Path integral for gravity and supergravity

Nuclear Physics B245 (1984) 436-448 © North-Holland Pubhshmg Company

PATH INTEGRAL FOR GRAVITY AND SUPERGRAVITY Kazuo FUJIKAWA

Research lnsntute for Theoretwal Physws, Hlroshtma Umversltv, Takehara, Htroshtma 725, Japan Osama YASUDA

Department of Physws, Umverslt) oJ Tokyo, Bunkvo-ku, Tokvo 113, Japan Recewed 23 February 1984 (Revised 29 May 1984) It is shown that the specification of the path integral variables m quantum gravity, which was prewously formulated on the basis of the anomaly conslderatmn and the BRS supersymmetry, can also be understood by the more famflmr conslderatmn of the unit .lacoblan approach We describe how to derive the same prescription in the framework of the basic path integral formahsm wtthout recourse to the BRS supersymmetry, which is one of the consequences of the Faddeev-Popov effective actmn A novel feature of quantum grawty (and the gauge theory m curved space-time m general) xs that a proper choice of the invanant gauge orbit volume element becomes essential to ensure the gauge independence of the partition functmn We also briefly note that the conformal (Weyl) symmetry is generally spoiled by the anomaly in quantum theory The connectmn between the present "anomaly-free" prescrtptlon and the true grawtatmnal anomaly by Alvarez-Gaum6 and W~tten Is also clarified

1. Introduction It h a s b e e n r e c e n t l y p o m t e d o u t t h a t t h e p a t h i n t e g r a l v a r i a b l e s i n q u a n t u m g r a v i t y are specified essentially uniquely if one imposes the (nawe) anomaly-free condition on the BRS supersymmetry [1,2].

To be preose,

dimensions)

associated with the general coordinate transformations

the path

integral measure

is g i v e n b y ( i n n s p a c e - t i m e

[2] d/z = d / z ( g ) lq ~[g~"+~'/4"rl'~]co@~,~°,@Bu~[g'/4S],

(1)

w h e r e t h e g r a v i t a t i o n a l p a r t d/z ( g ) is g i v e n b y ( d e p e n d i n g o n t h e c h o i c e o f i n d e p e n dent variables) [l ~

@[gkg~],

I] ~[g~g~],

d/x ( g ) = '

~t3

1-I ~ [ g kh~],

[I

k ,~

a ~ [ g ha], 436

k=

k= k =

n-4 4n n+4 4n

'

n-2

n+2

k-

4n

'

(2)

K Fujikawa, 0 Yasuda / Path integral for gravltv

437

wRh h~~ the vierbeln field, h'~(x)hat3(x) =- g~t3(x) and g-= det [g~t3(x)]/> 0 in the euclidean metric we use in the present paper. The variables r/~ and ~, correspond to the F a d d e e v - P o p o v ghost and anti-ghost fields, respectively, and B~ to the auxiliary field to be specified below The field S(x) collectively stands for any world-scalar quantity such as the ordinary scalar fields, spInor fields, and the vector fields projected on the local frame An(x)=- h'~(x)A~(x) The weight ½ field S ( x ) gl/4S(x) m (1) has been used as an integration variable in the background gravitational field [3, 4] ( I f one uses the vierbein h.~(x) In (2), for example, the F a d d e e v Popov ghost fields Cmn(x) associated with the local Lorentz transformation appear These fields should also be treated as the world-scalar quantity) The novel aspect of the prescription (1) compared with the previously known path integral [5-7] (and also canonical [8]) formahsm is that the integration varzables are specified. For example, it shows that g~(x) or x/gg"t3(x) should be used as the dynamical variable in the Einstein gravity in n = 4 (As for the explicit form of the p a r t m o n function, see eq (30) below ) Formula (1) has been successfully applied to the path integral of the relatwisUc string theory [1] in n = 2 and more recently to the quantum aspects of the generalized Kaluza-Klein theories [9, 10] We here note that the local measure for the gravitational variable which corresponds to (2) (for n = 4) has also been specified in ref [11] on the basis of a different reasoning, the treatment of the ghost variables however appears to be different from (1) In the course of these applications, it came to our attention that the naive umt jacobian approach [ 12, 13], which gives the correct local measure for the supergravity, gives the prescription different from (1) for more general class of theories In the present note we first show that the treatment of the jacoblan factor in ref [13] Is not justified from the distribution view point If one corrects this point, the prescription (1) becomes the unit jacobian condition on each variable separately The prescription (1) can thus be understood on the basis of the more familiar approach of the unit jacobian The remnant of the anomaly consideration appears in the unit jacobian c o n d m o n on each varmble separately We then show how to derive the measure (1) in the path integral formalism without recourse to the BRS supersymmetry This is quite satisfactory since the BRS supersymmetry [14] should be regarded as one of the consequences of the F a d d e e v - P o p o v effective action [6]

