r a t i o n of t h e very s p e c i a l r o l e

p layed by g e n e r a l r e l a t i v i t y w i t h i n t h e framework of c l a s s i c a l ( v i z .

non-quantum) p h y s i c s . In any conven t iona l f i e l d theo ry t h e space - t ime

s t r u c t u r e i s f i x e d and t h e f i e l d p ropaga tes in t ime on t h i s background.

In g e n e r a l r e l a t i v i t y however t h e k i n e m a t i c a l and dynamical a spec t s of

t h e theo ry a re t i g h t l y i n t e r l a c e d through t h e medium of t h e g r a v i t a t i o n a l

f i e l d , which , on t he one hand, s p e c i f i e s t h e geomet r i ca l p r o p e r t i e s of

s p a c e - t i m e , and on t h e o t h e r f u l f i l l s t he c l a s s i c a l t a s k of a f i e l d by

p r o p a g a t i n g a p h y s i c a l f o r c e . Convent iona l quantum t h e o r y , however, i s

fo rmula t ed on a r i g i d l y f i x e d space - t ime background, Euc l idean t h r e e -

space in t h e case of non r e l a t i v i s t i c quantum mechanics and Minkowskian

space - t ime in t h e cose o f r e l a t i v i s t i c quantum f i e l d t h e o r y . From t h i s

viewpoint i t can be expec ted t h a t any a t tempt t o u n i f y g e n e r a l r e l a t i v i t y

and quantum mechanics w i l l i n e v i t a b l y l e a d t o t e c h n i c a l and concep tua l

3

problem:!. One of t h e main mo t iva t i ons f o r s t u d y i n g quantum g r a v i t y hoo

always been t h a t t h e r e s o l u t i o n of t h e s e problems w i l l l e ad t o u

fundamenta l ly new i n s i g h t i n t o p h y s i c s .

I t i s not a p r i o r i c l e a r p r e c i s e l y what would be regarded au a

q u a n t i s a t i o n of g e n e r a l r e l a t i v i t y . The mathemat ica l s t r u c t u r e o f t h e

c l a s s i c a l t h e o r y c o n t a i n s a number of f e a t u r e s any of which might

perhaps be expec ted t o become s u b j e c t t o quantum laws. The p r i m o r d i a l concept i s t h a t of a p o i n t s e t whose mathemat ica l p o i n t s a re t o be

r e l a t e d in some way wi th p h y s i c a l space - t ime e v e n t s . This s e t i s t hen

equipped wi th a topology and then with a d i f f e r e n t i a b l e s t r u c t u r e which

makes i t i n t o a fou r -d imens iona l man i fo ld . F i n a l l y a m e t r i c t e n s o r i s

c o n s t r u c t e d on t h i s mani fo ld in such a way as t o s a t i s f y t h e E i n s t e i n

e q u a t i o n s . One might a t tempt t o i n t r o d u c e q u a n t i s a t i o n a t any one of

t he se l e v e l s . In p r a c t i c e most of t h e work which has been done t a k e s

t h e e a s i e s t r o u t e and f i x e s e v e r y t h i n g but t h e m e t r i c . Thus a

di f f e r e n t i a b l e mani fo ld i s s p e c i f i e d once and f o r a l l and t h e m e t r i c

t e n s o r i s r ega rded as an o p e r a t o r d e f i n e d on t h i s space . (Ac tua l l y i f

c a n o n i c a l q u a n t i s a t i o n i s be ing used then t h e r e l e v a n t mani fo ld may be

t h r e e , r a t h e r than f o u r , d i m e n s i o n a l ) . This i s c l e a r l y t h e a t t i t u d e t o

q u a n t i s a t i o n which i s c l o s e s t t o t h a t p r e v a l e n t in conven t iona l quantum

f i e l d t h e o r i e s , n e v e r t h e l e s s when one c o n s i d e r s t h e r o l e played by t h e

l i g h t c o n e s t r u c t u r e in t h e s e t h e o r i e s i t i s c l e a r t h a t a l r e a d y a major d i f f e r e n c e has emerged - t h e l i g h t c o n e s t r u c t u r e of gene ra l r e l a t i v i t y

i s i n d i s p u t a b l y dynamical and not p a r t of t h e f i x e d background.

However, t h e op in ion i s f r e q u e n t l y vo iced t h a t t h e q u a n t i s a t i o n

procedure should t a k e p l a c e a t a more fundamental l e v e l . Two of t h e

p r i n c i p a l advocates o f t h i s l i n e have been P r o f e s s o r s J . Wheeler and

R< Penrose . Vfheeler has f o r many y e a r s emphasised t h e need t o q u a n t i s e

the t o p o l o g i c a l as v e i l as t h e m e t r i c s t r u c t u r e of space - t ime and, wi th

h i s r e c e n t t hough t s on t h e r o l e played by formal l o g i c in quantum

g r a v i t y , has t aken t h e q u a n t i s a t i o n l e v e l r i g h t back t o t h e b a s i c

e lements of mathemat ics . S i m i l a r l y Penrose has f r e q u e n t l y argued t h a t

space - t ime i t s e l f , r a t h e r than j u s t t h e m e t r i c f i e l d , should be

i n t i m a t e l y l i n k e d wi th quantum t h e o r y . I t was t h i s p o i n t o f view which ( 7 )

p a r t l y mot iva ted h i s c o m b i n a t o r i a l s p i n network theory as we l l as (8)

h i s r e c e n t work on t w i s t o r s . Most peop le would agree t h a t a deeper

look a t t h e problem of quantum g r a v i t y a t t h i s type of very b a s i c l e v e l

i s p robably mandatory i f any r e a l l y ma jo r advance i s t o be ach ieved . However, i t i s a l s o impor tan t t o under s t and how f a r conven t iona l

q u a n t i s a t i o n (by which i s meant m e t r i c f i e l d q u a n t i s a t i o n ) can be pushed.

In p a r t i c u l a r , i t i s e s s e n t i a l t o d i s t i n g u i s h c a r e f u l l y between t h o s e

problems which a r e p e c u l i a r t o quantum g r a v i t y and t h o s e which a r e shared

by a l l quantum f i e l d t h e o r i e s . Hand in glove wi th t h i s must go an

a p p r e c i a t i o n of t h e p r a c t i c a l a p p l i c a t i o n s of t h i s type of q u a n t i s a t i o n

and t h e i r i m p l i c a t i o n s f o r r e a l i s t i c p h y s i c a l sys tems. In t h i s a r t i c l e

I s h a l l c o n c e n t r a t e mainly on t h e m e t r i c q u a n t i s a t i o n schemes and r e f e r

t he r e a d e r t o t h e b i b l i o g r a p h y f o r m a t e r i a l on some of t h e o t h e r a s p e c t s

of quantum g r a v i t y .

Many d i f f e r e n t approaches t o q u a n t i s i n g t h e g r a v i t a t i o n a l f i e l d

have evolved s i n c e t he s u b j e c t was f i r s t cons ide red in t h e e a r l y 1 9 3 0 ' s . These t e n d t o be c l a s s i f i e d under two h e a d i n g s , ' c o v a r i a n t ' ('

5

a l i t t l e m i s l ead ing bu t s i n c e they a re widely used they w i l l be r e t a i n e d

h e r e . Canonical q u a n t i s a t i o n i t s e l f w i l l be s p l i t up i n t o ' t r u e '

canon ica l q u a n t i s a t i o n (5) and super space -based q u a n t i s a t i o n (6) .

There i s a t endency , a t l e a s t among p a r t i c l e p h y s i c i s t s , t o suppose t h a t

the whole o f quantum g r a v i t y can be n e a t l y accommodated by t h e not ion of

t h e g r a v i t o n . This h e l i c i t y two, mass less p a r t i c l e i s then thought of

as i n t e r a c t i n g with i t s e l f in a way which i s more o r l e s s conven t iona l

a l though i t l e ads t o a t h e o r y which i s probably h i g h l y non reno rma l i s ab l e .

This i s t h e p r i n c i p l e concept which a r i s e s from t h e c o v a r i a n t q u a n t i s a t i o n

scheme bu t i t l eads t o a r a t h e r r e s t r i c t e d view of quantum g r a v i t y and

indeed of quantum f i e l d theo ry in g e n e r a l .

The p a r t i c l e i n t e r p r e t a t i o n , w i th i t s cor responding s e t of p a r t i c l e -

based o b s e r v a b l e s , of a quantum f i e l d t h e o r y , which t h e no t i on of a

g r a v i t o n e p i t o m i s e s , may not always be t h e most a p p r o p r i a t e one. There

i s in f a c t an impor tan t a l t e r n a t i v e p h y s i c a l i n t e r p r e t a t i o n of what i s

e s s e n t i a l l y t he same mathemat ics , even in t h e case of an o rd ina ry f l a t -

space quantum f i e l d t h e o r y . As t h i s a l t e r n a t i v e view i s t h e one which

i s most commonly used in quantum g r a v i t y (mainly in t h e canon ica l

approaches) i t i s worth d i s c u s s i n g i t h e r e , a t l e a s t i n a h e u r i s t i c

manner. For t h e sake of s i m p l i c i t y c o n s i d e r a f r e e n e u t r a l s c a l a r f i e l d

$(x) in o rd ina ry f l a t Minkowski space - t ime . The conven t iona l

q u a n t i s a t i o n of t h i s sys tem us ing Fock s p a c e , with t h e cor responding

p a r t i c l e i n t e r p r e t a t i o n , i s we l l known (see 2 f o r more d e t a i l s ) . On

t h e one hand i t can be ob t a ined by q u a n t i s i n g t h e s c a l a r f i e l d $(x) per

se and look ing f o r a s u i t a b l e r e p r e s e n t a t i o n ( i n t h e Schrod inger p i c t u r e

say) of t h e c a n o n i c a l commutation r e l a t i o n s

[ ( x ) , = i K 6 ( 3 ) ( x - i ) (1.1)

On t h e o t h e r hand one can begin v i t h o n e - p a r t i c l e s t a t e s , t w o - p a r t i c l e

s t a t e s e t c . d e s c r i b e d i n terms of o rd ina ry quantum mechanics and c o n s t r u c t

a l a r g e s t a t e - s p a c e which accommodates them a l l , namely Fock space .

' A n n i h i l a t i o n ' and ' c r e a t i o n ' o p e r a t o r s can then be d e f i n e d which connect

t o g e t h e r t h e s e va r ious f i n i t e p a r t i c l e subspaces and from which a quantum

f i e l d $(x) can be r e c o n s t r u c t e d . However, i t i s i n t e r e s t i n g t o ask how

t h i s s imple problem of q u a n t i s i n g a f r e e f i e l d looks from t h e v iewpoint

of c o n v e n t i o n a l quantum mechanics . I f a c l a s s i c a l sys tem has a

Eucl idean c o n f i g u r a t i o n space Q wi th g l o b a l c a r t e s i a n c o o r d i n a t e s q ^ . - . q ^

co r respond ing t o n degrees of f reedom, then t h e b a s i c problem of quantum

theo ry ( i n t h e Sch rcd inge r p i c t u r e ) i s t o f i n d a r e p r e s e n t a t i o n of t h e

canon ica l commutation r e l a t i o n s

[q. , .] = i K 6. . x,4-l...a

[q. , = 0 ( 1 . 2 )

[Pi P j ] = o

wi th s e l f - a d j o i n t o p e r a t o r s on a H i l b e r t space of s t a t e s . Then t h e

dynamical equa t ion

H ( W - a n ; P , . P 2 - " P n ) * t = i " J T ( 1 " 3 )

must be solved f o r t h e t ime e v o l u t i o n of t h e s t a t e v e c t o r ^ in terms

of t h e q u a n t i s e d Hamil tonian o p e r a t o r H.

