On the new effective action in quantum field theory

Nuclear Physics B312 (1989) 700-714 North-Holland, Amsterdam

ON THE NEW EFFECTIVE ACTION IN QUANTUM FIELD THEORY P. ELLICOTT and D.J. TOMS Department of Theoretical Physics, The University, Newcastle upon Tyne, NE1 7R U, UK

Received 22 October 1987 (Revised 25 August 1988)

The formulation of the effective action, in a way which is covariant under general field redefinitions, is considered. It is shown how Vilkovisky's ansatz for the effective action may be derived in the case of a flat field space. If the field space is curved, it is shown in a simple non-perturbative way why Vilkovisky's effective action is not correct. The same method is used to obtain DeWitt's result for the effective action. A generally covariant definition which includes both Vilkovisky's and DeWitt's definitions as special cases is given. We also discuss some of the features which arise in the case of a curved field space, including the analogue of the usual Legendre transformation which relates the effective action to the partition function.

1. Introduction T h e b a c k g r o u n d - f i e l d m e t h o d as f o r m u l a t e d b y D e W i t t [1, 2] is n o w a s t a n d a r d t e c h n i q u e o f q u a n t u m field theory. T h e basic object of interest in this a p p r o a c h is the effective action. A l t h o u g h the resulting effective action is gauge invariant, it is n o t gauge i n d e p e n d e n t ; n o r is it i n d e p e n d e n t of the choice of field p a r a m e t r i z a t i o n . T h e d e p e n d e n c e o n the gauge c o n d i t i o n is clear from Jackiw's [3] calculation of the effective p o t e n t i a l for scalar electrodynamics. A n e x a m p l e which illustrates the d e p e n d e n c e o n field p a r a m e t r i z a t i o n is given in ref. [4]. R e d e f i n i t i o n s of the classical field are equivalent to s o m e w h a t m o r e c o m p l i c a t e d c h a n g e s in the i n t e r p o l a t i n g field which is c o u p l e d to the source term in the f u n c t i o n a l i n t e g r a l defining the effective action. D e p e n d e n c e u p o n the i n t e r p o l a t i n g field arises f r o m the fact that v a c u u m correlations of different o p e r a t o r s will in g e n e r a l differ. This is natural, since different o p e r a t o r s m a y c o r r e s p o n d to different p h y s i c a l observables. In a similar way, gauge d e p e n d e n c e of the off-shell effective a c t i o n follows f r o m the use of i n t e r p o l a t i n g fields which are n o t gauge invariant. In d i f f e r e n t gauges these s t a n d for different invariant o p e r a t o r s , so that the resulting v a c u u m - r e s p o n s e functionals a n d the associated effective actions represent different i n v a r i a n t q u e s t i o n s which should a n d d o have different answers. W h a t the a p p r o p r i a t e i n t e r p o l a t i n g field a n d gauge are d e p e n d s u p o n the use for w h i c h the effective action is intended. F o r example, the S - m a t r i x is i n d e p e n d e n t of 0550-3213/89/$03.50©E1sevier Science Publishers B.V. (North-Holland Physics Publishing Division)