2. Unit jacobian condition The prescription (1) 1s based on the BRS transformation for a scalar field S(x) -~ gkS(x), for example, S(x) ~ S(x) + th[rt ° OpS(x) +2k(Oo~TP)S(x)],

(3)

with h a G r a s s m a n n parameter [1, 2], and the vanishing of the jacoblan factor (or the " a n o m a l y " ) for k = ¼ occurring at Tr O(x, y) ~ Tr [r/P(x) OoS(x - y) + l(oo~°(x) )8(x -y)] = 0

(4)

K Fujtkawa, 0 Ya~uda / Path mtegralJor gravm,

438

We note that the BRS transformation of a scalar field IS obtained from the general coordinate transformation by replacing the coordinate variation e" (x) by e" (x) = tAr/g(x), with r/" the Faddeev-Popov ghost field Eq (4) Implies that det [6S'(x)/6S(y)] = exp [1A Tr O(x, y)] = 1 for transformation (3) On the other hand, the prescription of the umt jacoblan in ref [13] is based on the assumption Tr C3(x, y) -= Yr [r/° (x)

O~8(x-y) +(OprtV(x)),3(x- y ) ]

=0

(5)

These (contradicting) expressions (4) and (5) involve the singular objects such as the derivative of the &function, and they should be carefully treated Following Schwartz, these objects should be regarded as the distributions We thus examine the actions of O(x, y) and (}(x, y) on test functions f(x) and g(y)

O(j, g) =-f

d"yf(x)O(x, y)g(y)

d'x

=~ d"x~l°(x)[-(Oof(x))g(x)+f(x)(@g(x))],

d(f,

g) -= f d"x

(6)

d"yf(x)O(x, y)g(y)

= f d"x[-rlV(x)(@f(x))g(x)]

(7)

For any orthonormal set of basis vectors {f. (x)} such as suitable products of Hermlte functions, the diagonal elements of O and (9 become

O(fm, f,.)=O,

(8)

but r 1 | d(fm, f . ) = ~ j d

n

p

2

x(OvrI (x))jk(x) #0,

(9)

in general Therefore Tr

O(x, y) = ~ O(fm,f,.)

= 0,

m

which justifies (4), whereas Tr O(x, y) = ~

O(f.,,fm)

m

--'½f d°x(ao~(x))~°(°)

(10)

K Fujikawa, 0 Yasuda / Path integral for gravity

439

It lS tempting to identify ( 1 1) as zero, but it should not be set to zero; If one Identifies (11) as zero, the discussion of the umt jacoblan becomes meaningless to being with The basic idea of the unit jacoblan is to retain the factor such as (1 1) and impose the cancellation of these factors arising from various integration variables [13] From the anomaly viewpoint [1, 2, 4], one assigns a more realistic meaning (1 e as the anomaly factor to a suitably evaluated ~ fn (x)fn (x), and the umt jacobian condition becomes physically more transparent On the basis of these mathematical prehmlnarles, one can recognize that the (nmve) anomaly-free c o n d m o n (1 e the vamshlng of the jacobmn factor under the BRS transformation assocmted with the general coordinate transformation) in refs [1, 2] is essentmlly the unit jacobian c o n d m o n on each variable separately ff one adopts (4) the anomaly-free consideration [1, 2] and the unit jacoblan approach [13] appeared to be qmte different just because of the assumption (5) made in ref [13] The local measure for the Einstein gravity m n = 4 is obtained by rewriting (1), for example, as d/x = det

