Coordinate-invariant regularization

ANNALS

OF PHYSICS

178,

272-312 (1987)

Coordinate-Invariant

Regularization*

M. B. HALPERN Lawrence Berkeley Laboratory and Departmenl University of Caiifornia, Berkeley, CatifDrnia

of Physics, 94720

Received April 28, 1987

A general phase-space framework for coordinate-invariant regularization is given. The development is geometric, with all regularization contained in regularized Dewitt superstructures on field deformations. Parallel development of invariant coordinate-space regularization is obtained by regularized functional integration of the momenta. As representative examples of the general formulation, the regularized general non-linear sigma model and regularized quantum gravity are discussed. 0 1987 Academnc Press. Inc.

1. INTRODUCTION The goal of the continuum regularization program [l-4] is uniform invariant non-perturbative continuum regularization across all quantum field theory. Successful applications have been given for the scalar prototype [2], gauge theory [3,5-S], gauge theory with fermions [9,10], and supersymmetric gauge theory [ 111. Recently, a coordinate-space framework for coordinate-invariant regularization [ 121 of theories with Dewitt’s measure [13] was announced, with introductory applications to the general non-linear sigma model and Euclidean gravity. This formulation is a geometric generalization of the previous regularizations of the program, which are seen as special cases in flat space and flat superspace [ 131. The present paper develops in some detail a still more general phase-space framework for coordinate-invariant regularization, which includes the coordinatespace formulation of Ref. [ 121 as a special case, after regularized integration [lo] of the momenta. The development is fundamentally geometric, with all regularization contained in regularized supervielbeins and supermetrics [ 133 on held deformations. Given certain covariant spacetime Laplacians, the phase-space formulation should provide non-perturbative invariant regularization for any functional integral with Liouville measure. Standard background gauge-field anomalies are expected as in Ref. [9]. Real-time Hamiltonian formulations, for example, canonical gravity [14], are naturally included. The applications men* This work was supported by the Director, Office of Energy Research, Oftice of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SFOOO98 and National Science Foundation under Research Grant PHY-85-15857. 272 OOO3-4916/87 $7.50 Copyright 0 1987 by Academic Press. Inc. All rights of reproductmn ,n any form reserved.

COORDINATE-INVARIANT

REGULARIZATION

273

tioned in [12] are expanded here, including a non-perturbative geometric characterization of the Weyl anomaly in the presence of the regulator for the general twodimensional non-linear sigma model. The phase-space approach also has a number of important technical advantages. In the first place, the formal covariant phase-space stochastic processes and Schwinger-Dyson systems with which I begin have no divergences in their formal structure. This is in contrast to formal covariant coordinate-space formulations, which are generally as singular as the coordinate-space measure itself. As a result, the regularization of the phase-space formulation proceeds according to a relatively straightforward geometric generalization of the first principles of the program. Another advantage is that the phase-space formulation involves a minimum of supergeometry. I learned most of what I know about this subject by watching it unfold from the simple phase-space processes. Moreover, conventional coordinate-space stochastic processes generally require a choice of stochastic calculus [ 151, while the phase-space processes are stochastically unambiguou.s, that is, insensitive to choice of stochastic calculus, as discussed below. Because of these technical complications at the coordinate-space level, I was originally unable to guess the form of the coordinate-space regularization, obtaining it first by regularized functional integration of the more general phase-space formulation in special cases, as developed below. Further technical advantages of the phase-space formulation, such as the phase-space stress-tensor, will also be encountered below. The coordinate-covariant generalization of the usual regulator function of the program R(d), with d an appropriate covariant spacetime Laplacian, appears linearly in the regularized supervielbein and quadratically in the regularized supermetric. The Laplacians of general relativity suffice for regularization of Einstein invariance. The Laplacian for the combined Einstein and reparametrization invariance of the general non-linear sigma model is also easily constructed, and I will remark on more general cases below. The heat-kernel [6] form of the regulator R(d) = exp(d//i’), with n the cutoff, is guaranteed to regularize any Euclidean theory. On the other hand. Euclidean-Minkowski rotation at finite cutoff, or direct Minkowski-space regularization, will require the original theory-dependent powerlaw regulators [l-3]. I shall generally suppress the details of gauge-fixing until it is necessary for the explicit example of Euclidean gravity. Zwanziger’s ghostless gauge-fixing [ 16, 171 is natural at the stochastic and Schwinger-Dyson levels, and an appendix is included which follows this approach generically through the developments of the text. In particular, Zwanziger’s gauge-fixing is applicable on unextended phase-spaces, but no difficulty is expected in applying the phase-space approach to the Grassmannextended Liouville measures of more conventional gauge-fixing [ 181. The Schwinger-Dyson bonus [ 121 for bottomless actions, conceptually independent of the regularization, is also not discussed explicitly until it is seen in Euclidean gravity. The stochastic formulations of this paper are expected to equilibrate as usual only when the action is properly bounded, although there are