P. Ellicott, D.J. Toms / New effective action in QFT

701

the gauge and depends only on the Botchers' class [5] of the interpolating field. It is therefore unaffected by local, invertible field redefinitions. Similarly, the value of the effective action at its minima - which in some theories can be related to the vacuum energy or particle production r a t e - is unchanged by field redefinitions which preserve the asymptotic boundary-value data. For other purposes the choice of an appropriate field variable or gauge may be more circumscribed. It may sometimes be useful to compute the effective action associated with one gauge choice using the Feynman rules of another. Similarly, it may be useful to define the effective action in a way so that, like the classical action, it transforms as a scalar under field redefinitions. A technique for accomplishing this was proposed by Vilkovisky [6,7] and later modified by DeWitt [8]. The distinction between DeWitt's definition of the effective action and that of Vilkovisky has been clarified recently by Rebhan [9] who showed that Vilkovisky's original scheme, when applied to Y a n g - M i l l s theory at two-loop order, led to non-local counterterms which are eliminated if DeWitt's definition is used. The basis of the Vilkovisky-DeWitt formulation resides in viewing fields as local coordinates of points in the space of fields, and then introducing a metric and connection on this space. Instead of coupling the source directly to the field, Vilkovisky couples it to the derivative of the geodetic interval between the background field and the quantum field giving a manifestly covariant formulation. DeWitt's [8] construction is more complicated, and a simple approach is described in sect. 3. The outline of our paper is the following. In sect. 2 we discuss the problem of the reparametrization invariance of the effective action. If the space of the fields is flat, it is shown in a very simple way how Vilkovisky's ansatz for the effective action may be derived. In sect. 3 we consider the more interesting case of a curved field space, present a general ansatz for the effective action which contains both Vilkovisky's and DeWitt's definitions as special cases, and discuss the question of consistency of the new effective action. In sect. 4, we formulate a covariant definition of one-particle irreducibility and give a covariant Legendre transformation. We impose some natural requirements on the covariant formalism which guarantee that the one-particle irreducible conditions hold. The general spirit of sects. 3 and 4 is that we want to generalize the flat-field-space case of sect. 2 but are left with arbitrariness if only general covariance is used as a guide. By imposing some restrictions which are natural generalizations of the usual case, we remove some of this arbitrariness.

2. Derivation of Vilkovisky's effective action Suppose that there exists a parametrization of the fields f~ (assumed to be bosons) such that in this parametrization the metric tensor in field space becomes Tab where ~'ab is field-independent. We are using DeWitt's [1] condensed notation here. In this parametrization both the Christoffel connection irffc and the curvature tensor Rabcdvanish identically. This means that we are restricting our attention to a

702

P. Ellicott, D.J. Toms / New effective action in QFT

flat field space. The line element in the parametrization above is ds 2 = d f a Y~b d f b ,

(2.1)

and the geodesic equation is

dV" ~d s

= 0.

(2.2)

Because everything so far is standard, the usual result for the effective action holds. If f~ represents the coordinates of the background fields, then the effective action is defined by i r[fl=-ihlnfdt~tflexp-~{S[f+f]-f

a

r °[:1}.

(2.3)

F , a [ f ] represents the functional derivative of F [ f ] with respect to the background field. The functional measure d/~[f] is

d/~[f] =

(Det ~,ob)1/2I-I d f a ,

(2.4)

(/

which is just the invariant volume element for field space. An important point to emphasize here is that because in the parametrization f a the space of fields is a vector space, the sum f a + fa which enters eq. (2.3) is well-defined. The loop expansion is generated by expanding the classical action S [ f + f ] in powers of the quantum field f and solving eq. (2.3) iteratively. If we write

slY+i] = s[l] + : : a s . I ~ ]

+

J s °.[/]I" 1

a

-[- Sint [ f , f ] ,

(2.5)

where

si°,[:, i ] =

1 y/,,.../°.s .......... [:], n~3

(2.6)

-

is to be treated as an interaction term, then the loop expansion of the effective action is found from F[fl=s[fl+½hlnDetS

,a

b[f]

:,1 1/

703

P. Ellicott, D.J. Toms / New effective action in QFT

where ( . . . ) denotes that the quantity enclosed by angular brackets is to be Wick-reduced keeping only one-particle irreducible graphs. The reason why only one-particle irreducible graphs contribute is due both to the presence of the logarithm in eq. (2.3) and the occurence of f ° F a in the argument of the exponential. The exponential in the last term of eq. (2.7) may now be expanded to the desired order in h. The Wick reduction of products of fields is found using the basic relation

~yaf6) = iAO6,

(2.8)

S, ab[f]a°~[f] =3co.

