introduction to quantum gravity
DESCRIPTION
Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in mathematics. The various competing theories, e.g. string theory and loop quantum gravity, have still to be checked against observations.TRANSCRIPT
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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.
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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
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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
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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 ' ('
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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
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[ ( 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.
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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
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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
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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 )
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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 . )
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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
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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
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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 .
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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 )
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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
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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)
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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 .
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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