7

By v i r t u e o f t h e Stone-Von Neumann t h e o r e m , t h e unique s o l u t i o n

(up t o u n i t a r y t r a n s f o r m a t i o n s ) i s t h a t i n vn ich t h e s t a t e space in t h e

s e t o f a l l complex va lued f u n c t i o n s of Q which a r e s q u a r e i n t e g r a b l e wi th

r e s p e c t t o t h e Lebesgue measure d q j d q ? . . . d q ^ . The o p e r a t o r s q . , P j a r e

t h e n r e p r e s e n t e d by

(4 . ( q ^ . - q j = ^ . . . q j (1.1)

j

and any o t h e r r e p r e s e n t a t i o n of eqn ( 1 . 2 ) ( o r more p r e c i s e l y o f t h e

e x p o n e n t i a t e d Weyl form) i s u n i t a r i l y e q u i v a l e n t t o t h i s one . The wave

f u n c t i o n has t h e i n t e r p r e t a t i o n t h a t i f B i s any B o r e l s e t i n R.n t h en

PB = j l ^ q ^ . - q j l * d q 1 . . . d q n ( 1 . 6 ) ' B

i s t h e p r o b a b i l i t y t h a t i f t h e sys t em i s i n t h e s t a t e ty and a

measurement i s made on t h e sys tem of t h e v a l u e s o f q . . . q^ ( i . e . o f t h e

c l a s s i c a l c o n f i g u r a t i o n of t h e sys tem) t h e n t h e y l i e i n B. Now a

c l a s s i c a l f i e l d t h e o r y can be r e g a r d e d as a c l a s s i c a l mechanica l sys t em

wi th i n f i n i t e l y many d e g r e e s o f f reedom. E s s e n t i a l l y , an o r t h o n o r m a l

b a s i s s e t o f f u n c t i o n s on / f t 3 , {e^( jc)} s a y , i s chosen ( t y p i c a l l y w i t h

p r o p e r t i e s in r e l a t i o n t o t h e Hami l t on i an which s i m p l i f y t h e dynamica l

e v o l u t i o n problem) and t h e f i e l d s a r e expanded as

(x,t) = I q ( t ) e , ( x ) ( 1 . 7 ) i = l 1

oo

Tt(x,t) = I p ^ t ) e . ( x ) i = l

(1.8)

in which ( q j . . ; P J . P , . . . ) cor respond t o t h e i n f i n i t e number of modes o r degrees of freedom of t he system. Thus t h e commutation r e l a t i o n s in

eqn ( 1 . 2 ) would s t i l l be expec ted t o be t r u e but now with i , j r ang ing from 1 t o

9

In a d d i t i o n , i f d n ( ) denotes t he analogue of dq ^ . . . dq^ and the system

i s in a quantum s t a t e Y, then i f a measurement i s made of t he c l a s s i c a l

f i e l d c o n f i g u r a t i o n , t h e q u a n t i t y

i s t he p r o b a b i l i t y t h a t t h e r e s u l t w i l l l i e in t h e ( i n f i n i t e d imens iona l )

s e t B. This i n t e r p r e t a t i o n of t h e s t a t e v e c t o r i s c l e a r l y d i f f e r e n t from

the usual p a r t i c l e one and i s e v i d e n t l y wel l s u i t e d t o s i t u a t i o n s where

c l a s s i c a l l y t h e f i e l d has some n a t u r a l meaning. (The two i n t e r p r e t a t i o n s

a re p a r t i a l l y l i n k e d through the theo ry of coherent s t a t e s ) . I t must be

emphasised t h a t t h e t r e a t m e n t above i s very crude and i n f a c t as i t s t ands

i s mathemat ica l ly i l l - d e f i n e d . For a s t a r t i t i s no t c l e a r e x a c t l y what

t h e c l a s s i c a l c o n f i g u r a t i o n space Q should b e . Should i t be a l l C

f u n c t i o n s on UL3, a l l C f u n c t i o n s on IK3 with compact suppor t . . . ?

Not ice t h a t in terms of t h e (q j .q , , . . . ) v a r i a b l e s t h e exac t way in which t h e q ^ ' s behave f o r l a r g e i de termines t h e type of o b j e c t t o which t h e sum in eqn ( 1 . 7 ) converges . Hot u n r e l a t e d t o t h i s i s t h e f a c t t h a t

u n f o r t u n a t e l y an i n f i n i t e d imensional analogue of Lebesgue measure does

not e x i s t . However, a l l t h e s e problems can be r e so lved and Fock space

i t s e l f can be shown t o be u n i t a r i l y e q u i v a l e n t t o a c e r t a i n L2(Q,dy)

space i n which p i s a gauss ian measure (which does g e n e r a l i s e t o i n f i n i t e

dimensions) and Q inc ludes not only f u n c t i o n s on IR.5 but a l s o ( 9 ) ( 1 0 )

d i s t r i b u t i o n s ! I t would not be a p p r o p r i a t e he re t o dwell any

f u r t h e r on t h i s t o p i c except t o re -emphas i se t h a t mathemat ica l Fock

space admits of two complementary p h y s i c a l i n t e r p r e t a t i o n s : t h e usua l

(1 .13 )

1 0

p a r t i c l e one and the one based on f u n c t i o n spaces which h inges round

eqns ( 1 . 1 2 ) and ( 1 . 1 3 ) .

The mo t iva t ion f o r t h i s d i s c u s s i o n was t h a t much o f t h e l i t e r a t u r e

on quantum g r a v i t y uses the second , p o s s i b l y more u n f a m i l i a r , p i c t u r e .

C e r t a i n l y in s i t u a t i o n s in which p o t e n t i a l g r a v i t a t i o n a l c o l l a p s e i s

invo lved i t i s t he more immediately a p p r o p r i a t e i n t e r p r e t a t i o n . Indeed

assuming t h a t t he quantum g r a v i t y analogue of t he d i s c u s s i o n above

invo lves I ^ C s ^ f * )J I 2 as t h e a p p r o p r i a t e p r o b a b i l i t y d e n s i t y

( t h i s i s a c t u a l l y not q u i t e c o r r e c t , s ee 5U.55). then t h e behav iour of

t h e s t a t e f u n c t i o n a l in t h e v i c i n i t y of m e t r i c s which c l a s s i c a l l y

cor respond t o s i n g u l a r i t i e s would have a d i r e c t b e a r i n g on t h e g r a v i t a t i o n a

c o l l a p s e o r o therwise of t h e quantum system. Also of course t h e no t ion of

p a r t i c l e in conven t iona l f i e l d t h e o r y i s c l o s e l y l i n k e d wi th t h e Po inca re

group. The absence of such a group in t h e case of t he g r a v i t a t i o n a l

f i e l d i s ano the r good reason f o r look ing c h a r i t a b l y a t n o n - p a r t i c l e

i n t e r p r e t a t i o n s . There i s one f u r t h e r remark t o make i n t h i s c o n t e x t .

As emphasised above t h e d i f f e r e n c e between t h e p a r t i c l e and f i e l d p i c t u r e s

of a conven t iona l f i e l d theory r e a l l y i s only a d i f f e r e n c e i n i n t e r -

p r e t a t i o n o f e s s e n t i a l l y t h e same mathemat ical s t r u c t u r e . Which

i n t e r p r e t a t i o n i s r e l e v a n t t o any given s i t u a t i o n i s de te rmined b a s i c a l l y

by what obse rvab l e s a r e be ing measured. However, t h e s i t u a t i o n in

quantum g r a v i t y i s s l i g h t l y d i f f e r e n t . In t h e cova r ion t approaches

( 5'

11

of the m e t r i c t e n s o r are genuine c a n o n i c a l v a r i a b l e s and because of

t h i n the c o v a r i a n t q u a n t i s a t i o n of a l l t en components of the me t r i c

t enoo r ( c f . Gup ta -Bleu le r in quantum e l ec t rodynamics ) l e ads t o a

p a r t i c l e p i c t u r e which i s s l i g h t l y d i f f e r e n t from the one above. On the

o t h e r hand i n t h e ' t r u e ' canon ica l q u a n t i s a t i o n scheme (55) two e q u i v a l e n t

i n t e r p r e t a t i o n s could be reasonably expec ted t o e x i s t (assuming t h a t t h e

p a r t i c l e no t ion makes sense a t a l l , which i n a h igh l y curved space i t may

no t ) bu t whether they a r e based in some sense on t h e same u n d e r l y i n g

mathemat ica l s t r u c t u r e as t he c o v a r i a n t scheme i s no t c l e a r . In t h e

' s u p e r s p a c e ' c anon ica l q u a n t i s a t i o n scheme (6) t h e f i e l d p i c t u r e i s

c e r t a i n l y dominant. Indeed superspace i t s e l f i s a type of g r a v i t a t i o n a l

analogue of t h e Q-space i n t r o d u c e d above b u t , however, not in t h e s t r i c t

canon ica l s e n s e . Superspace c o n t a i n s a d d i t i o n a l degrees of freedom over

and above t h e t r u e canon ica l ones and us a r e s u l t t h e equ iva lence of t h i s

mathemat ica l scheme t o e i t h e r t h a t o f t r u e canon ica l q u a n t i s a t i o n o r of

c o v a r i a n t q u a n t i s a t i o n i s not a t a l l c l e a r .

F i n a l l y l e t us no t e t h a t even a t t h i s s t r a i g h t f o r w a r d l e v e l of

me t r i c q u a n t i s a t i o n o n l y , t h e r e e x i s t problems of a very deep , and

l a r g e l y u n r e s o l v e d , concep tua l n a t u r e . The Copenhagen i n t e r p r e t a t i o n o f

quantum mechanics i s founded f i rmly on t h e concept of an e x t e r n a l

o b s e r v e r . I f one a t t empt s t o ex tend quantum g r a v i t y t o i nc lude t h e whole

u n i v e r s e (as i s f r e q u e n t l y done) r a t h e r than j u s t c o n s i d e r i n g some smal l l o c a l quantum e f f e c t , then i t i s i n e v i t a b l e t h a t many t r a d i t i o n a l (and

c h e r i s h e d ) views on quantum theory must be overhau led . One famous

example of such a r e t h i n k i s t h e E v e r e t t - W h e e l e r ' 1 1 ' i n t e r p r e t a t i o n of

quantum mechanics which a t t empts t o i n t r i n s i c a l l y i n c o r p o r a t e t h e

obse rve r in t h e sys t em, a s t e p which i s obvious ly n e c e s s a r y i f t h e

system i s t h e u n i v e r s e !

I t has become u n f a s h i o n a b l e t h e s e days f o r much n o t i c e t o be t aken

of t h e s e concep tua l p roblems, most people p r e f e r r i n g t o work on t h e more

' r e s p e c t a b l e ' t e c h n i c a l d i f f i c u l t i e s . However, in quantum g r a v i t y t h e

concep tua l and t e c h n i c a l problems f r e q u e n t l y go hand in hand and i t i s

p o s s i b l e t h a t by n e g l e c t i n g t he former one i s r e n d e r i n g i r r e l e v a n t t he

l a t t e r .