(2.9)

where

Aab[f] is the Green function in the presence of the background field f. For example, to two-loop order it is found that

F[f] = S[f] + ½ihlnDetSob[f] ~-~2{l~AalblAaab2Aa3b3g--(ala2a3)S2(blb2b3 ) -- 81A . . . . -Aa3a4~ ~,(aia2a3a4) ~\

-1- 0(~31 (2.10 t

The round brackets denote a symmetrization over all indices with a normalization factor of 1/N! where N is the number of indices. The overbar on ~(,,~o~) and ~(OlO~O~o,) signifies that the expression is evaluated at the background field f. The second term in eq. (2.10) arises from performing a gaussian functional integral. The position of the indices as S ob is important and comes from the factor of (Det 3,oh)1/2 in eq. (2.4) (S,o b= yb~S o~). The expression obtained for the effective action is divergent in general and requires both regularization and renormalization. This is done by regarding S[f] as the bare action and adding on counterterms which are given an expansion in powers of h. The background field also gets renormalized. We may write

s [ f ] = s¢°~[f] + hs~l~[fl + h2s'2~[fl + o ( h 3 ) ,

(2.111

where the superscript denotes the order of the relevant term in the loop expansion. Substitution of eq. (2.11) into eq. (2.10) and working to order h 2 gives

F[f] = S(°)[f] + h{½i lnDet S(°),J'[f] + S(1)[f] } q _ ~ 2 f 1 AalblAa2b2Aa3b3~(O) ~(0) %AalO2Ao3a4~(O) "* ~ 0 ~ 0 ~ 0 ~",(ala2a3)~,(b]bzb3)- 8~0 ~0 ~',(ala2a3a4)

+ ,;aab~-~l~2._o~ oh--* ~7~2~} + O(h 3) , (2.121

704

P. Ellicott, D.J. Toms / New effective action in QFT

where A~b is the Green function for S!°)b. An alternative way to derive this result is to go back to eq. (2.3), substitute in eq. (2.11) at the beginning and work consistently to order h 2. For details of how this works in a particular example, see Toms [10]. The procedure described above is fine so long as the basic requirement that the parametrization of the theory be such as to correspond to a metric in field space which is field independent, be maintained. However, there is no compulsion to be so rigid in the allowed choice of variables. Let ~Cdenote an arbitrary set of fields which parametrize the same theory as f~. We may regard f~ as a functional of ~ and vice versa. The metric in field space for this arbitrary parametrization may be obtained immediately from eq. (2.1) by making the transformation from f~ to ~. This leads to ds 2 = dq0i gij depj ,

(2.13)

gij =fa, iTabfb,j,

(2.14)

where

is the metric. The geodesic equation transforms to

d~

d2epi dq0J ds 2 +FJ/k[q°] ds ds - 0 ,

(2.15)

Fj~ [q0] = qY,a f a, jk"

(2.16)

where

This shows that in general coordinates there is a non-zero connection which must be included. It is straightforward to show that Fjk as defined in eq. (2.16) is just the Christoffel connection for the metric (2.14). (See Weinberg [11], for example.) If we want the effective action in terms of the general coordinates ~i which parametrize the background field, all that we need to do is to take the result (2.10) which we know to be valid, and make the change of coordinates fa__+ ~i. The classical action S[f] is a scalar functional, and hence is invariant under such a field redefinition. Consider next the one-loop term which involves S, b [ f ] . By use of the chain and product rules it follows that S, a b [ f ] = s i k [ ~ ] ? b, k~i,a q- S i [ ~ ] ~ i . a b .

(2.17)

A straightforward computation using eqs. (2.14) and (2.16) shows that ~i, ab = __ ~l, afb, kgkjlljl "

(2.18)

P. Ellicott, D.J. Toms / New effective action in QFT

705

Using this in eq. (2.17) leads to S ab ~i -b -- j , = ,af ,jS;i ,

(2.19)

where ;t

,i

(5

*ki~,l

(2.20)

"

From eq. (2.19) it then follows that Det S, b = Det S ; / .

(2.21)

Thus, to one-loop order the effective action for a general field parametrization becomes r[gp I = S [4] + ½ih In Det S ; / + O(h2).