274

M.

B. HALPERN

limited exceptions in the case of gravity. The Schwinger-Dyson formulations, on the other hand, are expected to be generally viable, in Euclidean or Minkowski space, even with a bottomless action. Two further applications of this paper are in preparation. Bern and I have completed a comparative perturbative study of the phase-space and coordinate-space regularizations [ 19) in the simple cases of the scalar prototype and gauge theory, explicitly confirming a number of the non-perturbative mechanisms discussed here. Moreover, Chan and I have completed a detailed perturbative study of coordinatespace regularized Euclidean gravity [ZO]. The weak-coupling expansion of the theory provides a representative example of a generic geometrization of regularized Schwinger-Dyson rules, the previous rules in the program being special cases in flat space and flat superspace. The structure of the geometrization will be broadly applicable in coordinate-invariant regularization. We have also applied the rules to complete a non-trivial one-loop explicit check of Einstein invariance of the regularization. The organization of this paper is as follows. Section 2 discusses the general formal phase-space functional integral of interest, introducing the relevant geometry and supergeometry. Section 3 gives the equivalent formal coordinate-covariant phase-space processes whose invariant regularization is discussed in Section 4. The derivation of the covariant regularized phase-space Schwinger-Dyson systems in Section 5 is designed to highlight the remarkable fact that the phase-space processes are stochastically unambiguous. Sections 6 and 7 work out further structure in the special case of the general theory with Dewitt measure. In particular, Section 6 discusses the regularized supergeometry of associated second-order coordinate-space processes, which are also stochastically unambiguous. Section 7 begins a study of moment relations for this class of theories. The technique and results of this section form the basis for the later sections on Weyl anomalies and regularized functional integration. Sections 8 and 9 are primarily a further specialization to the general nonlinear sigma model. The covariant spacetime Laplacians of the d-dimensional model are constructed explicitly in Section 8. With the help of the phase-space stress-tensor in Section 9, a non-perturbative geometric characterization of the general Weyl anomaly in the presence of the regulator is obtained in d = 2 dimensions. Invariant coordinate-space regularization is studied in the remaining Sections 10, 11, and 12. In particular, Section 10 performs the regularized functional integration in the case of the general theory with Dewitt measure, obtaining the regularized coordinate-space Schwinger-Dyson system of Ref. [12]. Section 11 is a guide to simple applications in coordinate-space, and includes the solution of a simple regularized background gravitational field problem. Finally, Section 12 applies the program to regularized and stabilized Euclidean gravity, including some details beyond those of Ref. [12]. Two technical appendices are also included. Appendix A traces generic Zwanziger gauge-fixing from phase-space to coordinate-space, while Appendix B gives equivalent covariant regularized coordinate-space Stratonovich processes.