2. QUANTUM FIELD THEORY ON A FIXED BACKGROUND

I t i s l o g i c a l l y compel l ing t o p recede t h e i n v e s t i g a t i o n o f t h e

f u l l quantum g r a v i t y problem with a d i s c u s s i o n of f i e l d q u a n t i s a t i o n on

a f i x e d background. The s i m p l e s t example (which i s t h e one cons ide red

h e r e ) i s o f a s c a l a r f i e l d $ d e f i n e d on a f i x e d f o u r - d i m e n s i o n a l pseudo-

Riemannian mani fo ld and s a t i s f y i n g t h e c l a s s i c a l e q u a t i o n s of motion

3 ( ( - de t g ) J gUU ) - m 2*(- de t g)> = 0 ( 2 . 1 )

de r ived from t h e l ag rang ian d e n s i t y

L(x) = | ( g , 1 V ( x ) 3 ( * ) 3 y *(x ) - m2 $ 2 ( x ) ) ( - de t g)* . ( 2 . 2 )

The q u a n t i s a t i o n of t h i s s c a l a r f i e l d c o n s t i t u t e s a t h e o r y which, from

the quantum g r a v i t y p o i n t o f view, i s d e f i c i e n t in t h e f o l l o w i n g two

1 3

ronpec t3 .

i ) The q u a n t i s a t i o n of t h e m e t r i c t e n s o r i t s e l f i s completely

negle c t e d .

i i ) Even i f t h e me t r i c were unquant i sed t h e r e should be a

r e a c t i o n back on i t v i a E i n s t e i n ' s e q u a t i o n s , from quantum

e f f e c t s i n $ (such as p a r t i c l e p roduc t ion by a t ime vary ing

N e v e r t h e l e s s , t he model above has c o n s i d e r a b l e i n t e r e s t . From a p r a c t i c a l

s t a n d p o i n t t h e r e a re va r ious s i t u a t i o n s in a s t r o p h y s i c s and ' e a r l y

u n i v e r s e ' cosmology in which t h e r o l e of an unquant i sed g r a v i t a t i o n a l

f i e l d producing r e a l p a r t i c l e s would be of g r e a t impor tance . From a

t h e o r e t i c a l po in t of view a thorough unde r s t and ing of t h i s s imple rtodel

would seem a n a t u r a l p r e r e q u i s i t e t o a t t e m p t i n g t o q u a n t i s e t h e m e t r i c

t e n s o r i t s e l f . I t i s t h e r e f o r e perhaps s u r p r i s i n g t h a t , whereas u

c o n s i d e r a b l e amount of e f f o r t has been expanded over t h e l a s t twenty

f i v e y e a r s on t he f u l l quantum g r a v i t y t h e o r y , only a r e l a t i v e l y smal l

(12)(13> amount of work has appeared d e a l i n g with t h i s s i m p l i f i e d problem.

The f i r s t ques t i on t o ask i s wha t , from a p h y s i c a l p o i n t of view,

the l ag rung ian in eqn ( 2 . 1 ) could be expec ted t o d e s c r i b e ? In t he f l u t

space c a s e , in which the met r ic t e n s o r i s simply t h e cons t an t

Minkowski t e n s o r 1 t h e answer i s we l l known. Indeed t h e t h e o r y

degene ra t e s i n t o a f r e e massive s c a l a r f i e l d ( in t h e c o n v e n t i o n a l sense )

wi th t h e two major i n t e r p r e t a t i o n s - f i e l d and p a r t i c l e - which were o u t l i n e d in 51. I t i s r easonab le t o suppose t h a t f o r f i e l d s g which

do not ' d e v i a t e t o o v i o l e n t l y ' from f l a t space two such i n t e r p r e t a t i o n s

w i l l again be p o s s i b l e . In p a r t i c u l a r from the p a r t i c l e po in t of view i t

It

i s n a t u r a l t o expec t t h a t the e f f e c t o f the me t r i c f i e l d g w i l l be uv

s i m i l a r t o t h a t o f , f o r example, an o rd ina ry e x t e r n a l e l e c t r o m a g n e t i c

f i e l d , l e a d i n g t o t he p roduc t ion o f ^ - p a r t i c l e s . ' 1 ** " 1 J ' However,

extreme ca re needs t o be e x e r c i s e d in c o n v e r t i n g t h i s p l a u s i b l e - s o u n d i n g

s t a t emen t i n t o an unambiguous p iece of t h e o r y . The p a r t i c l e i n t e r -

p r e t a t i o n of s t a n d a r d quantum ( f r e e ) f i e l d theo ry a r i s e s from two main

s o u r c e s , t h e r e p r e s e n t a t i o n o f t h e canon ica l conmutation r e l a t i o n s (CCR)

and t h e i n v a r i a n t a c t i o n of t he Po incare group. The s t e p s l e a d i n g t o ,

and a s s o c i a t e d w i t h , t h i s i n t e r p r e t a t i o n a re b r i e f l y :

1) Choose an i n e r t i a l frame of r e f e r e n c e (and hence a choice o f t ime)

in Minkowski space . Cons t ruc t t h e momentum n ( x , t ) which i s

c o n j u g a t e t o $ ( x , t ) in t h i s frame and p o s t u l a t e t h a t t h e r e s u l t i n g quantum f i e l d s s a t i s f y t h e equal t ime CCR

[ ( x , t ) , ( i , t ) J = i K a ( 3 ) (x - jr.) ( 2 .3 )

which a r e fo rmal ly c o n s i s t e n t with t h e dynamical e q u a t i o n s f o r

$ ( x , t ) :

' ( x , t ) - (V2 - m2) ( x . t ) = 0 . (2.It)

(N.B. In a gene ra l quantum f i e l d t h e o r y t h e f i e l d s must be smeared in

x and t in o r d e r t o correspond t o genuine o p e r a t o r s . Thus equa t ion ( 2 . 3 )

(which impl i e s smear ing in only) would be mean ing less . However, f o r

t h e f r e e f i e l d , and i t i s expec ted a l s o f o r t h e f i e l d on t h e f i x e d

background, t h e procedure i s j u s t i f i e d . )

15

Kind in e x p l i c i t r e p r e s e n t a t i o n of eqn ( 2 . 3 ) ( a t some i n i t i n l t ime

t = 0 say ) by s e l l ' - a d j o i n t o p e r a t o r s ( a f t e r s u i t a b l e smearing) on a H i l b e r t space in which the Hamil tonian i s a genuine s e l f - a d j o i n t o p e r a t o r which g e n e r a t e s t ime e v o l u t i o n in t h e sense t h a t

i / j j Ht - i / f i Ht $ ( x , t ) = e $ (x ,0 ) e

( 2 . 5 ) i / K Ht - i / j . Ht

n(2,t) = e n(j,0) e

The s t a n d a r d procedure i s t o s e p a r a t e t h e s o l u t i o n s t o t h e o p e r a t o r

equa t ion (2.It) in t h e form (fi = l )

f - i E . t iE . t .

x , t ) = du( j ) [e J b ( j ) 4,.(x) + e J b T ( j ) ^ . ( x ) ] ( 2 . 6 ) J J J

where

1 6

(2i,)

( i , j ) = 6 ( 3 ) ( k - k '>

d p ( i ) = d3k . (2 .10 )

( I f t h e system were be ing q u a n t i s e d in a box with p e r i o d i c boundary

c o n d i t i o n s then t h e s p a t i a l i n t e g r a l would become an i n f i n i t e sum.) In

p a r t i c u l a r eqn ( 2 . 6 ) l eads t o an expansion of t h e Cauchy d a t a $(.x,0)

and n ( x , 0 ) i n terms of t he normal modes (x_) wi th o p e r a t o r c o e f f i c i e n t s . J

The t = 0 CCR eqns (2 .3 ) a re e q u i v a l e n t t o

I" el' \ (2.11) [ a . , a^J = 5 ( j , k )

where

a . = >e7 b . . (2 .12 ) 0 J O

The usua l s t e p new i s t o choose t h e Fock r e p r e s e n t a t i o n which i s

c h a r a c t e r i s e d by t h e e x i s t e n c e of a unique c y c l i c s t a t e t h a t i s

a n n i h i l a t e d by a l l t h e a . . 0

3) The o p e r a t o r s N. H a . ' a . have i n t e g e r e i genva lue s and t h e J J J

co r r e spond ing e i g e n v e c t o r s a r e mapped ' up -one ' o r 'down-one'

by a . ^ and a . . The l a t t e r a r e t h e r e f o r e i d e n t i f i e d as o p e r a t o r s J J

which c r e a t e o r a n n i h i l a t e quan ta whose wave f u n c t i o n s in t h e

conven t iona l o n e - p a r t i c l e quantum-mechanical s ense a r e t h e normal

modes i^ . (x) . In so f a r as t h e s e quan ta con be i d e n t i f i e d as 0

17

p h y s i c a l p a r t i c l e s t h i s i s t h e s t a g e a t which the p a r t i c l e

concept f i r s t appea r s . In p a r t i c u l a r t he c y c l i c s t a t e mentioned

above i s c a l l e d t h e 'vacuum' or ' n o - p a r t i c l e ' s t a t e ,

'i) H i s shown t o be a w e l l - d e f i n e d o p e r a t o r on Fock s p a c e , a f t e r

normal o r d e r i n g , w i th t he p rope r ty of

i ) a n n i h i l a t i n g t h e vacuum s t a t e ( i n t h e Schrod inger

p i c t u r e t h i s means t h a t t h e vacuum s t a t e does not

change wi th t ime i . e . t h e r e i s no p a r t i c l e

p r o d u c t i o n ) .

i i ) commuting wi th t h e 'number ' o p e r a t o r s H . , which a re J

t h e r e f o r e c o n s t a n t s of t h e motion. In p a r t i c u l a r an

n - p a r t i c l e s t a t e always evo lves in t h e Schrod inger

p i c t u r e i n t o an n - p a r t i c l e s t a t e (aga in no p a r t i c l e

p roduc t ion o r a n n i h i l a t i o n ) .

The d i s c u s s i o n so f a r has been f o r a f i x e d choice of t ime c o o r d i n a t e

and by v i r t u e o f t h e s e p a r a t i o n of v a r i a b l e s in eqn (2 .6 ) f o r a d e f i n i t e

choice of p o s i t i v e and nega t i ve f r e q u e n c i e s . However, c l e a r l y , one should

ask what happens i f a d i f f e r e n t choice of t ime ( i . e . , a d i f f e r e n t

i n e r t i a l frame) i s made. S ince any two i n e r t i a l r e f e r e n c e frames a r e

r e l a t e d by a Po inca r group a c t i o n t h e ques t i on i s r e a l l y now t h e

Po inca r group a c t s on t h e o r i g i n a l Fock space . The answer i s t h a t Fock

space c a r r i e s a u n i t a r y r e p r e s e n t a t i o n of t h e Po incar group P which has

t h e p r o p e r t y t h a t

i ) The c r e a t i o n o p e r a t o r s t r a n s f o r m c o v a r i a n t l y anorigst

themselves as do t h e a n n i h i l a t i o n o o e r a t o r s .

30

i i ) the vacuum s t a t e i s i n v a r i a n t under t he group ( i t

i s a n n i h i l a t e d by a l l t h e g e n e r a t o r s o f P ) . In f a c t

the t ime t r a n s l a t i o n group g e n e r a t o r i s p r e c i s e l y t h e

Hamiltonian cons ide red a l r e a d y , i . e . , t ime

t r a n s l a t i o n i s t ime e v o l u t i o n ,

and i i i ) any n - p a r t i c l e s t a t e i s mapped i n t o ano the r n - p a r t i c l e

s t a t e by t h e group.

These t h r e e p r o p e r t i e s (which a r e c l o s e l y l i n k e d ) imply in e f f e c t t h a t

t h e no t ion of p a r t i c l e or quanta i s e s s e n t i a l l y independent of i n e r t i a l

obse rve r and t h a t t h e n - p a r t i c l e s t a t e s behave t h e same, as f a r as t h e

Po inca re group i s concerned , as they do in t h e u sua l r e l a t i v i s t i c n -

p a r t i c l e quantum t h e o r y . At t h i s s t a g e in t h e conven t iona l t ex tbook

t r e a t m e n t o f quantum f i e l d t h e o r y i t i s t a c i t l y assumed t h a t t h e pu re ly

mathemat ica l ' q u a n t a ' d i s c u s s e d so f a r cor respond t o r e a l p h y s i c a l

p a r t i c l e s which could be measured wi th an a p p r o p r i a t e p i e c e of equipment

and which accord in some way wi th our i n t u i t i v e f e e l i n g s of what a

' p a r t i c l e should b e ' . This connec t ion between mathemat ics and phys i c s

i s one of t h e v i t a l s t e p 3 in t h e p h y s i c a l i n t e r p r e t a t i o n of a q u a n t i s e d

f i e l d bu t i s f r e q u e n t l y g los sed o v e r . The i n v e s t i g a t i o n of t h i s

connec t ion t u r n s out t o be o f paramount importance in t h e case of an

a r b i t r a r y background.