(2.22)

This is exactly the modification of the one-loop effective action suggested by Vilkovisky [6, 7]. Note that the position of the indices on S ah[f] in the one-loop part of eq. (2.10) is crucial for obtaining eq. (2.22) since the Jacobian of the transformation cancels with the inverse jacobian when the determinant is taken (see eq. (2.19)). This would not have been the case if the one-loop term had been written as In Det S ,b. We again wish to emphasize that the reason that the index placement occurs as it does naturally within the formalism is due to the presence of (Det ~ab) 1/2 in (2.4). The significance of this for quantum field theory in curve spacetime is discussed in ref. [121. It is clear now why the standard background-field method will obtain an incorrect answer when applied to theories which have a metric which is field-dependent. Application of the method directly to S[q0] would lead to F [ ~ ] = S[gp] + ½ih lnDet S/[Up] + O(h2),

(2.23)

which is manifestly non-covariant and therefore field-parameter dependent. No account is taken of the non-trivial connection. Finally, consider higher loop contributions to the effective action. First of all, the background-field propagator for the general parametrization is obtained immediately from eqs. (2.9) and (2.19)

[1].

(2.24)

The transformation of the vertex function is S,(al...a,,)

=-i~ ~) , a 1 .. " gpio, a n S ; ( i l . . . i , , )

.

(2.25)

P. Ellicott, D.J. Toms / New effective action in QFT

706

(This is easily proved by induction.) It is then obvious that higher loop contributions to the effective action are all invariant under the transformation from f " to Cp~. To two-loop order it follows from eq. (2.10) that F [ ~ ] = S [ ~ ] + ½ih lnDet S ; / [ ~ ] h2fl~Ai,j,Ai2j2Ai3J3~

+ -- ~, 12 .

.

.

.

~ ;(ili2i3)~;(jlJ2J3)

1-aili2Ai3i4s --

8 ~

~

....

~ + O(h3).

;(11121314) f

(2.26)

The extension to higher loops is straightforward. Vilkovisky's [6, 7] ansatz for the effective action is i /~[Up] = - i h l n f dt~[~o]exp ~ { S[qo] + o ' [ ~ , qo]P;i[~]},

(2.27)

where o[~p, ep] denotes the geodetic interval (which is one-half of the square of the geodetic distance between the points ep and Cp) and o~[C~,q0] = ( 6 / ~ ) 0 [ ~ , q~]. (We do not assume/~ = F here, but will prove it below.) The result in eq. (2.27) reduces to the usual definition (2.3) for the case of a flat field space provided that the theory is parametrized using Cartesian-like coordinates. Because or[Up, ep] transforms like a vector with respect to coordinate changes at ~ and like a scalar with respect to coordinate changes at % eq. (2.27) is a manifestly covariant definition for the effective action. A covariant version of the loop expansion may be obtained from the covariant Taylor expansion of the classical action

s[ep] = ,=0 E (-1)" n! S;(i .... i")°i~[~'q~]"'oi"[C~'q~]"

(2.28)

m

(The overbar on S;(i . . . . in) indicates that it is to be evaluated at ~.) Transforming the variables of integration from q~i to o~[Cp,q0] in the functional integral (2.27) leads to

= (g[

l)l/2H do',

(2.29)

i

in the case of a flat field space. (If the field space is curved then the functional measure becomes more complicated [7].) It is then obvious that the loop expansion will generate exactly the same expression as obtained by the direct transformation method described above. This proves that P[Fp] = F[~], and hence shows that, in the case of a flat field space, Vilkovisky's ansatz for the effective action may be derived from the usual definition rather than postulated a priori. 3. Curved field spaces When the field space is curved (where the curvature is computed from the Christoffel connection of the field space metric as usual), it does not appear as if there is any simple way of proceeding which is analogous to that of sect. 2. If we are

P. Ellicott, D.J. Toms / New effective action in QFT

707

initially motivated solely by considerations of general covariance then there is no need to be as restrictive as Vilkovisky's [6,7] definition is in eq. (2.27). More generally, we may take as the basic definition [13] i