COORDINATE-INVARIANT REGULARIZATION

275

2. PHASE-SPACE FUNCTIONAL INTEGRALS

In this section, I fix ideas and notation in a discussion of the general phase-space functional integral whose regularization is studied in succeeding sections. On a d-dimensional spacetime t”‘, introduce generic field coordinates d”“(t) and conjugate momenta n,,,,(t), with possible Einstein tensor indices included in the generic index M. The formal functional integral of interest is

where Liouville

measure is specified with the symplectic volume form (2.2)

I shall refer to the general phase-space action H as the Hamiltonian, since it plays the role of classical Hamiltonian in a classical partition function. The functional integral B possesses a formal reparametrization invariance under the group 9’ of local symplectic transformations (4, rc) + (&,7c) for which

and the Poisson bracket (2.4)

with (di;) = ddr is preserved. I focus on the subgroup G, of local point-transformations or held-coordinate diffeomorphisms 4”(t) = F”(&5)). Under G,, coordinate deformations and momenta transform respectively as contravariant and covariant vectors,

(2.5b)

and H is a G,-scalar, as in (2.3). The averages of the theory in the (4, n) frame are given by (2.6)

M.B.HALPERN

276

and G,-reparametrization

invariance implies for example that @>c~,, = W,,

(2.7)

for any G&-scalar F[$, it] = F[cj, rc]. Such statements, and G,-invariance of the S-matrix, require G,-invariant regularization in a general quantum field theory. Still more important are the true invariances of the theory, such as gaugeinvariance or the group G, of Einstein diffeomorphisms 5” =f”(<) in the presence of gravity, with Einstein metric g,,(t) and vielbein e,,(t). In manifestly G,-covariant formulations, the field-coordinate 4” is in general a Gs-tensor density of Gs-weight o = v, that is 4” -e” x Ge-tensor, with e = det”‘[ g,,]. I shall speak of @” as a GE-tensor only when its Gs-weight is zero. When both G, and G, are of interest, the situation is more involved. 1 consider only the subgroup of G, which preserves the Gs-rank but may change the Gs-weight of I$“. A v-frame is defined by a further subgroup of G, which maintains the Ge-weight v of @“, but, in general, G, may move among v-frames, as in the gravitational example 4” = g,, + 6” = gmn = e- ‘g,,,. In any such v-frame, the canonical structure of the theory implies that the pairs 64” - c?/&c~ and nM - SjSb” have o = v and 1 - v, respectively. I also introduce a number of G,- and Gs-covariant Dewitt [13] superstructures on field deformations. I intend to use these superstructures as convenient auxiliary quantities or covariant kernels [21], independent of the explicit form of H, to maintain manifest covariance of the alternative stochastic and Schwinger-Dyson formulations below. The ultralocal supermetric $MN with w = 1 - 2v defines a G,- and Gt-invariant inner-product on deformations (2.8a) (2.8b) while the inverse supermetric gMN, gMRgRN = SE, has o = 2v - 1. I will also need the associated supervielbein gMA with o = -v + $, gMN($(t))

~MAMO)

=

8MA(d(t))

=$

&‘A(d(t))

(2.9a)

(5) ENA(d(S)),

(2.9b)

and its inverse &‘MA, bMAgNA = St, with w = v - 1. It is easy to construct Gg-tensor superstructures for each v-frame by scaling out appropriate powers of e. For example, EMA s e 19 MA EM, 3 e- ‘12t: MA, (2.10a) G MN3e -9

MN,

G MN = egMN

(2.10b)

COORDINATE-INVARIANT

277

REGULARIZATION

are Gs-tensors in the v = 0 frame with 4”” a tensor. As an explicit example, the most general ultralocal supermetric for gravity [ 131 is $cJMN= gWt; TS= &mn; ?S

(2.1 la)

Gmm;rs = &( ,mr,ns + g”“g”‘) + yg”ng’”

(2.1 lb)

in the frame with g,, a tensor. I turn now to discuss some phase-space theories of particular simplest example is the G,-covariant and Gs-invariant Hamiltonian 1 H=(&) TC,,,,~~~TT~+ S[q5] 2s whose kinetic term involves the Dewitt momentum integration in (2.1) gives

inverse supermetric

interest. The

explicitly.