[g(x)-3/28(X - y ) ]

H ~g,13(x)@~7"(x)~,(x)~B~,(x)

(12)

x

This local measure disagrees with g3/2 in ref [13] because of the relation (5) used in ref. [13]. As for supergravlty, one can confirm that (4) and (5) give rise to the same overall local measure due to the (accidental) cancellation of various jacoblan factors among bosons and fermlons This explains why the prescription in ref [13] gives the correct local measure for supergravlty [2] but falls for the more general class of theories

3. Path integral for gravity and supergravity We now want to derive the measure (1) in the framework of the basic path integral f o r m a h s m [5, 6] without recourse to the BRS supersymmetry. By this way we can get a deeper understanding of the weight factor for the ghost field ~" in (1). For this purpose we start with the discussion of the scalar field S ( x ) The transformation law of the scalar field S ( x ) under the coordinate transformation x '~ = x ~ - e ~ ( x ) ,

(13)

S'(x') = S ( x )

(14)

is given by

One might be tempted to conclude that the invarlant measure is thus given by II ~ S ( x ) = H ~ S ' ( x ' ) 1¢

(15)

x'

This naive expectation is not correct, the reason is that the symbol Hx becomes non-trivial in curved space-time, just as the volume element dnx is transformed

440

K Fupkawa. 0

non-trivially.

The weight

Yasuda / Path mtegral for gravri)

4 variable

s”(x) = g”4S(x)

instead

defines

an invariant

measure fl Ss”( x) = n &(x’) X X This result (16) has been derived by several analyse this problem here from the functional

(16)

different integral

methods [3,4] vtewpomt

We further

We start with the relation $(x)&(x) and rewrite

4(x) = g”“+(x)

The relation

Y

n GB[g”“4(x)] I

exp

dx

1

= const ,

(17)

(17) then becomes 4(x)‘&

$z [I

dx

(18)

1

= const

Although the transformation law of 4(x) IS arbitrary at this stage, one can assign the scalar transformation law to 4(x) to transform (18) defined m one coordinate system to another The “action” m (18) IS obviously mvartant under the coordmate transformatton, and thus the measure m (18) should also be invariant We thus recover (16). One can similarly recover the measure for the contravtant vector in (1) on the basis of the sequence {det

=

k(x)~(x-y)l)F”2 I

=

= I

n 9AJL(x)exp Y

n 9[g”“A”(x)] X

{det [g(x)S(x x

I

k‘(x)g,Jx)A”(x)

$z [I

3

exp

AP(x)g,,A”(x)&dx

+z [

I

-Y)}~“*

n 9[g(“‘z”4”A”(x)] X

exp

where we used the fact that we are constdermg obtain

I

dx

\

n %g (n+2”4nAp(~)] exp

iz [I

[I $z

A~g+,A”&

n-dtmenstonal

A’g,,A”Jg

dx

dx

1 1,

space-time

1

= const ,

(19) Thus we

(20)

and we recover the invariant measure for the contravariant vector in (1) and (2) A more systematic and rigorous way to specify the invariant measure is to define the variation of the generic field 4(x) at the same x-variable 64(x) and construct

the measure

mvartant

5%4’(x) - 4(x), under

the transformation

(21) ( 13) by multtplymg

K FuJlkawa, 0

by a suitable

power of g(x);

of the unit Jacobian. the degrees

relation

The coordinate

of freedom

that the invariant is given by

n @*P(x) . “P (2) if -apsagPP

(4) with T* replaced

by .sP 1s used as a criterion

x@ now plays the role of the parameter indexing

m the field variables

One can then confirm n-drmensronal space-time

as in &p a,iap

441

Yasuda / Path Integral for graulty

measure

for the metric

g”” = ga” m

)