F[~] =

-ih ln f dl~[q~]exp~ { S[w] + Ti[C~, ep]F;i[C~])

(3.1)

where Ti[~, qD] is any quantity which transforms as a vector at ~ and a scalar at ¢p. The only thing we know for certain about T~[~, ~] is that if the space of fields is flat, then the results of sect. 2 prove that Ti[~,cp] = o~[~, qo]. However, in the general case there is no reason why T'[~, ¢p] could not involve the curvature of the field space. This is in fact the case for DeWitt's [8] definition. If we require the effective action to reduce to the classical action when h ~ 0, then T*[~, ~] = 0. There is one further assumption made by Vilkovisky [6, 7]; namely,

(3.2)

( o i [ ~ , f~])o=O,

where the angular brackets around any functional F [ ~ , ¢p] denotes its functional average defined by

(F[~,ep])r=exp

('

-~F[~]

)fdl*[q~]F[~,q~]exp~(S[q>]+T'[~,q~]F;,[~]}. (3.3)

It is crucial that eq. (3.2) holds because it is a vital ingredient in the proof that the effective action is independent of the choice of gauge-fixing condition in gauge theories. This is why one could not adopt Vilkovisky's ansatz for the effective action with the requirement (3.2) dropped. Given a definition of the effective action, such as that taken by Vilkovisky (which is eq. (3.1) with Ti[~, ~] = o'[Fp, ~]), it is possible to calculate (o~[~, ~o])o order by order in the loop expansion. For Vilkovisky's effective action this results in

( oi[ ~, ep])o= -- !it, AiJa""o k, . j . ~ ; 3 . . . . .

k + O(h2)

•

(3.4)

(Relevant expansions may be found in ref. [13].) A'J is the propagator defined as the inverse of S; ij. This result has been noted independently by Rebhan [14]. The result of eq. (3.4) is a clear indication that Vilkovisky's [6, 7] definition of the effective action is inconsistent with the requirement of eq. (3.2). This inconsistency may be seen in a non-perturbative way as well. An added advantage of this view is that it leads in a very simple way to DeWitt's definition of

708

P. Ellicott, D.J. Toms / New effective action in QFT

the effective action. Covariant differentiation of eq. (3.1) with respect to Up leads to

O~ ( I zJ;i[~, ~9])T-- ~Ji) F;j -1- ( TJt~.~CP])TF; j i ,

(3.5)

where the overbar denotes an expression evaluated at ~. By taking T~[~, cp] = ai[~, q~], which is Vilkovisky's proposal for the effective action, and by imposing eq. (3.2), it is a consequence of eq. (3.5) that the following relation must hold (3.6) Again, because this can only be satisfied in a flat field space, this shows that Vilkovisky's proposal for the effective action cannot be true for a curved field space. Consider again a general T~[C~,tO]. This functional may be given a covariant Taylor expansion in powers of oi[~,cp]. Because we require Ti[~,~] = 0 , the simplest possibility for Ti[~, cp] is just the lowest order term in its Taylor expansion

Ti[ F~, q~] = Fij[ ~ ]oJ[ C~, cp] .

(3.7)

Here F i j [ ~ ] is some unknown tensor functional which we wish to determine. The only information which we know at this stage is that for a flat field space, F~[~] = 8ij, eq. (3.7) is of course not the most general possibility for Ti[~, qv] (and in general there is no obvious reason why there could not be higher order terms added to eq. (3.7)), but is sufficiently general to include both DeWitt's [8] and Vilkovisky's [6, 7] definitions. The first conclusion to be drawn from eq. (3.5) is that if we restrict ourselves to the on-shell effective action, then we must have

=0=0.