(2.12) A trivial

(2.13a) (2.13b) so this H is Dewitt’s supermetric prescription for the coordinate-space measure in a theory with action S. The general non-linear sigma model and Dewitt-measure quantum gravity are included in this class. The Hamiltonian (2.12) has also been studied in [22,23] as the “Gibbs average” corresponding to the coordinate-space formulation (2.13). Other cases of interest include real-time Hamiltonian and constrained Hamiltonian systems. As a representative example, consider canonical gravity [ 141 with @‘= (g,, A,), rcM = (rc”, rP), and H = -i

1

(d5)[d

8, g, - A, T”(n”,

g,)]

g mn: g,, goj = A;, g,, = /l;il' - A;.

(2.14a) (2.14b)

Here i, j are spatial indices and I have given the standard identification of the full metric g,, in (2.14b). This formulation is manifestly invariant only under the group GF of spatial Einstein diffeomorphisms: gii and 1, are Gc-tensors in the frame of (2.14), and general G,--weights are counted as above for G,. With (2.8b) and (2.14b), the components of the full auxiliary Dewitt supermetric (2.11) are easily obtained in the (gii, 1,) frame. As mentioned in the Introduction, Zwanziger’s ghostless gauge-fixing [16, 171, discussed generically in Appendix A, is natural on unextended phase-space at the stochastic and Schwinger-Dyson levels discussed below. A next step to Grassmannextended phase-spaces, incorporating the developments of Refs. [9-111, will allow

M.B.HALPERN

278 the study of more conventional BRST invariance.

ghost gauge-fixing

3. COORDINATE-COVARIANT

[ 181, including

regularization

of

PHASE-SPACE PROCESSES

The original stochastic process constructed by Langevin [24] in 1908 was in fact a phase-space process, but this formulation has been largely neglected in favor of the coordinate-space stochastic processes popularized by Parisi and Wu [25]. I am aware of two further studies in formal phase-space stochastic quantization [26] of simple systems, both of which were helpful in developing the formal coordinatecovariant phase-space processes of this section. I begin by stating the general formal phase-space process

(3.lb) (3.lc) whose covariance and formal equivalence to the general phase-space integral (2.1) will be checked below. Here yap is Gaussian noise; t is a Markov time with [t] = length; overdot is Markov time-derivative, and the positive parameter /?, with [fi] = inverse length, generally controls the rate of equilibration. As promised in Section 2, the Dewitt superstructures gMA and g,+,, appear in (3.1) as convenient auxikary quantities or covariant kernels [21], independent of the specific form of H. As a matter of orientation, I mention that the a=0 form of these equations has been studied with the Dewitt H Eq. (2.12) as the “microcanonical” formulation [27,23] of the quantum field theory (2.13). The addition of the noise and the viscous term --/K!JMN 6H/dx, at non-zero values of p improves thermalization at a practical level and provides the opportunity for continuum regularization below. For simplicity, I will assume in this and the following section that the phasespace process (3.1) is defined with Stratonovich calculus [ l&28,29, l-31, so that the formal rules of calculus including chain rule apply. I will however return after regularization to check in Section 5 that the process is insensitive to choice of stochastic calculus. Formal Gg-covariance of the process (3.1) is checked with essentially standard Hamiltonian methods. The chain-rule identities (3.2a)

COORDINATE-INVARIANT

REGULARIZATION

279

(3.2b) are easily obtained from the defining relations (2.5) of the general local pointtransformation. The G,-transformation of (3.1 b) (3.31

follows immediately from (3.2a), but the covariance of (3.la) is somewhat more involved. With FjA= qA, (3.4)

and Eqs. (3.2a) and (3.2b), the result

(3.5)

is obtained after some algebra. It follows that the phase-space process (3.1) is formally G,-covariant. I remark that the super tangent-space vector noise qa is a G,-scalar of weight 4 independent of v-frame, as required by (3.1~). When H is manifestly Gs-invariant, it follows that the G,-weights in any v-frame of the terms in (3. la) and (3.1 b) match, so the process (3.1) is G,-covariant. In particular, the equivalent form e -Ii M= -

p+

6H &P,

[ 1 M

- PGMN g

+ &EMA

N

(“) 6

(3.6a) (3.6b)

is manifestly GE-covariant in the frame where #J” ‘v 6/&P’ and [S/&#J”],. E e-‘S/&Y are G,-tensors. The same GF-weights are found and manifest Grinvariance of (3.6) is seen for canonical gravity with (2.14). I also briefly indicate the formal equivalence of the phase-space process (3.1) to the original theory (2.1). Detailed derivation after regularization will be given in Section 5.