S ince t h e u l t i m a t e aim i s t o q u a n t i s e t h e s c a l a r f i e l d i n eqn ( 2 . 2 )

t h e next obvious s t e p i s t o couple an e x t e r n a l sou rce t o t h e f r e e

s c a l a r f i e l d j u s t cons ide r ed . The a p p r o p r i a t e l a g r a n g i a n i s

L(x) = | (nWU 4>(x) 3v *(x) - m V ( x ) ) + j ( x ) 2(x) ( 2 . 1 3 )

1 9

here j ( x ) d e s c r i b e s t h e e x t e r n a l unquant i sed source which , by v i r t u e ol' t h e form of i t s i n t e r a c t i o n , might be expec ted t o produce

p a i r s of ijr-mesons. Indeed one obvious way of t r e a t i n g eqn (2 .13) i s t o nopa ra t e o f f j ( x ) $ 2 ( x ) and view i t as an i n t e r a c t i o n term which we hope cun be d e f i n e d as on o p e r a t o r on t h e o r i g i n a l Fock space wi th i t n maim

in quan ta . I f t h i 3 t e c h n i c a l s t e p can be performed then t h i s i n t e r a c t i o n

term can c e r t a i n l y l ead t o t he p roduc t ion of p a i r s of ' p a r t i c l e s ' of the

o r i g i n a l t ype . ( In o t h e r words t he Fock vacuum i s no l o n g e r a n n i h i l a t e d by

t h e f u l l Hami l ton ian ) . I t i s , however, q u i t e p o s s i b l e t h a t t h e

Hamiltonian cannot be d e f i n e d as an o p e r a t o r on t h e o r i g i n a l Fock space a t

a l l . This s i t u a t i o n might be r ecogn i sed h e u r i s t i c a l l y by t h e p roduc t ion

in time o f on i n f i n i t e number of quan ta . In t h i s c a se t h e Schrod inger

p i c t u r e i s not very a p p r o p r i a t e ; however, a He i senbe rg - type p i c t u r e

might s t i l l e x i s t bu t wi th t h e dynamical e v o l u t i o n be ing d e s c r i b e d by a

n o n - u n i t a r i l y implementable automorphism of t h e o p e r a t o r o b s e rv ab l c s

r a t h e r than by eqns ( 2 . 5 ) .

In g e n e r a l te rms i t i s not c l e a r t h a t t h e quanta which occur can be

r ega rded as having t h e same p h y s i c a l s i g n i f i c a n c e as b e f o r e . The whole

problem of r e n o r m a l i s a t i o n and t h e i d e n t i f i c a t i o n of p h y s i c a l obse rvab le3

r e a r s i t s head a t t h i s s t a g e . As an extreme example, i f j ( x ) were a U2

c o n s t a n t - ^ , then t h e p roduc t ion of t h e mass m quanta would have t o

be such as t o g ive a f i n a l t h e o r y which i s a f r e e f i e l d wi th mass

vm2 + u 2 . In t h i s case i t i s c l e a r t h a t t h e 'wrong' Fock space has been

chosen i n i t i a l l y bu t t h e s i t u a t i o n f o r genera l s o u r c e s i s c o n s i d e r a b l y

more compl ica ted than t h i s and t h e problem of t he c o r r e c t p h y s i c a l

2 0

i n t e r p r e t a t i o n i s n o n - t r i v i n l . In f a c t t h e r e i s no u n i v e r s a l l y

agreed p a r t i c l e i n t e r p r e t a t i o n f o r a gene ra l e x t e r n a l f i e l d . One major c o n t r i b u t o r y f a c t o r i s t h a t in g e n e r a l t h e theo ry i s no l onge r i n v a r i a n t

under t h e Po incare group. This of course i s a f e a t u r e shared by t h e

s c a l a r f i e l d d e f i n e d on an a r b i t r a r y background (which, g e n e r i c a l l y ,

w i l l have no group of symmetr ies) and f o r t h i s reason i f f o r no o t h e r

the system d e s c r i b e d by eqn (2 .13 ) i s worth s t u d y i n g c a r e f u l l y . C e r t a i n

problems remain even i f t h e c u r r e n t j ( x ) i s s t a t i c (when a t l e a s t t h e t ime t r a n s l a t i o n group e x i s t s ) .

With t h e s e c a u t i o n a r y remarks in mind l e t us now t u r n t o t h e

s i t u a t i o n d e s c r i b e d by eqn ( 2 . 1 ) . Numerous d i f f i c u l t i e s can be

a n t i c i p a t e d in p roceed ing with t h e analogue of any of t h e s t e p s ske tched

above f o r t h e f r e e , f l a t - 3 p a c e f i e l d . One n a t u r a l approach perhaps i s

t o s e p a r a t e ou t the Minkowski m e t r i c n and w r i t e yv

g (x) = n + h (x) (2.114) yv yv yv

where h ) y ( x ) d e s c r i b e s t h e d e v i a t i o n of t h e geometry from f l a t n e s s . The

b i g advantage of t h i s scheme i s t h a t i t reduces s u p e r f i c i a l l y t he problem

t o one s i m i l a r t o t h a t posed by t h e e x t e r n a l sou rce i n eqn ( 2 . 1 3 ) . In

p a r t i c u l a r t h e e x i s t e n c e of t h e f l a t background wi th i t s Po inca re group

of motions and p r e f e r r e d c l a s s of i n e r t i a l r e f e r e n c e f rames should lend

t o t h e same s o r t of p a r t i c l e i n t e r p r e t a t i o n . However, t h e r e a r e a number

of o b j e c t i o n s t o t h i s p o i n t of view. For example:

i ) The a c t u a l background mani fo ld may not be remotely

21

Minkownkian in e i t h e r i t s t o p o l o g i c a l o r

m e t r i c a l p r o p e r t i e s , in which case the s e p a r a t i o n

in eqn (2 . lU) (with i t s co r respond ing s e p a r a t i o n

of t h e Hamil tonian i n t o f r e e and i n t e r a c t i o n te rms)

i s comple te ly i n a p p r o p r i a t e ,

i i ) Even i f eqn ( 2 . l i t ) i s j u s t i f i e d (from t h e po in t o f view of i ) ) t he p rocedure i s s t i l l dubious because

t he l i g h t c o n e s t r u c t u r e of t h e p h y s i c a l space t ime i s

d i f f e r e n t from t h a t of Minkowski space . For example

i f t h e f i e l d cj> has some s o r t of m i c r o c a u s a l i t y p r o p e r t y with r e s p e c t t o t h e m e t r i c then t h i s i s

not e q u i v a l e n t t o m i c r o c a u s a l i t y wi th r e s p e c t t o t h e

f i c t i t i o u s Minkowski background.

Thus i t i s very d e s i r a b l e t o avoid any f i e l d s e p a r a t i o n and,

c o r r e s p o n d i n g l y , t o cons ide r t he l a g r a n g i a n in eqn ( 2 . 1 ) as a s i n g l e

e n t i t y . However, w i t h i n t he framework o f conven t iona l quantum f i e l d

theory t h i s poses a number o f problems. F i r s t l y t h e r e i s now, in

g e n e r a l , no symmetry group of t h e m e t r i c which can p lay t h e r o l e o f

t h e Po inca re group. In p a r t i c u l a r t h e r e a r e no p r e f e r r e d c l a s s e s of

t ime and one would expec t a p r i o r i t o have t o cons ide r t h e CCR of eqn

( 2 . 2 ) d e f i n e d ove r an a r b i t r a r y s p a t i a l t h r e e - s u r f a c e . There i s no

n a t u r a l d e f i n i t i o n of n e g a t i v e and p o s i t i v e f r e q u e n c i e s and even i f

some analogue o f eqn ( 2 . 6 ) i s c o n s t r u c t e d t h e r e i s no reason why t h e

r e s u l t i n g c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s which cor respond t o

d i f f e r e n t cho ices o f t h r e e - s u r f a c e should l e a d t o e q u i v a l e n t n o t i o n s

of p a r t i c l e . In g e n e r a l any pure a n n i h i l a t i o n (o r c r e a t i o n ) o p e r a t o r

2?

w i l l evolve i n t ime i n t o a mix tu re of such o p e r a t o r s , as indeed i t

does f o r t h e s imple e x t e r n a l source case in eqn ( 2 . 1 3 ) . This i s not

in i t s e l f s u r p r i s i n g as i t cor responds t o t h e expec ted phenomenon of

p a r t i c l e p r o d u c t i o n , bu t t h e problem of dec id ing in what sense t h e

r e s u l t i n g quan ta a c t u a l l y cor respond t o p h y s i c a l l y measurable p a r t i c l e s

i s n o n - t r i v i a l . Another d i f f i c u l t y i s t h a t t h e Hamil tonian has t o be

normally o rde red and t h i s depends on t h e exact r e p r e s e n t a t i o n of t h e

CCR which i s chosen. Two d i f f e r e n t cho ices can e a s i l y lead t o

Hamil tonian o p e r a t o r s which d i f f e r from each o t h e r by an i n f i n i t e c o n s t a n t .

This might mean t h a t in some r e a l p h y s i c a l sense an i n f i n i t e amount of

energy i s produced by the background in t h e form of p h y s i c a l - p a r t i c l e s

but i t could a l s o simply be t h e r e s u l t of choosing a p h y s i c a l l y

i n a p p r o p r i a t e CCR r e p r e s e n t a t i o n , i n which case t h e i n f i n i t e answer

would no t n e c e s s a r i l y have any more s i g n i f i c a n c e than t h e r e n o r m a l i s a b l e

u l t r a v i o l e t d ivergences of some c o n v e n t i o n a l quantum f i e l d t h e o r i e s .

I f t h e m e t r i c i s s t a t i c o r s t a t i o n a r y , s o t h a t t h e r e e x i s t s sort

g l o b a l t i m e l i k e K i l l i n g v e c t o r wi th i t s a s s o c i a t e d group of symmetr ies ,

some p r o g r e s s can be made. S i m i l a r l y , i f t h e m e t r i c i s a s y m p t o t i c a l l y

f l a t ( o r , remembering t h a t i t need not s a t i s f y E i n s t e i n ' s equa t i ons in

any s e n s e , f l a t o u t s i d e of some f i n i t e r eg ion ) then t h e f r e e ' i n ' and

' o u t ' f i e l d s can be used t o g ive some s o r t of p r e f e r r e d p a r t i c l e

i n t e r p r e t a t i o n , wi th a co r respond ing s e t of p h y s i c a l o b s e r v a b l e s (which

i nc lude in p a r t i c u l a r , e n e r g y ) . ' 1 5 ' I t i s t empt ing t o s p e c u l a t e t h a t

i t may be in t h e r o l e of c o n s o l i d a t i n g t h e concept o f a p a r t i c l e t h a t

t h e BMS group (whose r e p r e s e n t a t i o n s have r e c e n t l y been worked o u t ) ' 1 6 '

inivy f i n a l l y come i n t o i t o own. Even in t h e s e s i t u a t i o n s however, a

luimlinr of problems remain and the i n t e r e s t e d r eade r i s r e f e r r e d t o t h e

ox:Uont t h e s i s und papers of S. F u l l i n g ' 1 2 " 1 3 ' f o r d e t a i l s of t h e s e .