(3.8)

Off-shell, it does not follows from eq. (3.5) that eq. (3.8) holds. However, for the traditional effective action, as well as Vilkovisky's effective action in the case of a flat field space, eq. (3.8) does hold if we remove the restriction F;i = 0. Furthermore, although we are not considering gauge theories in this paper, it seems as if ( T J [ ~ , ¢p])T= 0,

(3.9)

will be required to prove independence of the off-shell effective action on the choice of gauge-fixing condition analogously to the way in which (oJ[~, q~]), = 0 is needed for DeWitt's effective action [15]. Accordingly, we will consider eq. (3.9) as a natural requirement even off-shell. This condition also guarantees that the effective action is

P. Ellicott, D.J. Toms / New effective action in Q F T

709

comprised of one-particle irreducible vacuum graphs as the proof in ref. [16] clearly shows. Using eq. (3.9) in eq. (3.5) leads to the following condition which the functional TJ[~p, qo] must satisfy

(Ta;i[~,q~l)r=ai,.

(3.10)

With the simplest choice (3.7) for TJ[~, cp] it follows from eq. (3.10) that Fiy[Up] = C - l i j [ ~ ] ,

(3.11)

where i

oi

--

(3.12)

This corresponds precisely to DeWitt's [8] definition of the effective action. In the flat-field-space case, it is easy to verify that F~[Cp] defined in eqs. (3.11), and (3.12) has the correct value of 8ij. 4. T h e Legendre transform and one-particle irreducibility

In sect. 3 we gave a fully covariant effective action for a general curved field space. In order that the effective action be a useful object, there is an additional feature which it must share with the traditional effective action; namely, it must be the generator of one-particle irreducible graphs. The standard way of proving that the traditional effective action generates one-particle irreducible graphs is by relating it to the generator of connected graphs by means of a Legendre transformation. (See, for example Abets and Lee [17].) We now wish to discuss the covariant generalization of the Legendre transformation. (For earlier approaches, see refs. [16,18-21].) For the traditional effective action, if W [ J ] is the generating functional for connected graphs, then / ' [ ~ ] = W [ J ] - J i U p i,

w[J]

_ _

84

_ ~i,

(4.1) (4.2)

defines the Legendre transformation. A consequence of eqs. (4.1) and (4.2) is that = -J,.

(4.3)

It then follows by differentiation of eqs. (4.2) and (4.3) that ~, ik W , kj = _ 6ji ,

(4.4)

P. Ellicott, D.J. Toms / New effective action in QFT

710

where W' kj =

F, ijk

=

-- F,

~2W[j]/6jj ~Jk. Subsequent

differentiations of eq. (4.4) lead to

(4.5)

il ln, jrnF kn W ' Iron,

[', ijkl = F, im F, jn F, kpF, tqW" mnpq jr_ F, imF, jnF, kpF, lqF, rsW . . . . W ' pqs q- F, im F, jpF, kn F, tqF, rsW . . . . W" pqs q_ F, im I', jqF, kpF, in F, rsW . . . . W ' pqs . (4.6) Higher order relations are found in a like manner. Evaluation of both sides at J~ = F ~= 0, gives the one-particle irreducible relations for the n-point functions. In order that this method be generalized to the new effective action, it is important that a covariant version of eq. (4.1) be found, since as we have already noted, J ~ is not a covariant expression. The obvious generalization of eqs. (4.1) and (4.2) is /'[c~]

=

w[Jl-J,v'[c~],

(4.7)

~W[J] ,~J,

-

viii],

(4.8)

-

-

where v~[~p] (as well as J~) transforms like a vector under redefinitions of the background field. The relation analogous to eq. (4.3) which is implied by eqs. (4.7) and (4.8) is

T';i = -JkS*;i.