280 The equivalent equilibrium,

M.B.HALPERN

phase-space

Schwinger-Dyson

(SD)

equations

at formal

(3.7) are easily obtained by standard methods [2,3] from the process (3.1). Here F[& rc] is a general functional at fixed Markov time and the bracket is Poisson bracket, defined in (2.4). With the transformations (3.2), it is easily checked that the SD operator L (0 = (LF)) is scalar under G,. Moreover, all the terms in L have G, (or Gc)-weight zero, so the SD equations are G, (or GF)-covariant, and manifestly G, (or G&invariant forms of L are easily found for each v-frame. The SD equations (3.7) are identities in the original theory (2.1) 0=

i

9co(eCH, F}

(3.8a) (3.8b)

verifying that the correct d-dimensional theory is formally obtained independent of the auxiliary superstructures and the parameter /?. Moreover, the auxiliary role of the Dewitt supermetric as a covariant kernel is clearly seen in (3.8b). I also note that the SD equations (3.7), derived with Stratonovich calculus, are in a characteristically Ito form [15, 30,63 with no field-derivatives of the superstructures. This is the key to the fact that the phase-space process (3.1) is stochastically unambiguous, but I prefer to discuss this point in Section 5 after regularization. For completeness, I finally mention the equivalent G, and G, or (Gr)-invariant phase-space Fokker-Planck equation

(3.9) which admits the formal equilibrium solution peel‘v exp( - H), as it should. Since the regularization scheme does not regularize the Fokker-Planck formulation [2], Eq. (3.9) will not be employed below.

4. PHASE-SPACE REGULARIZATION The fully coordinate-covariant phase-space process (3.1) was designed with no divergences in its formal structure. It follows that the process may be regularized by a geometric generalization of the first principles of the regularization program

COORDINATE-INVARIANT

[l&4], that is, by fully coordinate-covariant the /?-family’ of Markovian regularizations,

tiM(5)= -$

281

REGULARIZATION

displacement

of the noise. I propose

(5)- P%N(4(5)) g (5)+ fi 1 (d5’)a”,,,,&(r)

l”(4)=-p

(4.la) (4.lb)

M

(f?,4(5r1) rls(5’, f’)> = 26,4, et where the regularization

- 5’) &t - f)>

(4.lc)

is contained entirely in the regularized supervielbein c&; A$’- R(d”)MC;N5’~NNA(~(4’)).

(4.2)

Here the regulator R(A) is a function of abstract spacetime covariant Laplacian and I specify the particular matrix representation (A tw), = (&&i,%(5) mh4t :Nr’ E (d”$/ ,jd(t - 5’)

A,

(4.3a) (4.3b)

on objects 8, N nM which transform under G, and G, (or Gf) like the momenta. As checked below, it is a general rule that the regularization is coordinateinvariant under the covariances of A. The level of difficulty in explicit construction of such Laplacians depends on an interplay of G, and G,. For example, the Gs-covariant Laplacians A = g”“D, D, of general relativity, with Einstein connection P&J g), provide Gs-invariant regularization in a particular v-frame. The same A constructed with the full g,, of (2.14b) is adequate for regularization of canonical gravity as well. I call such Laplacians provisional, since it would be preferable to maintain manifest covariance under G, and G, simultaneously. The original gaugecovariant Laplacian employed in the regularization of gauge theory [ 1, 31 is provisional in the same sense. When such provisional GE-covariant Laplacians are employed, the checks of G,-covariance below do not apply. In the same way, G,-covariant Laplacians with superconnection are easily constructed in the absence of local symmetries. In Section 8, I construct the G,- and G,-covariant Laplacian of the general non-linear sigma model, which involves both connections. The construction of A in more general cases deserves further study. For example, I have not succeeded in developing Laplacians which are simultaneously covariant under G,- and G,-transformations which change the G;-weight of CJP, as in the gravitational example g,, = e - ‘g,,,,,. In this connection, I recall that existence of a covariant Laplacian does not preclude the usual anomalies in background gauge-field problems [9]: Standard axial and chiral anomalous Ward identities have been obtained explicitly in such models, and their Noether structure understood in terms of equivalent effective ’ Equation (4.1) defines a family of regularizations [3,6, lo] parametrized by 8. Such structure corresponds to latticeization ambiguities, and large /? is the simplest of the family, as seen below.