I t nhould be emphasised t h a t t h i s problem i s not merely of pure

t h o o r c t i c a l i n t e r e s t . I t i s p e r f e c t l y p o s s i b l e t h a t t h e p roduc t ion of

p a r t i c l c o by a g r a v i t a t i o n a l f i e l d cou ld p rov ide a fundamental

i ' -"iolution o f t h e whole problem of g r a v i t a t i o n a l c o l l a p s e . This i s

lctnrly shown in t h e e x c i t i n g r e s u l t s of S.W. Hawking ' 1 7 ' which a r e

imported i n t h e p r e s e n t volume. He c o n s i d e r s an a s y m p t o t i c a l l y f l a t

g r a v i t a t i o n o l l y col lapsomg system and shows t h a t i t can l o s e an i n f i n i t e

iimount of energy by t h e mechanism of p a r t i c l e p r o d u c t i o n . (Note t h a t

the amount o f energy r a d i a t e d could be p e r f e c t l y w e l l d e f i n e d even i f

t h e r e i s no unambiguous i n t e r p r e t a t i o n of t h e p r e c i s e form in which

i t i s r a d i a t e d ) . This i s not a complete r e s u l t in t h e p h y s i c a l s ense

because i t ignores t h e r e a c t i o n back of t h e p a r t i c l e s on the g r a v i t a t i o n a l

f i e l d and, as might be expected from t h e remarks above , t h e p r e c i s e

choice of t ime and hence p a r t i c l e o p e r a t o r s i s a d e l i c a t e one.

Neve r the l e s s Hawking1s work i s of g r e a t i n t e r e s t and one can a n t i c i p a t e

t h a t a c o n s i d e r a b l e amount of e f f o r t w i l l be expended in t he f u t u r e on

pursu ing t h i s approach.

There w i l l c e r t a i n l y be some i n s t a n c e s when bhe ' i -e jd r a t h e r t h a n t h e p a r t i c l e i n t e r p r e t a t i o n of t h e theory i s m, any , ca se more l i k e l y t o

be t h e a p p r o p r i a t e ; or - ",\ '' ? r f' w '

one. However, one can c o n f l d S h t l y p r e d i c t t h a t most

.... _ , tliL of the problems which a r i s e in t h e p a r t i c l e i n t e r p r e t a t i o n w i l l r e a p p e a r in d i f f e r e n t g u i s e s . At a deeper l e v e l i t i s p o s s i b l e t h a t "-the t)ieory

shou ld be modelled on t h e C - a l g e b r a approach t o conven t iona l quantum

f i e l d t h e o r y in which the l o c a l o b s e r v a b l e s play a dominant r o l e , r a t h e r

than the more usua l approach used above i n which a H i l b e r t space of s t a t e s

i s chosen as t h e b a s i c e n t i t y . Indeed even t h e s imple problem of an

e x t e r n a l source coupled t o a s c a l a r f i e l d can be u s e f u l l y t r e a t e d in t h i s

way. In so f a r as a man i fo ld l o c a l l y resembles Minkowski-space (by v i r t u e

of i t s very d e f i n i t i o n ) , t h e i d e a of c o n c e n t r a t i n g on l o c a l obse rvab le s

i s an a t t r a c t i v e one.

Of course one can always t a k e r e f u g e in t h e a s s e r t i o n t h a t t h e

problem stems b a s i c a l l y from not q u a n t i s i n g t he me t r i c t e n s o r f i e l d and

can only be r e s o l v e d by a f u l l quantum g r a v i t y t h e o r y . I t i s d i f f i c u l t

however t o b e l i e v e t h a t quantum g r a v i t y i t s e l f could r e a l l y be r e l e v a n t (17) (18) (19)

t o t he t ype of c a l c u l a t i o n s which Hawking , Unruh and Ford

have been making and t h e problem of s u c c e s s f u l l y and unambiguously

q u a n t i s i n g a 3 c a l a r f i e l d i n an a r b i t r a r y bu t f i x e d background must

remain an impor tan t c h a l l e n g e .

3. QUAHTUM FIELD THEORY ON A BACKGROUND WITH BACK REACTION

One p o s s i b l e t h e o r e t i c a l development of t he scheme d i s c u s s e d in

2 i s t h a t in which t h e quantum ( s c a l a r ) f i e l d a c t s as t h e a c t u a l

sou rce of t h e ( s t i l l c l a s s i c a l ) background. In o t h e r words t h e r e a c t i o n

back on t h e g r a v i t a t i o n a l f i e l d caused by t h e p roduc t ion o f s c a l a r

p a r t i c l e s i s i n c o r p o r a t e d as p a r t of t h e dynamics. To ach ieve t h i s i t

i s necessa ry t o i n c l u d e in some way t h e energy-momentum of t h e q u a n t i s e d

s c a l a r m a t t e r - f i e l d as t h e r i g h t - h a n d s i d e of E i n s t e i n ' s e q u a t i o n s . The

equa t ion

G. J e ) = T ( m a t t e r , g) ( 3 . 1 )

In not s u i t a b l e as i t s t ands s i n c e i t equa tes an o p e r a t o r and a

(-number. The obvious m o d i f i c a t i o n i s t o w r i t e

G y v (g ) = (3 .2 )

where < > denotes t h e e x p e c t a t i o n va lue of the q u a n t i s e d system in some

- v . , ^ ^ (20) n u i t a b l e s t a t e .

This i s t h e system of equa t ions which w i l l be d i s cus sed in t h e

p re sen t s e c t i o n . The s i t u a t i o n i s c l e a r l y a t l e a s t as complicated as

t h a t d i s c u s s e d in 2 but wi th t h e a d d i t i o n a l f e a t u r e t h a t t h e g r a v i t a t i o n a l

f i e l d i s now in t roduced as a dynamical v a r i a b l e r a t h e r than as a f i x e d

background.

The obvious ques t i on which a r i s e s i s what p r e c i s e l y i s meant by

a ' s u i t a b l e s t a t e ' ? There i s no reason t o suppose t h a t , f o r example,

a r e a l i s t i c c o l l a p s i n g system would be d e s c r i b e d simply by a pure s t a t e

and i n gene ra l one must al low f o r < > t o correspond t o a mixed,

s t a t i s t i c a l (p robably n o n - e q u i l i b r i u m ) s t a t e of t h e sys tem. T h i s ,

however, r a i s e s t h e immediate po in t t h a t such a s t a t e w i l l almost

c e r t a i n l y i t s e l f depend on t h e me t r i c t e n s o r g ^ ( t h i n k f o r example of

any g e n e r a l l y c o v a r i a n t - l o o k i n g ve rs ion of t h e Gibbs ensemble) . C l e a r l y

the t h e o r y i s a good deal more non l i n e a r thun i s ev iden t from a

cursory glance a t eqn ( 3 . 2 ) and t h e f u l l i m p l i c a t i o n s of t h i s approach . , (2 l ) (22)(23}(2

6

a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s , normal o r d e r i n g e t c , which were

d i s cus sed in 52 s t i l l apply h e r e . In p a r t i c u l a r t h e normal o r d e r i n g of

the energy momentum t e n s o r w i l l e v i d e n t l y p lay a major r o l e in t h e c o r r e c t use of eqn ( 3 . 2 ) . Not ice t h a t t h e a d d i t i o n of a c o n s t a n t t o

t h e energy momentum t e n s o r has a r e a l p h y s i c a l e f f e c t on t h e g r a v i t a t i o n a l

f i e l d . Thus the fundamental f e a t u r e of gene ra l r e l a t i v i t y , t h a t t h e

a b s o l u t e r a t h e r than r e l a t i v e va lue of t h e energy-momentum t e n s o r has a

meaning, i s s h a r p l y r e f l e c t e d in t h i s quantum t h e o r y . T h i s p rov ides an

a d d i t i o n a l f a c e t t o t h e p r e v i o u s l y mentioned problem of t h e CCR

r e p r e s e n t a t i o n dependence of normal o r d e r i n g .

The most i n t e r e s t i n g r e s u l t s which have been ach ieved so f a r u s i n g ( 2 7 )

t h i s approach a re probably t h o s e of L. Pa rke r and S. F u l l i n g . They

c o n s i d e r a massive s c a l a r f i e l d q u a n t i s e d in t h i s way v i a eqn (3 .2 ) bu t

in which t h e m e t r i c t e n s o r i s r e s t r i c t e d t o be of t h e Robertson-Walker

form

3

ds 2 = d t 2 - R ( t ) 2 I S.. ( x 1 , x 2 , x 3 ) d x i d x j (3 -3) i , j = l 1 J

where S . (x 1 ,x 2 , x 3 ) i s t h e ( f i x e d ) m e t r i c o f a t h r e e - s p h e r e .

Ev iden t ly a very s p e c i a l choice of s t a t e > in eqn ( 3 . 2 ) must be

made t o r ende r t h i s system of equa t ions s e l f c o n s i s t e n t . A g e n e r a l s t a t e

would not be compat ib le wi th t h e s imple me t r i c t e n s o r in eqn (3 -3 ) and

t h e main s t e p in Pa rke r and F u l l i n g ' s work i s t h e c o n s t r u c t i o n of a

s u i t a b l e s t a t e . The r e s u l t of t h e i r c a l c u l a t i o n s i s an e x p l i c i t form

f o r t h e f u n c t i o n R( t ) which pos se s se s t h e remarkable f e a t u r e t h a t t h e

27

ygtom docs not e x h i b i t t h e c l a s s i c a l g r a v i t a t i o n a l c o l l a p s e bu t

r a t h e r 'bounces o f f t h e s i n g u l a r i t y a t R = 0 wi th t h e r ad iu3 R( t )

a ch i ev ing a minimum of t h e Compton wavelength of t h e massive s c a l a r

p a r t i c l e s .

This r e s u l t i s p o t e n t i a l l y of fundamental importance t o t he s u b j e c t of g r a v i t a t i o n a l c o l l a p s e . I f t h e s c a l a r f i e l d d e s c r i b e d say p i o n s , i t

would mean t h a t t he quantum e f f e c t s on t h e c o l l a p s e s t a r t e d a t a d i s t a n c e

of 10 cms r a t h e r than t h e c h a r a c t e r i s t i c Planck l e n g t h v 10 cms -13 . . C

of pure quantum g r a v i t y . The 10 cms r e s u l t imp l i e s of course t h a t t h e

f i n e d e t a i l s of t h e behav iou r o f t h e system a t t h e t u r n - a r o u n d p o i n t

depend s i g n i f i c a n t l y on t h e s t r o n g i n t e r a c t i o n s . However t h e thought -

provoking p o s s i b i l i t y remains t h a t i t may no t be n e c e s s a r y t o q u a n t i s e

the g r a v i t a t i o n a l f i e l d i t s e l f in o r d e r t o avoid g r a v i t a t i o n a l c o l l a p s e .

In e f f e c t t h e v i o l a t i o n of t he Hawking-Penrose energy c o n d i t i o n s by t h e

e x p e c t a t i o n va lue of t h e q u a n t i s e d m a t t e r ' s momentum t e n s o r may be

s u f f i c i e n t .

There a r e many t e c h n i c a l problems remaining in t h e P a r k e r - F u l l i n g

work (as t he au tho r s themselves po in t ou t ) concern ing t h e choice of t h e

3 t a t e > a n d t h e r e n o r m a l i s a t i o n of t he t h e o r y , which a r e , t o some e x t e n t ,

connected wi th t h o s e of t h e s imp le r t ype of system d i s c u s s e d in 2.