(4.9)

Note that for the traditional effective action v~[C~]= ~ and hence 5 k, ~= 8k~. The one-particle irreducible relations which we demand that the new effective action satisfy are the covariant generalization of eqs. (4.4)-(4.6) F; ~kW; *J = - 8 J~,

F; ijk

=

- - F ; ill'; j m I ' ; k n W ; tmn ,

(4.10)

(4.11)

P; ijkl = F; imI'; jnF; kp['; lqW; mnpq + F; imF; jnI'; kp-F; lqF; rsW; mnrw; pqs q- F; imF; jpF; knF; tqF; rsW; mnrw; pqs + F; ira1"; jqI'; kpF; lnF; rsW; mnrw; pqs

(4.12)

P. Ellicott, D.J. Toms / New effective action in QFT

711

Just as before we want these relations to follow from the Legendre transformation (4.7)-(4.9). Covariant differentiation of eqs. (4.9) and (4.8), respectively, gives F; is = -Jd5k; is - Jk; jvk; i,

(4.13)

W; ~j= 5k D~I ;/DYj "

(4.14)

Here D ~ I / D J j denotes the formal inverse of @;/. It is clear from eq. (4.13) that for a general vector vi[~] we cannot even satisfy the first relation (4.10). If the field space is flat, then we can always choose a Cartesian field parametrization for which ~i J = 3ij. But as 3ij is an invariant tensor, it follows that, for a general field parametrization in a flat field space (see the arguments in sect. 2), we must have ~i;j = 3i s . Although this restriction space, it is natural to try now sufficient to ensure follow from the Legendre simplifies to

(4.15)

on gi[~] can only be derived in the case of a flat field to adopt it in the general case as well. This condition is that the one-particle irreducible relations (4.10)-(4.12) transformation as is easily shown. Note that eq. (4.9) now

r ., I. . .

~,

(4.16)

if eq. (4.15) is used, a result which is identical to that found for the traditional method. It is easy to see however that eq. (4.15) is restrictive and implies that the curvature of the field space must satisfy = 0. This means that the space must admit a homothetic Killing vector. Using the functional-integral representation (3.1) for F[~] we have from eqs. (4.7) and (4.16) that the generating functional is given by

Rijklvl W[J]

i i exp~WIS]=fdl~[cPlexp ~{S[q~l

+ J~(vi[ ~] -

Ti[u~,qo])) .

(4.17)

It is interesting to note that the argument of the exponential has an explicit dependence. From eq. (4.17) we have

3j i

=

(-Lja,Ok;j-Tk; j[Up'cp]) r"

(4.18)

712

P. Ellicott, D.J. Toms / New effective action in Q F T

Using eqs. (4.14)-(4.16) this becomes

8W --

vi[gp]-(Ti[gp, qa]+F;kW;J~(8~j

-- T k

; j[qo, -- ep])} T"

(4.19)

The term ( .-- ) in eq. (4.19) may be recognized as the constraint (3.5) and hence vanishes (even off-shell). We therefore end up with

8W[J] -

84

-

v'[ep],

as we must. Note that we do not need to impose (Ti}T 0 separately. To summarize, for Vilkovisky's [6, 7] effective action, which is valid in a flat field space only, W[J] is given by =

i exp ~ W[ J ] =

f d#[~0]exp ~i { S[q0] + 4 ( v i [ ~ ]

- o~[~, ~1)},

(4.20)

where d[~p] is a vector which satisfies eq. (4.15). Note that this has the correct transformation properties in contrast to the non-covariant expression written down by Vilkovisky [6]. For DeWitt's [8] effective action, we have i exp~W[J]=

i fdt,[,~]exp-~{S[,p]+4(d[~]-C-15[~]oJ[~,W])} (4.21)

where C~j[~] is defined in eq. (3.12). The relations (4.10)-(4.12) hold even off-shell (i.e. even without setting 4 = F;~ = 0). However, we are often interested in on-shell quantities, and in particular require the connected propagator defined by (1/i)(82W/MiMj)lj=o . In order to obtain it, we differentiate eq. (3.5) a further time and then use eqs. (3.9) and (3.10) to find that on-shell we have ]( ( T i T J > T F ; j k

} j~:i=o _- -

- - ~ k i-

(4.22)

However, because we are insisting on the one-particle irreducible relation (4.10), we conclude that 1i 8J,.SJj 82W J=O

=(TiTJ}TIF:i=°"

(4.23)

For the traditional effective action, we have Ti[~, ~0] = ~i - ~ and then eq. (4.23)