= Xr& 9[g(n+4)‘4”gnp(X)] . -

(22)

remembers the variation Sg”e(x)= 8[g(x)kg”P(x)]= on: variables -ap~p~up +2k a,~~$@ (Other choices of independent

can be treated similarly consider the “resolutton

) The basic idea of the path of unity”, for example,

integral

approach

[5] is to

(23) which corresponds to the gauge condrtron a,$’ p = 0; w”(x) is a classical quantity whrch describes the gauge orbit correspondmg to the general coordinate transformation G(w) (We here ignore the Gribov-type complications [15] which may arise in the present case also) A novel aspect of quantum gravrty (and the gauge theory in curved space-time in general) is that formula (23) needs to be carefully applied by choosing the appropriate gauge orbit volume element To explain this pomt, we first observe that the general field configuratron Q(x) (such as g”’ in (22)) 1s specified by the representative field configuratron 4(x) correspondmg to the ortgm of the gauge orbit and the gauge transformatron G(w) as Q(x)= G(w)4(x), 4(x) may satisfy a suitable gauge condrtron The mfimtesrmal change of the gauge conditron corresponds to an infinitesimal gauge transformation of the representatwe field configuration as cb’(x) = G(e)@(x) (24) Here, for srmphcny, we consider the ordinary linear gauge transformatron G(E) only; the non-linear transformatron will be drscussed later The measure for the general field configuration Q(x), whrch is invariant under the transformation G(E)@(X), can be constructed on the basis of our analyses mvarrance of the measure m (22) for an arbitrary c-number shows that the full configuratton

in (22) covers the freedom

in eqs (16)-(22) The gauge transformatron correspondmg

to the

gauge transformation also The ultimate arm of the path integral IS to define the gauge mdependent measure for the representative field configuration, I e fl[94] = n [9+‘]. We formally define such a measure by drvrdmg the invarrant measure for Q(x) by the gauge orbit volume element which is mvariant under the transformatron G(w) + G(w’) specified by G(E)@ = G(E)G(o)G(E)-‘Go = G(w’)4’(x)

(25)

K Fuflkawa, 0 Yasuda / Path mtegral for grawtv

442

The transformation law of to ° is most simply obtained by considering the action of G(w) on the scalar field S(x) In this case

G(w) = exp [w"(x) 0.] for the infinitesimal

(26)

wU(x) and G(e)G(o))G(e) -1 = exp [G(e)w" a.G(e) 1] = exp [w'"(x) a . ] ,

(26')

with

o)'"(x) = oJ"(x) +e"(x) G w " ( x ) - (Ge~(x))wP(x)

(27)

for an Infinitesimal eU(x) Eq (27) coincides with the transformation law of the contravarlant vector Therefore the lnvarlant volume element is given by (cf eq (20)) [1 Do3"(x)--= H x

D[g'"+2)/4%)'(x)]

(28)

x

By extracting the lnvarlant volume element (28) from (23), which amounts to dividing the measure in (22) by the measure in (28), the path integral measure is defined by (for pure gravity)

dl2 =- H @~,~°(x)6(O~g,~°(x)det [ 8(O~'~'(x))] .... ~ &5"(y) J

= H

I

xexp{f{ ,B~3G~'~3+'¢°L~

jr/ ] _'

(29,

which agrees with (1) apphed to the gauge condition O ~ ~13= 0 The partition function for Einstein gravity, for example, is thus given by

. ~ rs(o~g~,) ]

dx /

The ghost field r}" introduced here is related to the conventional ghost field r/" by

~"(x) =-g{"+2)/4"r/'(x) ,

(31)

which is a reflection of the fact that we describe the gauge orbit volume by the parameter o3" m (28) From the present viewpoint, the peculiar BRS transformation law of r/" is understood as the one which preserves the measure H @~}" invariant [2] As for the supergravity [13], prescription ( 1) gives rise to [2] (in n = 4, for example) m ~ d# = H ~(g I / 8 hz)~(g

x @(g3/Sr/.