However, from a p r a c t i c a l ( a s t r o p h y s i c a l ) p o i n t o f view t h i s approach t o

'quantum g r a v i t y ' i s very promis ing und w i l l undoubtedly be t h e s u b j e c t of a f a i r l y s u b s t a n t i a l r e s e a r c h e f f o r t i n t h e f u t u r e .

so

It. COVARIANT QUANTISATION

We new t u r n our a t t e n t i o n t o t h e problem of q u a n t i s i n g t h e

g r a v i t a t i o n a l f i e l d i t s e l f . H i s t o r i c a l l y t h e va r ious approaches t o t h i s

have t ended t o be c l a s s i f i e d as e i t h e r ' c a n o n i c a l ' o r ' c o v a r i a n t ' and i t

i s t he l a t t e r approach which i s cons ide red in t h i s s e c t i o n . Two s l i g h t l y

d i f f e r e n t p o i n t s of view have emerged over t h e l a s t f i f t e e n y e a r s . Both

of them s t a r t by f i x i n g in advance t h e fou r -d imens iona l space - t ime

manifo ld upon which t h e m e t r i c t e n s o r g i s r ega rded as be ing d e f i n e d as

an o p e r a t o r - v a l u e d d i s t r i b u t i o n . The o p e r a t o r i s s e p a r a t e d i n t o a

c l a s s i c a l background p l u s a quantum c o r r e c t i o n in t h e form

g = 6 + J ( fc . l ) p v Uv UV

which i s then i n s e r t e d i n t o t h e E i n s t e i n a c t i o n S = j J-% R(g) d^x (assuming f o r s i m p l i c i t y t h a t no m a t t e r f i e l d s a r e p r e s e n t ) . The f a c t

t h a t a l l t en components of a r e a f f o r d e d o p e r a t o r s t a t u s (as

opposed t o j u s t t he two canon ica l v a r i a b l e s ) means t h a t t h e approach b e i n g fo l lowed has something in common wi th t h e Gupta -Bleu le r

q u a n t i s a t i o n of t h e e l e c t r o m a g n e t i c f i e l d , as opposed t o , s a y ,

r a d i a t i o n gauge canon ica l q u a n t i s a t i o n . In t h e p i o n e e r i n g work of (2 8}

De Wit t , Schwinger ' s a c t i o n p r i n c i p l e i s mod i f i ed t o g ive Green ' s

f u n c t i o n s by va ry ing t h e a c t i o n wi th r e s p e c t t o t h e background f i e l d .

(The e x t e r n a l sou rces of Schwinger ' s o r i g i n a l theory a r e d i f f i c u l t t o

use when a non -abe l i an group i s p r e s e n t . ) De W i t t ' s work i s very

29

comprehensive and i n c l u d e s a d i s c u s s i o n of what c o n s t i t u t e s on

olinnrvobXo in t h e theory and t h e problem of i t s measurement. The key

I r c l i n i c a l t o o l in t h i s i s t h e P e i e r l s - P o i s 3 o n b r a c k e t which enab les a

m l u t i o n t o be s e t up between t h e quantum commutator o f obse rvab les

tuid t he Green ' s f u n c t i o n s . Do Witt works e x c l u s i v e l y in c o n f i g u r a t i o n

npuce and h i s formalism i s , a t l e a s t a t t h e h e u r i s t i c l e v e l , m a n i f e s t l y

c o v a r i a n t under t h e va r ious gauge groups which ac t in t h e t h e o r y . The

absence of a momentum space in h i s approach ( t h e r e i s of course no

n a t u r a l d e f i n i t i o n o f F o u r i e r t r a n s f o r m in an a r b i t r a r y Riemannian

manifo ld wi th m e t r i c gjj^ ) means t h a t some of h i s t echn iques seen u n f a m i l i a r t o anyone who i s p r i m a r i l y t r a i n e d in conven t iona l p a r t i c l e

p h y s i c s . P a r t l y f o r t h i s reason a s l i g h t l y d i f f e r e n t p o i n t of view has

a r i s e n in which t h e s e p a r a t i o n in C i . l ) i s performed always wi th r e s p e c t

t o t he Minkowski space background. When t h e r e s u l t i n g f i e l d i s

s u b s t i t u t e d i n t o t h e E i n s t e i n l a g r a n g i a n , a very non l i n e a r ( t y p i c a l l y

non-polynomial) i n t e r a c t i o n i s o b t a i n e d between mass less sp in two

g r a v i t o n s ( i . e . r e p r e s e n t a t i o n s of t h e Poincarc group) p ropaga t ing i n

t h i s Minkowski space . The dubiousness of such a s e p a r a t i o n has a l r eady

been d i s cus sed in 52 in t h e con tex t of an e x t e r n a l g r a v i t a t i o n a l f i e l d

and t h e comments made t h e r e apply h e r e . However, i t does have t h e

advantage of r educ ing t h e t e c h n i c a l problem, a t l e a s t s u p e r f i c i a l l y ,

t o t h e s o r t of s i t u a t i o n which has been encountered b e f o r e in p a r t i c l e -

p h y s i c s - o r i e n t e d quantum f i e l d t h e o r y . This approach was i n i t i a t e d by ( 2 9 ) ( 3 0 ) ( 3 7 )

R. Feynman and S. Gupta and has been e n t h u s i a s t i c a l l y

adopted by a number of (mainly European) p a r t i c l e p h y s i c i s t s in r e c e n t ( 3 ' i ) ( 35 ) y e a r s . ' The s t r u c t u r e i s s i m i l a r in many r e s p e c t s t o t h a t of t h e

Yong-Millo theory in t h a t t h e genera l c o o r d i n a t e i n v a r i a n c e m a n i f e s t a

i t s e l f through the e x i s t e n c e of a non -abe l i an gauge group. The main

t a s k s t o be performed a re indeed the same in bo th c a s e s . Namely:

i ) Cons t ruc t t h e c o r r e c t Feynman r u l e s which l ead t o a

u n i t a r y S - m a t r i x .

i i ) Cons t ruc t t h e analogue of t h e Ward-Takahashi i d e n t i t i e s

which should r e f l e c t t h e gauge i n v a r i a n c e ,

i i i ) Find an a p p r o p r i a t e ( i . e . gauge i n v a r i a n t ) r g u l a r i s a t i o n

scheme.

i v ) I n v e s t i g a t e t he re normal i s a t ion of t h e t h e o r y ,

v) Find t e chn iques f o r summing up u s e f u l s e t s of Feynman

g raphs .

These q u e s t i o n s a re a l l connected and w i l l be d i s c u s s e d a t l e n g t h in

K . J . D u f f ' s c h a p t e r . I t i s t h e r e f o r e s u f f i c i e n t t o remark h e r e

t h a t t he t echn ique which i s u n i v e r s a l l y used t h e s e days t o g e n e r a t e a

p e r t u r b a t i v e expansion of t h e Green ' s f u n c t i o n s and hence t o c o n s t r u c t ( 3 8 )

t h e Feynman r u l e s , i s t h a t of a f u n c t i o n a l pa th i n t e g r a l . In a

s imple s c a l a r f i e l d t h e o r y t h e b a s i c n - p o i n t t ime o rdered product (which

g ives t h e n - p a r t i c l e S - m a t r i x v i a t he LSZ formalism) can be expressed by

a f u n c t i o n a l i n t e g r a l as

= H | (d$) ( x , ) . . . $ ( x n ) e " '

(U.2)

J=0

where t h e vacuum-vacuum ampl i tude in t h e p resence of t he e x t e r n a l

31

nource J ( x ) in

i

O U t < 0 | 0 > i " = N | (d*) e ( I t .3)

n n d II i s s o m e n o r m a l i s a t i o n c o n s t a n t . I f t h e l a g r a n g i a n I ( $ ) i s

n o p a r a t e d i n t o t h e s u m o f a f r e e p a r t L q p l u 3 a n i n t e r a c t i o n p a r t

XV (X i n a c o u p l i n g c o n s t a n t ) , i . e .

LU) = Lo(*) + XV(t), (k.k)

t h e n t h e b a s i c a m p l i t u d e (.3) c a n b e w r i t t e n a s

i/K f [Lq(4)+ XV() + ^ = N j (d*)

N e . ( d $ ) e i Ct.5)

Now the f u n c t i o n a l i n t e g r a l

f Z o ( J ) = I (d*) e (I t .6)

i s in Gaussian form ( s i n c e i-Q() i s b i l i n e a r in and i t s d e r i v a t i v e ) and

can be e x p l i c i t l y computed t o be

- \ j d"x d V J ( x ) A F (x -y ) J (y ) Z Q ( J ) = e (I t .7)

32

where Ap(x - y ) i s t h e a p p r o p r i a t e Feynman p ropaga to r f o r t h e f r e e $ i / K | XV(K/. 6 / 6 J ) d * x

f i e l d . I f t h e o p e r a t o r e in eqn (U. 5) i s now

expanded in term3 o f powers of X then a p e r t u r b a t i v e form f o r ~ ! 1 u t

i s o b t a i n e d in which each power of X i s m u l t i p l i e d by v a r i o u s space - t ime

i n t e g r a l s over p roduc t s of t h e Aj,(x - y) p r o p a g a t o r s . In f a c t t h i s expansion i s e x a c t l y t h e same as t h a t ob t a ined from t h e Feynman-pyson

i n t e r a c t i o n p i c t u r e and i s t h e modern way of o b t a i n i n g t h e co r respond ing

Feynman r u l e s . The u l t r a v i o l e t d ive rgence problem i s o f cour se t h e same

in bo th approaches .

The s i t u a t i o n f o r a t h e o r y which admits a gauge group i s more

compl ica ted . Not a l l components of t h e f i e l d a re t r u e dynamical v a r i a b l e s

and so t h e v a l i d i t y of t h e i r use as v a r i a b l e s t o be inc luded in t h e Feynman

i n t e g r a l over p h y s i c a l pa th s i s not a p r i o r i c l e a r . In t h e case of

e l e c t r o m a g n e t i s u t h e r e a re in f a c t no major problems, p r i m a r i l y because t h e gauge group i s a b e l i a n . However, f o r t h e Yang-Mills o r g r a v i t a t i o n a l

t h e o r i e s , which are d i s t i n g u i s h e d by p o s s e s s i n g n o n - a b e l i a n gauge g roups ,

the s i t u a t i o n i s more complex. The a c t u a l form of t h e f u n c t i o n a l i n t e g r a l

(31) was f i r s t e x h i b i t e d by De Witt and then by Fadeev and Popov and has

been e x t e n s i v e l y i n v e s t i g a t e d s i n c e t h e n . The main s u r p r i s e i s t h a t ,

when exp res sed i n Feynman diagram l anguage , loops of ' f i c t i t i o u s ' quanta

appear which do not occur in t h e o r i e s wi thout non a b e l i a n gauge groups .

The n e c e s s i t y f o r such f i c t i t i o u s loops was f i r s t demonst ra ted by Feynman

who found t h a t t he na ive p e r t u r b a t i v e r u l e s l e a d t o n o n - u n i t a r y (and non-

g a u g e - i n v a r i a n t ) S - m a t r i x e lements .

The r g u l a r i s a t i o n which i s mainly in use a t p r e s e n t i s t h a t of

3 3

dimensional r e g u l a r i a a t i o n . This i s mo3t a p p r o p r i a t e f o r t h e momentiim

npaco approaches but i s c u r r e n t l y b e i n g adapted by De Witt t o h i s

con f i g u r a t i o n space -based t r e a t m e n t .

The c u r r e n t s t a t e of t h e computa t iona l a r t i s t h a t va r ious t r e e

graphs and one loop graph have been computed and have been c o v a r i a n t l y

r e g u l a r i s e d in t h e sense t h a t t h e f i n i t e remainders s a t i s f y t h e (32)

a p p r o p r i a t e Ward i d e n t i t i e s . The main ques t i on which has t o be

d i s c u s s e d i s whether o r not t he theo ry i s r e n o r m u l i s a b l e . C e r t a i n l y a

s u p e r f i c i a l power count l e a d s t o a h i g h l y d ive rgen t t h e o r y , a r e s u l t

which seems t o be borne out by e x p l i c i t c a l c u l a t i o n f o r t h e combined

E i n s t e i n p lus m a t t e r - f i e l d l a g r a n g i a n s . The s i t u a t i o n i s not comple te ly

w a t e r t i g h t because i t i s p o s s i b l e t h a t miraculous c a n c e l l a t i o n s nay s t i l l

occur ( p o s s i b l y only f o r c e r t a i n cho ices of m a t t e r l a g r a n g i a n ) .