P. Ellicott, D.J. Toms / New effective action in QFT

713

becomes the familiar result

1 82W s=0 i 84 aJs

(4.24) =

-

Hence, eq. (4.23) gives a covariant expression for the connected propagator which has the correct limit for a flat field space with a Cartesian field parametrization. For a general parametrization of a flat field space, we must use 1 82W

i ~Ji 6Js

= (°i°J)~ r,=0,

(4.25)

J=0

and for DeWitt's [8] effective action for a curved field space

1 82W v=o

i 8J~SJj

=(°"°m)r]L'=°c-l~"c-lJm'

(4.26)

where T / = c - l i j o j with C~ defined by eq. (3.12). This gives a covariant one-particle irreducible formalism which is consistent with the traditional method when it applies. In conclusion, we have developed a covariant formalism that guarantees one-particle irreducibility for an effective action defined by eq. (3.1), provided that

( T / [ ~ , c p l ) r = 0,

(4.27)

and W[J] is obtained from the covariant Legendre transformation described above. Vilkovisky's [6, 7] effective action does not satisfy eq. (4.27) for a general curved field space, whereas DeWitt's [8] does. However, without imposing some extra conditions, there is no a priori reason why an alternative T i could not be chosen which satisfies eq. (4.27) and hence also the one-particle irreducible relations, In view of the problems encountered with F[~], we feel that the auxiliary expressions used by DeWitt [8] will prove more useful. For example, there is no problem in defining a Legendre transformation for F [ ~ , , ~] where ¢p, is another point in field space. We are grateful to S.R. Huggins and G. Kunstatter for many useful discussions and comments. P. Ellicott would like to thank the SERC for financial support.

714

P. Ellicott, D.J. Toms / New effective action in Q F T

References [1] B.S. DeWitt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965) [2] B.S. DeWitt, in Quantum gravity II, ed. C.J. Isham, g. Penrose, and D.W. Sciama (Oxford University Press, Oxford, 1981) [3] R. Jackiw, Phys. Rev. D9 (1974) 1686 [4] G. Kunstatter and H.P. Leivo, Phys. Lett. 8183 (1987) 75; G. Kunstatter, in Super field theories, ed. H.C. Lee (Plenum, New York, 1987) [5] H.J. Borchers, Nuovo Cim. 15 (1960) 784 [6] G.A. Vilkovisky, in Quantum theory of gravity, ed. S.M. Christensen (Adam Hilger, Bristol, 1984) [7] G.A. Vilkovisky, Nud. Phys. B234 (1984) 125 [8] B.S. DeWitt, in Quantum field theory and quantum statistics, ed. I.A. Batalin, C.J. Isham, and G.A. Vilkovisky (Adam Hilger, Bristol, 1987) [9] A. Rebhan, Nucl. Phys. B288 (1987) 832 [10] D.J. Toms, Phys. Rev. D26 (1982) 2712 [11] S. Weinberg, Gravitation and cosmology (Wiley, New York, 1972) [12] D.J. Toms, Phys. Rev. D35 (1987) 3796 [13] D.J. Toms, Proc. Second Canadian Conf. on General relativity and relativistic astrophysics, ed. A. Coley, C. Dyer, and T. Tupper (World Scientific, Singapore, 1988) [14] A. Rebhan, private communication [15] S.R. Huggins, private communication [16] C.P. Burgess and G. Kunstatter, Mod. Phys. Lett. A2 (1987) 875 [17] E.S. Abers and B.W. Lee, Phys. Rep. 9C (1973) 1 [18] C.M. Hull, in Super field theories, ed. H.C. Lee (Plenum, New York, 1987). [19] D. Friedan, Ann. Phys. (NY) 163 (1983) 318 [20] P.S. Howe, G. Papadopoulos, and K.S. Stelle, The background field method and the non-linear sigma model, preprint (1987) [21] G. Kunstatter, in Field theory in two dimensions, ed. H.C. Lee and G. Kunstatter (World Scientific, Singapore), to be published