I/8

~ .a) ° d ) ( g l / 4 S ) ~ ( g ' / g P ) ~ ( g i / 4 A m )

)@(gl/4c. )~(gl/4Cm. ) X [multiplier fields],

(32)

K Fujtkawa, 0 Yasuda / Path integral for grawty

443

where the multiplier fields stand for the auxlhary fields such as B. and the anti-ghost field ~:. in (29), which appear for the three kinds of ghost fields r/', C" and C ~" corresponding to the general coordinate, super-gauge and local Lorentz transformations, respectively These multiplier fields do not contain any weight factor The weight factor for the ghost field C m" assocmted with the local Lorentz transformation, for example, is also fixed on the basis of the invariance of the gauge orbit volume element of the local Lorentz transformation under general coordinate transformations The general coordinate transformatmn properties thus umquely specify the path integral measure for supergrawty [13]

g~/4

4. Gauge independence By construction eq (29) is regarded as the invariant volume element (22) divided by the invarlant gauge orbit volume element (28), and thus it gives rise to a result independent of the speofic gauge c o n d m o n when integrated over the entire functional space with a gauge invarlant lntegrand We now examine how the gauge independence is realized in more detail We start with the generic measure defined by (see (29) above) d/2 -= [I @OS(F" ( ~ ) - a ~) d e t / ~ t ( O ) ,

(33)

where • stands for the general field configuration such as ~ ¢ m (22), and F ~ ( O ) the gauge fixing function such as 0eft ~" in (29) The determinant factor 1~7/(O) is defined by using the variable 03, which defines the lnvariant measure in (28), as

~,l(CI)) OF"(CI)) OF"(O) O~°r=-M(cI))O~°v -=

003~-

0 ~ o ~ 0038

a03~

(34)

The gauge independence of (33) means that one has the relation

I[I~08(F~(O)-a~)detlf4(O)exp[-S(O)+lIOJdx 1 =IHNO'8(F~(O')-a~-X~)detI(4(cI)')exp[-S(O')+tfOJdx],

(35)

with S ( O ) the gauge mvarlant action and A ~ an infinitesimal function. We note that the source term which breaks the gauge lnvanance contains the field • (= O' - F ( O ' ) M 1( O')A in the notation of (36) and (37) below) on both sides of (35) The customary and most direct way to prove (35) is to consider the non-hnear gauge transformation e ° specified by [16]

same

M;(O)e ~ = a a

(36)

The important point to be noted here is that relation (36) is defined only in the of the submanifold specified by the gauge c o n d m o n

mfimteszmalnetghborhood

K Fuflkawa, 0 Ya~uda / Path integral for gravm

444

F " ( 4 9 ) = a s ( I f one restricts 4) exactly to F~(49) = a s, the m e a s u r e In (33) b e c o m e s d i v e r g e n t ) The p a r a m e t e r e ¢ in (36) gives rise to the t r a n s f o r m a t i o n

49'=_ 49 + F( cI))e ,

(37)

with F ( q b ) =_ 6qb/8o) By c o n s t r u c t i o n

Fa( 49') = F"( q)) + M ~ e ~ = F ~ (49) + h ~

(38)

in the infinitesimal n e i g h b o r h o o d o f F a ( 4 9 ) = a", a n d in p a r t i c u l a r

3(F"(49) - a '~) = 6(F'~( 49') - a '~ - h a )

(39)

The v a r i a b l e q~' in (37) is c o n s t r a i n e d to the infinitesimal n e i g h b o r h o o d o f F a ( 4 9 ' ) = a " + h " W h e n 49 a n d 49' are thus c o n s t r a i n e d to the infinitesimal n e i g h b o r h o o d s o f F a ( 4 9 ) = a ~ a n d F " ( 4 9 ') = a ~ + h a, respectively, the f u n c t i o n a l m e a s u r e II @49 gives rise to a non-trivial j a c o b l a n factor u n d e r the t r a n s f o r m a t i o n (37) t h r o u g h the b - d e p e n d e n c e o f e in (36) At the s a m e time, £/(49) in (34) gives rise to an extra t r a n s f o r m a t i o n factor u n d e r (37), w h i c h s h o u l d cancel the j a c o b l a n factor in the m e a s u r e II @q) if relation (35) holds In the case o f the Y a n g - M l l l s t h e o r y In flat s p a c e - t i m e ( c o n s e q u e n t l y , o3 = co), It is s h o w n in ref [17] that the sufficient c o n d i t i o n s for r e l a t i o n (35) are