U n f o r t u n a t e l y t h e extreme complexity of t h e necessa ry c a l c u l a t i o n s ( a

two loop graph would be very h e l p f u l ) means t h a t a d e f i n i t i v e answer i s

not l i k e l y t o be for thcoming in t h e inmedia te f u t u r e . The r e a d e r i s

r e f e r r e d t o M.J. D u f f ' s c h a p t e r f o r f u r t h e r en l igh tenment on t h i s p o i n t

but perhaps i t i s worth commenting a l i t t l e on t h e s i g n i f i c a n c e of t h e

probable n o n - r e n o r m a l i s a b i l i t y o f t h e t h e o r y . I t i s p o s s i b l e t o t u r n

such a s i t u a t i o n t o p o s i t i v e advantage . The problem of q u a n t i s i n g non-

polynomial l a g r a n g i a n s (which by c o n v e n t i o n a l reckoning a r e c e r t a i n l y

non - r eno rmol i s ab l e ) r ece ived c o n s i d e r a b l e a t t e n t i o n a few y e a r s ago wi th

t he use of a method which in e f f e c t f i x e d , s i m u l t a n e o u s l y , t h e va lues o f

i n f i n i t e l y many s u b t r a c t i o n c o n s t a n t s in an S -ma t r ix e lement . Abdus Solum, ( 33) J . S t r a t h d e e and I a p p l i e d t h e s e t echn iques t o t h e i n t e r a c t i o n of t he

3 1 .

combined g r a v i t a t i o n a l , e l e c t r o m a g n e t i c and e l e c t r o n f i e l d s und

succeeded in o b t a i n i n g f i n i t e answers f o r c e r t a i n i n f i n i t e s e t s of

Feynman d iagrams , t h a t would i n d i v i d u a l l y be r ega rded as h i g h l y d i v e r g e n t .

The conven t iona l quantum f i e l d theo ry s i t u a t i o n i s s i m i l a r t o expanding - 1 / x - 1 / x 1

e around x = O a s e = 1 - ' / + t a k i n g t h e l i m i t X 2 !x 2

as x 0 from p o s i t i v e va lues and then announcing t h a t t h e r e s u l t i s

1 -

3 5

demands ol' quantum f i e l d theo ry e v e n t u a l l y l ead t o i t s demioe. However,

t.ho theory of g e n e r a l r e l a t i v i t y has a s i g n i f i c a n c e and v a l i d i t y q u i t e

a p a r t from t h e quantum t h e o r y and many r e l a t i v i s t s would very

reasonably o b j e c t i f E i n s t e i n ' s s t r u c t u r e was t o be j e t t i s o n e d p u r e l y on the grounds of n o n - r e n o r m a l i s a b i l i t y . Indeed they a r e more l i k e l y

t o i n s i s t t h a t t h e n o n - r e n o r m a l i s a b i l i t y imp l i e s t he r e j e c t i o n of quantum f i e l d t h e o r y ! Neve r the l e s s i t i s c l e a r t h a t a f a i r l y l a r g e amount of

e f f o r t w i l l be spen t in t h e nea r f u t u r e in t r y i n g t o f i n d a r e n o r m a l i s a b l e

theory of g r a v i t y which could p o s s i b l y s t i l l be g e n e r a l l y c o v a r i a n t and

achieve i t s ends by t h e s u b t l e i n t r o d u c t i o n of c e r t a i n m a t t e r - f i e l d t e rms .

A t h i r d r e a c t i o n t o t h e n o n - r e n o r m a l i s a b i l i t y of t h e c o v a r i a n t

theory i s t h a t t h e t r o u b l e has i t s o r i g i n in t h e s e p a r a t i o n o f t h e

g r a v i t a t i o n a l f i e l d i n t o a c l a s s i c a l background p lu s a quantum c o r r e c t i o n .

Such s p l i t s a r e very u n n a t u r a l w i t h i n t h e con ten t of t h e c l a s s i c a l theory

and i t i s on a t t r a c t i v e c o n j e c t u r e t h a t t h e problems of quantum g r a v i t y can be r e s o l v e d by avo id ing them. This i s one of t h e p o i n t s in f avour of

the canon ica l approaches which, because o f t h e i r r a t h e r s t r o n g e r

geomet r i ca l f l a v o u r , c e r t a i n l y do not n a t u r a l l y admit such decompos i t ions .

I t would be misguided however, t o t a k e t he r e n o r m a l i s a t i o n problem t o o

l i g h t l y and i t i s f a i r t o say t h a t t h e non-appearance of t h a t p a r t i c u l a r

problem in canon ica l q u a n t i s a t i o n i s due p r i m a r i l y t o t h e f a c t t h a t t h e

a p p r o p r i a t e c a l c u l a t i o n a l t e c h n i q u e s have not y e t been developed t o t h e

po in t where a c t u a l numbers a re o b t a i n e d , r a t h e r than t h a t t h e formal ism

i s i n t r i n s i c a l l y t r o u b l e - f r e e .

36

5. TRUE CANONICAL QUANTISATION

We now d i s c u s s t h e approach t o q u a n t i s a t i o n which i s c l o s e s t t o

t h e s t a n d a r d canon ica l p rocedure . The e s s e n t i a l i d e a i s t o e x t r a c t from

t h e m e t r i c t e n s o r t h o s e components which cor respond to genuine dynamical

( r a t h e r than gauge) degrees of freedom and then t o impose canon ica l

commutation r e l a t i o n s upon them and t h e i r c a n o n i c a l c o n j u g a t e s . T h i s approach i s o f t e n known in t h e l i t e r a t u r e as ' n o n - c o v a r i a n t 1 canon ica l

q u a n t i s a t i o n . The a l t e r n a t i v e superspace-based t e c h n i q u e (sometimes

c a l l e d ' c o v a r i a n t ' canon ica l q u a n t i s a t i o n ) w i l l be cons ide r ed i n 6. As

t h e r e a r e no o t h e r a r t i c l e s in t h i s volume which deal s p e c i f i c a l l y wi th

t h e s e t o p i c s ( u n l i k e f o r example c o v a r i a n t q u a n t i s a t i o n ) they w i l l be

d i s c u s s e d i n some d e t a i l h e r e .

The f i r s t problem t h a t must be i n v e s t i g a t e d i s t h e c l a s s i c a l

decomposi t ion of t h e E i n s t e i n t h e o r y i n t o canon ica l form. The a p p r o p r i a t e (39)

t echn ique i s we l l known fo l l owing t h e work of Dirac and Arnowi t t , Deser

and Misner (ADM)'110 ' . In o r d e r t o i l l u s t r a t e t he p r i n c i p l e i n v o l v e d ,

c o n s i d e r t h e simple example of a mass l e s s s c a l a r f i e l d t h e o r y in a f l a t

space- t ime wi th t h e l a g r a n g i a n

(5-1)

The co r re spond ing a c t i o n i s

(5.2)

37

which when va r i ed wi th reopec t t o

(5-9)

(5-10)

where Hqn (5-11) i s j u s t t he Hamil tonian d e n s i t y .

C l a s s i c a l l y , i f t h e canon ica l v a r i a b l e s 1(1 and tr a r e s p e c i f i e d on the

; p a c e l i k e h y p e r s u r f a c e t = t , then eqns (5-9) and (5 -10) a r e i n t e g r a t e d

.0 g ive t h e va lues on any l a t e r ( o r e a r l i e r ) t = t^ h y p e r s u r f a c e . The

luantum analogue i s t he Heisenberg p i c t u r e formalism in which the quantum

f i e l d s $ and 71 s a t i s f y equal t ime commutation r e l a t i o n s

(5 .11 )

( 5 . 1 2 )

md t h e f i e l d s a t t ime t a re exp res sed in terms of t h o s e a t t ime t by

39

-i/K llU,*) (t.-t) H(i,n)(t.-t ) n o n l o

(x,t ) = e (ito) e (5.13)

-i/K HU.wHtj-^) i/K H($,Tr)(t1-to) nfx.tj) = c *(x,tQ) e (5. ill)

A l t e r n a t i v e l y of cou r se one can use t h e Schrod inger p i c t u r e in which t h e

o p e r a t o r s have no t ime dependence and s a t i s f y

[(x), ;

1)0

t h e r e were in t h e Minkowski spuce ca se . Thus one i s more

o r l e s s o b l i g e d t o cons ide r t h e Cauchy problem over an

a r b i t r a r y s p a c e l i k e h y p e r s u r f a c e .

i i ) The E i n s t e i n equa t i ons G^1' = 0 do not invo lve t h e second-o rde r

t ime d e r i v a t i v e s of t h e m e t r i c t e n s o r g , t h u s a n t i c i p a t i n g cxB

t h a t , as in eqn ( 5 - 5 ) , " ^ 6 i t h e s e e q u a t i o n s w i l l dp ats reduce t o c o n s t r a i n t equa t i ons which t h e i n i t i a l da t a must

s a t i s f y , r a t h e r than t o genuine equa t i ons o f mot ion .

i i i ) The remaining e q u a t i o n s G^J = 0 do not de te rmine t h e t ime

e v o l u t i o n of a l l of t h e components of g ^ even i f t h e

c o n s t r a i n t s G 11 = 0 have been s a t i s f i e d , o

To i n v e s t i g a t e t h e s e p o i n t s f u r t h e r i t i s u s e f u l t o apply t h e s t a n d a r d

ADM techn ique and decompose t he m e t r i c t e n s o r as

-H 2 + W. N . ' 3 ^ , N. 1 J 0

V , = ( I ' 5 . 1 8 ) N i ' g i j

where w,v = 0 . . . 3 ; i , j = 1 , 2 , 3 . To see why t h i s p a r t i c u l a r form i s

chosen c o n s i d e r t h e ' 3 + 1 ' decomposit ion of space - t ime i n t o a family of

t h r e e - d i m e n s i o n a l s p a c e - l i k e h y p e r s u r f a c e s p a r a m e t e r i s e d by the va lue o f

an a r b i t r a r i l y chosen t ime c o o r d i n a t e x . The n a t u r a l m e t r i c induced on

a t y p i c a l e q u a l - t i m e h y p e r s u r f a c e i s simply g . . (x,x) and i t s i n v e r s e ^ J

(3) i " o

i s w r i t t e n as g 1 J ( x , x ) . The p rope r t ime dx between two s u r f a c e s

l a b e l l e d with t h e pa ramete r s x and x + dx w i l l ( f o r i n f i n i t e s i m a l dx)

be p r o p o r t i o n a l t o dx. Thus we w r i t e

1.1

dT = N ( x , x ) d x ( 5 . 1 9 )

where t h e f u n c t i o n N i s kncwn as t h e l a p s e f u n c t i o n . Now cons ide r t h e

cor responding normal v e c t o r , whose base has c o o r d i n a t e s ( x ' , x 2 , x 3 )

l y i n g in t he f i r s t h y p e r s u r f a c e . The t i p of t h i s v e c t o r can be connected

to t he p o i n t in t h e second s u r f a c e with t h e same s p a t i a l c o o r d i n a t e s

( x ' , x 2 , x 3 ) , by a v e c t o r O y i n g in t he second h y p e r s u r f a c e x

whose components can be w r i t t e n in t h e form (N 'dx 0 , N 2dx, N 3dx) . The

q u a n t i t i e s H1 (x,x) a r e known as s h i f t f u n c t i o n s and the s i t u a t i o n i s

ske tched in F ig . 1 . 1 .