595/178/2-7

282

M. B. HALPERN

action formulations. The identities correspond to regularized Noether transformations, according to an a priori regularized version of Fujikawa’s argument [ 311, while the naive Noether and background gauge-field transformations involve unregularized quantities. The situation is apparently quite different for regularized dynamical gauge-fields coupled to non-anomalous [ 1,3] or potentially anomalous [32] symmetries, since all Green functions are finite and regularized effective action formulations do not exist [2]. Development of Ward-identity techniques directly at the regularized stochastic and SD levels is an important open problem. See also the related remarks in Appendix A. Given such a d, the regulator is constructed by matrix multiplication to satisfy R(A)-A 1, or R(&,,g;NC’ n’ (4.4a) a”, Sd(5 - C’) G.f<; Ai;‘n’

(4.4b)

&MA(&5)) sdtr - 5’)

in order to guarantee the formal large-cutoff correspondence of the regularization (4.1) to the unregularized process (3.1). As an example, both the matrix multiplication and the formal large-cutoff behavior are seen in the heat-kernel option R(&iNt’

= (e&A2)Mg~N5’=

~“(5 _ (‘)

(e&A2)hN

(4Sa)

7 SN, dd( 5 - fy ) + (d/A *)MC;Nc’ +; j- (d~“)(d”/n2),,‘p~“(~/~2)p~,,;N~‘+

. ..

(4Sb)

and similar structure is seen for power-law regulators R(d) = (1 - d//1*)-“. Given a G+-covariant d (4.6)

the G,-covariance ing steps:

of the regularized process is then verified with (4.3) in the follow-

(W,),Q

g

(5)

(4.7a) (4.7b) (4.7c)

(4.7d)

COORDINATE-INVARIANT

283

REGULARIZATION

The last transformation property (4.7e) of the regularized noise term, analogous to (3.4), guarantees that the G,-covariance proof of Section 3 goes through for the regularized process (4.1) as well. When H is manifestly G,-invariant, manifestly Gs-covariant forms of the regularized process can be found for each v-frame, in analogy with (3.6). I will illustrate for the frame where 4” is a Cc-tensor and vM - xM has G,-weight 1. It is helpful to construct the matrix elements of the Laplacian (4.8a)

(A VMu)C= (~&Yv(5) (A,)$ (dJMy

=e-‘(t)(dt)iN4S)

(4.8b)

E (As)/

(4.8~)

S,.tt, t’)=e-‘(S)(J)M,‘Nt’

(4.8d)

SAL 5’)-e--‘(w”(5’-5’)

on V,=e-’ 8,, which has zero weight in this frame. Here 6,((, t’) is covariant &function, and it is easy to check, in analogy with (4.7), that (dc)Mr’Nr’ transforms as a Gs-bitensor in this frame. Covariant multiplication in the construction of the corresponding manifestly covariant regulator R(dc)M5iN5’ = e-‘(t) n’

R(d)M,‘Ni’

(4.9a) (4.9b)

s,“, 6, (5, 5’)

preserves the G,-bitensor property of the regulator and the simple relation to R(d). In the case of the heat kernel, for example, (4.10a)

+ 4 1 (cl(?) e(~“)(d,/n2)M5’P:~(d~/n2)p5,.;Nt’+ both of these properties are easily seen. With R(d,.), the manifestly Gt-covariant this frame is obtained, eC’7iJt)

. ...