[ 0~49s]

Tr k - - ~ - , .] = 0 ,

(40)

Tr [f'~e °] = 0 ,

(41)

w h e r e 849j s t a n d s for the v a r i a t i o n o f the Y a n g - M i l l s field u n d e r the ordinary l i n e a r t r a n s f o r m a t i o n a n d f~¢ IS the structure f u n c t i o n The c a l c u l a t i o n In the p r e s e n t q u a n t u m gravity p r o c e e d s just as that in ref [17] if one uses the g e n e r a l abstract n o t a t i o n in ref [17] Eq. (40) n o w c o r r e s p o n d s to the unit j a c o b i a n c o n d i t i o n for the i n v a r l a n t m e a s u r e in (22), a n d it is satisfied by c o n s t r u c t i o n C o n d i t i o n (41) is r e p l a c e d b y

Tr[-6W(x)-'W(z)6~6(x-z)+f

dyfo%(x,y,z)e°(y)]=O,

(42)

W ( x ) is an extra factor in £ / ( 4 9 ) arising from O~'~(x)/Owt3(y)=_ W ( x ) 6 ~ 8 ( x - y ) in (29), a n d the structure f u n c t i o n f~t~ is given by (see also the last

where

two terms in (27))

f ~ ( x , y, z) =_8(x - y ) 8 ( x - z)( 8~ O; - 8~, 0~)

(43)

O n e m a y rewrite (42) as Tr

~SW(x)W(x) l ~ ( x

-- Z) + W ( x )

f d y f o 3 ( x , y, z ) e P ( y ) W ( z ) '] = O, .J d (44)

K Fuflkawa,0 Yasuda/ Path integralfor gravtty

445

by noting ~W -1-- - W - 1 6 W W -~ We now observe that

f dz[6W(x)W(x)-16~6(x-z)+W(x) = aW(x)to

(x) +

f dyf~t3(x,y,z)eP(y)W(z)-l]o3~(z)

p opto"(x)- (af)to.(x)]

= 3[W(x)to"(x)]~- 3o3"(x),

(45)

which corresponds to the transformation of o3" defined in (25). Thus (42) is written as

I-(o3oi1

Tr L ao3 J = ° '

(46)

which is equivalent to the unit jacobian condition for measure (28) under the transformation (25) We note that the lnvarlance of [I ~O3 holds even for the non-hnear transformation e(q0) in (36), since q> and to are regarded to be independent when • is restricted to the infinitesimal neighborhood of FS(q~) = a s. Relation (35) can also be understood more directly from the lnvarlance of the measure (28) under the transformation G(to')= G(s)G(to)G(e) 1 in (25) with e defined by (36) The functional measures in (35) stand for the full measure I] @@ divided by H 903 in the infinitesimal neighborhoods of FS(qb) = a s and F"(q~ ') = a" +A s, respectively, this can be confirmed by integrating both sides of (35) by I1 ~O3. By constructmn, 11 ~O3 = [] @o3' and thus the equality (35) holds Here one may wonder why the non-trivial jacoblan factor appeared for the measure I] ~ under the transformation (36) in the discussion of the gauge independence in ref [17], whereas we now regard [1 @qb = l ] ~q>' w:thout the 6-functional constraints ~(FS(dP)-a s) and 6(F~(cIg')-a ~ - A " ) The answer to this question is that (36) is defined only in the infinitesimal neighborhood of F " ( @ ) = a s, and thus the transformation (37) becomes non-linear only for the configuration • in the infinitesimal neighborhood of F S ( q )) = a s. The infinitesimal neighborhood of the particular configuration F~(@) = a ~, for example, has the measure zero in the entire functional space H ~@ without a g-functional peak in the Integrand (One may also recall that the full measure [I @@ is invariant under any c-number gauge transformation, whereas relation (36) does not enjoy such an lnvarlance property) The transformation (36), which is the basic Ingredient of the discussion in ref [17], xs defined only in the theory where the gauge condition Is already specified The characteristics of the transformation (36) are neatly formulated by the BRS supersymmetry [ 14], and the notion of the infimtesimal neighborhood of F~(q)) = a s is realized by the use of the Grassmann parameter which has no proper magnitude In the conventional sense The BRS transformation involves the ghost fields explicitly, for example, ~:~(x) -> ~:~(x) + AB~ ( x ) ,