F ig . 1 . 1 ^ ^ >>

The s p a c e - l i k e v e c t o r AC = AB + BC = (N'dx + d x 1 , N2dx + d x 2 , N3dx + dx 3)

and t h e r e f o r e t h e length of DC i s :

1(2

ds2 = g dxP dxV = - N 2 (dx ) 2 + g. . (N1 dx + dx 1 ) (tH dx + dx"') pv l j

= ( - N2 + N. N1) (dx ) Z + 2 N. dx d x j + g. . dx1 dx . (5 -20) l J J

which, b e a r i n g i n mind t h a t t he Roman i , j i n d i c e s a r e t o be r a i s e d and

lowered by t h e induced m e t r i c on t h e t h r e e - s u r f a c e x , i s p r e c i s e l y eqn

( 5 - 1 8 ) .

The i n v e r s e m e t r i c t e n s o r can r e a d i l y be shown t o be

1 HJ

^ =

I I ( 5 . 2 1 ) hi -

A r n o w i t t , Deser and Misner found t h a t t h e s e c o n d - o r d e r E i n s t e i n a c t i o n

could be w r i t t e n i n terms of t h e s e f i e l d v a r i a b l e s as

I = f d"x ( d e t ( l , ) g ) ' 2 R(g) = f d"x { ( d e t { 3 ) g ) / 2 N [ ( K . . K i j - K 1 K . j ) J J -^ J ^ J

(3) [ I + Rj + a f o u r - d i v e r g e n c e ) (5 .22 )

where K. . s ( N . , . + N . , . - g . . ) . (5 -23) i j 2N l l j j l i l j ,o

In eqn (5-22) the s u p e r s c r i p t s and r e f e r t o q u a n t i t i e s computed with

t he induced t h r e e - s p a c e m e t r i c and t h e o r i g i n a l f o u r - s p a c e m e t r i c

U3

( 3 ^ r e s p e c t i v e l y . In p a r t i c u l a r v ' R(x ,x ) i s t h e c u r v a t u r e t e n s o r of the

h y p e r s u r f a c e l a b e l l e d by t h e pa ramete r x and hence d e s c r i b e s t h e

i n t r i n s i c c u r v a t u r e of t h i s s u r f a c e . On t h e o t h e r hand the t e n s o r K. . ij

in eqn (5 .23 ) ( i n which 1 r e f e r s t o c o v a r i a n t d i f f e r e n t i a t i o n wi th r e s p e c t

t o t h e t h r e e - m e t r i c ) i s , g e o m e t r i c a l l y , t h e e x t r i n s i c c u r v a t u r e of t h e

h y p e r s u r f a c e and as such d e s c r i b e s t h e manner in which t h a t s u r f a c e i s

embedded i n t h e su r round ing fouidimensional geometry.

The main advantage of t h i s form i s t h a t t h e t ime d e r i v a t i v e i s

i s o l a t e d and one can compute t h e ' c o n j u g a t e momentum' t o g. . as J

j i s _ _ = - (de t 3g)J r 0 , i j + N j l i - g i k g j * ^ 0 )

6 g i j

- ^ i ( 2 N k l k - ( 3 ) 6 k i g k , s 0 ) > . (5 .2. , )

Not ice t h a t t h e r e i s no N o r H. term in t h e l ag rang i an and as a r e s u l t a

formal c a l c u l a t i o n g ives

( 5'2 5 )

6N

1 -

6 1 n n = = 0

6N. l

(5 .26 )

The f i n a l r e s u l t of a l l t h i s t heo ry i s t h a t t h e E i n s t e i n a c t i o n p r i n c i p l e

can be w r i t t e n (ana logous ly t o eqn ( 5 . 1 1 ) ) in f i r s t - o r d e r v a r i a t i o n a l

form as

l i l i

= | d*x { i j y - Hu C" U i j , g ^ ) } (5 .27 )

where N = U and o

C ; ( d e t ( 3 ) 6 ) " J ( , i j . . - 1 b . 1 w . j ) - ( d e t ( 3 ) g ) J ( 3 ) R (5-28) J. J i. J

These t h r e e equa t ions form t h e s t a r t i n g po in t f o r a l l modern t r e a t m e n t s

of canon ica l q u a n t i s a t i o n . I f we vary t he a c t i o n wi th r e s p e c t t o it1J we

ob t a in g. . = g^ . ( g ^ n1""> which may be so lved as

= r s ' V ( 5 - 3 0 )

which in f a c t i s e x a c t l y eqn (5-2l i ) . On t h e o t h e r hand v a r i a t i o n wi th

r e s p e c t t o l eads t o an equa t ion of t h e form

= it , ) ( 5 . 3 i ) r s u

and e q u a t i o n s (5 .30 ) and (5 .31 ) a r e , t o g e t h e r , e x a c t l y t h e G. J = 0

E i n s t e i n e q u a t i o n s .

F i n a l l y i f H i s v a r i e d we ob t a in U

= 0 (5 .32 )

1,5

which t u r n out ( u s i n g eqn (5 -30) ) t o be t he remaining C^1' = 0 E i n s t e i n

e q u a t i o n s . There a r e a number o f p o i n t s worth ment ioning about t h e

r e s u l t s achieved so f a r a t t h e c l a s s i c a l l e v e l :

a) The non-appearance o f U^ (and hence t h e v a n i s h i n g of t h e

co r respond ing con juga t e q u a n t i t i e s in eqns (5 .25 ) ( 5 . 2 6 ) ) i s e s p e c i a l l y c l e a r in eqn ( 5 . 2 7 ) . Indeed N^ p l ays t h e r o l e ,

from t h e canon ica l v i e w p o i n t , of a Lagrange m u l t i p l i e r and

c e r t a i n l y does not count as a t r u e canon ica l v a r i a b l e . I t

i s t he g r a v i t a t i o n a l analogue of A^ in t h e Maxwell

e l e c t r o m a g n e t i c t h e o r y .

b) In s p i t e of i t s n o n - c o v a r i a n t - l o o k i n g form, t h e t h e o r y i s

s t i l l g e n e r a l l y c o v a r i a n t . In o t h e r words eqn (5-27) i s

a p p l i c a b l e t o any choice of space o r t ime c o o r d i n a t e s .

This means t h a t t he theo ry i s not ye t in t r u e canon ica l

form because (as we s h a l l see l a t e r ) f o u r ou t of t h e s i x

) I f C l ' (n ,g) = 0 on an i n i t i a l h y p e r s u r f a c e and eqns

( 5 . 3 0 ) (5 .31 ) a re s a t i s f i e d , then C u ( n , g ) = on any l a t e r

h y p e r s u r f a c e . In o t h e r words t h e c o n s t r a i n t s a r e

conserved in t ime - i n f a c t by v i r t u e of t h e Bianchi

i d e n t i t i e s .

) The converse i s a l s o t r u e . Namely i f n 1 J and g^ a r e

chosen s o t h a t C^t tt ,g) = 0 on a l l h y p e r s u r f a c e s then t h e

= 0 equa t i ons (5 .30 ) and (5-31) a r e a u t o m a t i c a l l y

s a t i s f i e d . In t h i s sense t h e dynamical G..J = 0 e q u a t i o n s

can be regarded as o c c u r r i n g t w i c e . ki.

I f a t t h e c l a s s i c a l l e v e l g.^ and it a r e regarded as b e i n g

c o n j u g a t e v a r i a b l e s , so t h a t a t some f i x e d t ime we have t he Poisson b r a c k e t r e l a t i o n s

{g. . ( x ) , - ' (x ) ) , P.B. (x " i ) , (5 -33)

then

(5-3IO

(5 .35 )

which shew t h a t j i s t h e g e n e r a t o r ( i n t h e c a n o n i c a l t r a n s f o r m a t i o n s e n s e ) of t h e i n f i n i t e s i m a l c o o r d i n a t e t r a n s f o r m a t i o n

xU x

p + p (x ) . The q u a n t i t i e s C1' s a t i s f y a Poisson b r a c k e t

It 7

a l g e b r a which i s a r e f l e c t i o n o f t h i s f a c t .

I t i s a t t h i s s t a g e t h a t t he ' c o v a r i a n t ' and ' n o n - c o v a r i a n t '

approaches t o canon ica l q u a n t i s a t i o n go t h e i r s e p a r a t e ways. In t h e

c o v a r i a n t approach ( see 6) t h e v a r i a b l e s in t he a c t i o n p r i n c i p l e eqn

( 5 . 2 7 ) , which as emphasised above i s s t i l l g e n e r a l l y c o v a r i a n t , a r e

q u a n t i s e d as they s t a n d . On t h e o t h e r hand in t h e n o n - c o v a r i a n t approach

t h a t i s b e i n g d i s c u s s e d h e r e , t he system i s reduced f u r t h e r c l a s s i c a l l y

b e f o r e q u a n t i s a t i o n . There a r e va r ious ways of do ing t h i s but t h e b a s i c ( i .0 ) ( ' i l )

ideas i s t o per form t h e f o l l o w i n g s t e p s :

1) Solve t h e e q u a t i o n s Cu(7i,g) = 0 e x p l i c i t l y f o r f o u r of t h e

twe lve ( g ^ j , " ) v a r i a b l e s . This i s p o s s i b l e in p r i n c i p l e , b u t , in p r a c t i c e , has only been achieved p e r t u r b a t i v e l y .

This l e a v e s e i g h t v a r i a b l e s in t h e s t r u c t u r e whose t ime

dependence i s d e s c r i b e d by e i g h t of t h e twe lve = 0

equa t i ons (5 -30) and ( 5 - 3 1 ) , t h e remaining ones b e i n g

i d e n t i c a l l y t r u e ( they a r e i n f a c t t h e Bianchi i d e n t i t i e s

in t h e form CU(ir,g) = 0 .

2) Choose a system of c o o r d i n a t e s . There a r e a number of

a lmost e q u i v a l e n t ways of doing t h i s . A sample s e l e c t i o n

i s :

a) Impose any f o u r ' g a u g e ' c o n d i t i o n s of t h e form

F1 J(n,g) = 0 . Thi3 removes fou r of t h e e i g h t

(g > 11 ) v a r i a b l e s which a re l e f t a f t e r s t e p 1 .

The e q u a t i o n s F l '(7i,g) = 0 , p lus e q u a t i o n s (5-30)

and ( 5 - 3 1 ) , can be used t o f i n d f o u r e l l i p t i c

1.8

d i f f e r e n t i a l e q u a t i o n s f o r N which can a l s o in M

p r i n c i p l e be s o l v e d , t h u s e l i m i n a t i n g t h e l ag range

m u l t i p l i e r s ti ^ from t h e t h e o r y . There i s on exac t

analogue of t h i s in t h e Maxwell t h e o r y where t he

equa t ions a r e D A - 3 ( 3 Av) = j . I f t h e U V v v

r a d i a t i o n gauge d iv A = 0 i s chosen t h e n c l e a r l y one

of t h e t h r e e A v a r i a b l e s i s e l i m i n a t e d . However, t h e 3

time component of t h e Maxwell equa t i ons p lu s Tr-ot

( d iv A) = 0 y i e l d s t h e e l l i p t i c equa t ion - V2 A = j which can be so lved a t once u s i n g t h e a p p r o p r i a t e

t h r e e d imens iona l Green ' s f u n c t i o n , as A = j . o o A t y p i c a l example in t h e g r a v i t a t i o n a l case would be

t h e c o n d i t i o n s

6 i . / o = 0

( ( d e t ^ g ) * g i j ) . = 0 >J

which l e a d t o t h e e l l i p t i c equa t ions

N , . / " ( 3 ) R ( g ) = 0

{ < d e t ( 3 ) g ) J ( N i l j + N j U - ( 3 y j N * ) + 2 N * i j } . = 0 IK 1J

which can , i n p r i n c i p l e , be so lved . However, i f t h e t h r e e -

space i