(4.10b

form of the regularized process (4.1) for

= -

(4.11b)

284

M.B.HALPERN

where I have defined the G,-tensor regularized supervielbein (4.12a)

Eit,g; A<’ 5 R(d,)Mc’““ENA(~(T’)) =e -‘(t) 7

(4.12b)

~~t~At;~e-“2(5’)

E/.44(S))

(4.12~)

S,.(L c;‘)

and checked correspondence of (4.11) with the formal process (3.6) in (4.12~). The forms (4.8)-(4.12) are also manifestly Ge-covariant for regularized canonical gravity in the corresponding (gii, A,) frame.

5. REGULARIZED PHASE-SPACE SCHWINGER-DYSON

SYSTEMS

Conventional coordinate-space processes of the Parisi-Wu type require a choice of stochastic calculus [ 151 when the noise is multiplied by a functional of the stochastic field. In contrast, the phase-space process (4.1) (and formally the process (3.1)) is unambiguous as it stands. This is best seen in a derivation of the regularized Schwinger-Dyson equations corresponding to the process. I will sketch the derivation with Stratonovich calculus, and then indicate why the same result is obtained with Ito calculus, or any other choice. The expression for the Markov time-development of any equal Markov-time functional F[f$, 7r]

(5.1) follows easily from the process (4.1). Standard manipulations [ 1, 21 with the Gaussian noise (4.1~) then allow the separation of the last term of (5.1) into twl distinct contraction types,

(5.2a

(5.2b) corresponding

respectively to contractions

of the noise with its own prefactor (here

COORDINATE-INVARIANT

285

REGULARIZATION

regularized supervielbein) in the process (4.1), and further contractions observable F. The explicit form of the noise-derivative

with the

(5.3b) is then obtained with the equal Markov-time

relations (5.4)

which follow directly from the process (4.1). The contraction types (5.2a) and (5.2b) are now easily evaluated. The regularized supervielbein is a function of the coordinates and not the momenta, so that

(dt)(dt’)& M &Lr:acl;a(r’) i =O.

(5.5)

This vanishing of the noise-prefactor contractions is the crucial observation for the uniqueness of the phase-space stochastic calculus, which I will discuss after obtaining the SD equations. Moreover, (5.6a) (56b) in which I have defined the regularized phase-space super-Laplacian the regularized supermetric

.A n in terms of

= I (&“I R(d),,.,t.

(5.7b)

‘Pr~R(~)~~,‘Qr”~fQ(~(~“))

n’ With (5.1) and (5.5~(5.6)

et

(5.7c)

- 5’) %4N(4(5)).

I obtain the superbasis-independent

evolution

equation (5.8)

286

M. B. HALPERN

and then, at formal equations

equilibrium,

the regularized

covariant

phase-space SD

provide a d-dimensional formulation of the regularization. With (5.7c), the formal large-n correspondence of the regularized system (5.9) to the formal system (3.7) is immediate. G,-covariance of the regularized SD system (5.9) is easily checked. From (4.7d), (56b) and (57a), the regularized super-Laplacian is a G,-scalar (510a)

.2ix=A.

(5.10b)

and so therefore is the regularized SD operator itself. Moreover, the formulation shows manifest G, (or Gf)-covariance in each v-frame. The form for the frame with 4” a tensor is

b2 mn=s (4 e(5)(&)45’)GA Mt;’N5’hN(5’)&,,,(5) G&; Nts =

‘PS”R(A,),,..Q5”G,p(1(4”)), s (dt;“) 45”) R(A,.JMt

where G&. N5f is a manifest G, (or Gf)-bitensor Uniqueness of Phase-Space

Stochastic

(51la) (5.1 lb)

in this frame.