(47)

with auxdiary fields ~, and B. in (29) and a Grassmann parameter A (we follow

K Fuflkawa, 0 Yasuda / Path integral]or gramO'

446

the n o t a t i o n s o f refs [1, 2]) It is k n o w n [1, 14] that the BRS m v a r l a n c e o f the m e a s u r e is s u f f i o e n t for the gauge i n d e p e n d e n c e o f the p a r t i t i o n functxon I f one e x p o n e n t l a t e s the g a u g e c o n d i t i o n a n d the F a d d e e v - P o p o v d e t e r m i n a n t by using the a u x i l i a r y fields as in (29), relation (35) w i t h o u t the source term J can be written as

[tB,~(F~'(@)-a")+~l~4(qb)~]dx} =fd~'exp{-S(q,')+f[tB.(F"(cIg')-a"-A")+,l~l(~')~]dx}

(~l-oo)-=f d~ exp{-S(q') +I

(48)

By e x p a n d i n g (48) in h", we o b t a m the relation 6

=

=fdtxB~(x)exp{-S(cb)+f[tB,~(F"(Cb)-a~)+,~l(@)~]dx} =0

(49)

We n o w w a n t to s h o w that (49) follows directly from the BRS l n v a r l a n c e If the m e a s u r e is BRS l n v a n a n t as is in o u r c o n s t r u c t i o n o f (1) a n d (29), the t r a n s f o r m a t i o n in (47) suggests

<~1~o (x)l-~> -- + A<~IBo(x)I-~>

(50)

We thus o b t a i n 6

' ~ T ~ x ) (o~ i-~>1~o ~o = (~IB~ (x)l-~> =- 0,

(51)

where the first e q u a l i t y in (51) is a s t a n d a r d result o f S c h w m g e r ' s a c t i o n p r m o p l e We note that S c h w l n g e r ' s action p r i n c i p l e c o m p a r e s two t r a n s i t i o n a m p h t u d e s g e n e r a t e d b y the different d y n a m i c s b e t w e e n the two (physical) states, in the present case, the v a c u u m states are s p e o f i e d by the d y n a m i c s with the g a u g e condmon F"(qb) = a ~ Here we e n t e r e d into some details o f the g a u g e i n d e p e n d e n c e W h a t we w a n t e d to s h o w is that the most f u n d a m e n t a l i n g r e d i e n t o f the g a u g e i n d e p e n d e n c e is the gauge orbit v o l u m e e l e m e n t (28) which is m v a r l a n t u n d e r the t r a n s f o r m a t i o n (25) In the case o f the Y a n g - M d l s t h e o r y m flat s p a c e - t i m e , st is easy to confirm that the c o n d m o n (41) is n o t h i n g but the l n v a r m n c e r e q u i r e m e n t o f the gauge orbit v o l u m e e l e m e n t u n d e r the t r a n s f o r m a t i o n c o r r e s p o n d i n g to (25) F r o m a g e n e r a l g e o m e t r i c a l view o f the local gauge s y m m e t r y , we c o n s i d e r that this s~mple charact e r l z a t m n o f the g a u g e i n d e p e n d e n c e by m e a n s o f the m v a r m n t gauge orbit v o l u m e e l e m e n t is quite satisfactory

gwen

K Fujlkawa, 0 Yasuda / Path integral for gravity

447

5. Conformal anomaly The condition of the unit jacoblan for each dynamical variable separatelybecomes Important when one considers the conformal (Weyl) anomaly since the Weyl transformation property of field variables critically depends on the weight factors [1, 4] In this respect, the absence of any more local measure other than the weight factor for each variable In (1) and (29) is essential The conformal anomaly was neatly dealt with in the apphcatton of (l) to the relativistic string theory [ l], whereas the applications in refs [9, 10] depend only on the ovelall local measure, in the latter two works [9, 10], the gauge independence of the partition function up to the one-loop level has been explicitly demonstrated* One of the important imphcatlons of (1) and (29) is that it is generally difficult to avoid the anomalies in the general coordinate transformation and the conformal (Weyl) transformation simultaneously, it is impossible to give the unit jacoblan to the Weyl transformation for each dynamical variable separately in (1). This is easily understood by recalling that the conformal (Weyl) transformation is defined by g~"(x)->e2~(~)g~(x) and g~(x)-> e 2'~('°gu~(X ) Thus ~ ' ~ ( x ) - ) e ""(')/2¢~"(x)

(52)

for the integration variable specified in (1) The variable ~'~ cannot be a Weyl scalar for any space-time dimensionahty n ~ 0, and this non-trivial transformation generally gives rise to an anomaly if the jacoblan factor is carefully estimated [4] The local conformal symmetry could survive in the quantum theory only when the conformal anomaly factors arising from various dynamical variables cancel each other We finally comment on the connection between the present "anomaly-free" prescrtptlon and the true gravitational anomaly discovered by Alvarez-Gaum~ and Witten [19]. As is seen from eqs (6) and (8), our (naive) anomaly-free criterion can be made more rigorous for the real orthonormal basis sets or for the complex orthonormal basis sets with their complex conjugates added in the jacobian factor, the former case arises for the gravitational variables and more generally for real fields, and the latter case for the Dirac fermions, for example. For the chiral fermlons, for example, the basic euclidean operator becomes non-hermitian and a more sophisticated analysis is generally required (see, for example, ref [20]) In fact, Alvarez-Gaum6 and Witten [19] discovered the true gravitational anomaly for chiral fermlons and other complex fields in certain space-time dimensions The present path integral prescription (as well as the one in ref. [2]) should therefore be regarded as the manifestly anomaly-free formulation of basically * We emphasize that this gauge independence Is not proved if one uses the convennonal non-covarmnt choace of the path integral measure, at least m the calculatlonal scheme adopted in refs [9, 10] The advantage of the present p r e s c n p t m n as that the gauge independence ~s ensured for a wide class of calculatlonal schemes The present prescription has also been a pphe d to a detailed analysas of the conformal anomaly m curved space-time m ref [18]

448

K Fujlkawa, 0 Yasuda / Path integral for grawty

a n o m a l y - f r e e theories. T h e interesting aspect o f q u a n t u m gravity is that this prescription s p e o f i e s each path integral v a r i a b l e separately, an d the c o n s i d e r a t i o n s o f c o n f o r m a l (Weyl) s y m m e t r y b e c o m e t r a n s p a r e n t since the c o n f o r m a l a n o m a l y is u n i q u e l y specified o n l y w h e n the general c o v a n a n c e is i m p o s e d Th e g au g e i n d e p e n de nc e is also en s u r ed for a wide class o f r e g u l a r l z a t l o n schemes, as was n o t e d a b o v e O n e o f the p r es en t authors ( K F ) benefited f r o m the discussions at the Erlce W o r k s h o p on Q u a n t u m Gravity, O c t o b e r 1-7, 1983 H e a c k n o w l e d g e s the s u p p o r t p r o v i d e d by the Y a m a d a Science F o u n d a t i o n for travel e x p e n s e s an d by E I du Pont de N e m e u r s & C o a n d I n t e r n a t i o n a l T e l e p h o n e an d T e l e g r a p h C o r p o r a t i o n for local e x p e n s e s at the W o r k s h o p

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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