Calculus

The regularized phase-space SD equations (5.9) were obtained with Stratonovich calculus, but, as at the formal level (3.7), they appear in the Ito form, with no lieldderivatives of the regularized superstructures. This fact is easily traced to the vanishing of the noise-prefactor contractions in (5.5) which in turn implies that phase-space processes of the type (3.1) or (4.1) are stochastically unambiguous, as long as the noise prefactor is not a function of the momenta: The general stochastic calculus multiplies the Stratonovich result for these contractions by a constant 0 < y < 1, where y = 1 is Stratonovich and y = 0 is Ito. Since the Stratonovich result is itself zero, the same result (5.8~(5.9) is obtained independent of the choice of stochastic calculus. In this sense, it is accurate to say that the phase-space stochastic calculus is unique. In the perturbative analysis of regularized conventional coordinate-space processes [ 1, 3,6], such noise-prefactor contractions have been called regulator vertex clusters with one incoming line (RVC,‘s). The gauge-invariant ambiguity 13, 63 in these contributions was traced to the equal Markov-time value of the Langevin-Green function, which involves evaluation of the step function e(t) at

287

COORDINATE-INVARIANTREGULARIZATION

t= 0: Stratonovich is the midpoint prescription e(O) = 4, while Ito sets e(O) = 0, which eliminates RVC ,‘s. Uniqueness of the phase-space stochastic calculus implies that RVC,‘s will be unambiguously suppressed in perturbative analysis of phase-space processes. Bern and I [19] have studied this phenomenon explicitly in simple models, confirming that the value of the relevant equal Markov-time Green functions are unambiguously zero. I shall be more explicit about this remark in the following section.

6. REGULARIZED

DEWITT

MEASURE:

SECOND-ORDER

PROCESSES

The coordinate-invariant phase-space regularization framework is now complete, but further structure may be seen in the case that H is a non-singular quadratic form in the momenta, which allows elimination of the momenta in favor of the generalized velocities. The resulting regularized covariant second-order2coordinatespace processes are also stochastically unambiguous, since they retain the phasespace (or second-order) stochastic calculus. As an example, I discuss these processes for the general theory with Dewitt measure. The general Dewitt-measure theory with coordinate-space action S, (6.1)

was given as an example in Section 2. The corresponding phase-space process (4.1)

form of the regularized

(6.2a)

lj”= ??“%N K25)

= j (4

allows elimination of the momenta covariant second-order process

(6.2b) (6.2~)

Gc; A5~~/,(5’) with

rcM =$,,,NdN,

and the regularized

(6.3a) (6.3b)

2 Simple formal second-order processes were also employed in Refs. [24,26].

288

M. B. HALPERN

is obtained after some algebra. The quantity L is the classical Lagrangian corresponding to the classical Hamiltonian H, and plays a major role in the Gibbs formulation [22, 231. More explicitly, the second-order process (6.3) may be written

& fj” E4” + f$.($q q@”= -pp + yMN --$+& in which the super-Christoffei

1

(6.4)

connection

(6.5)

makes its first explicit appearance to complete the G, and CC-covariant Markovtime derivative D/Dt. The presence of the momenta obviates the need for superconnection in the unregularized phase-space process, but I remind the reader that superconnection is generally present in the spacetime Laplacians of the regularization. The form SMNN$ of the regularized noise in (6.4) indicates a transition to a more natural spacetime Laplacian for coordinate-space regularization. This Laplacian is defined on objects V” = gMNPN = GMNV, which transform under G, and G, like coordinate deformations S4”,

tA ‘“)c - tA&!$ vN(t)

(6.6a)

= tA,)!$

(6.6b)

(A)“5;NC’

6% - t’)

=s”‘(~(r))(d”),,‘Q~‘~~~(~(~‘)).

(6.6c)

Here I have assumed that the supermetric is covariantly constant. This is true even with the provisional G,-covariant Laplacians of general relativity, since the relevant supermetrics are generally functions only of the Einstein metric, as in (2.11) and Section 12 below. The regulator on V”, (6.7a) (6.7b) (6.7~)

may also be constructed by ordinary matrix multiplication second-order process is most naturally expressed as

from (A)MS;Ny,. Then the

COORDINATE-INVARIANT

REGULARIZATION

289

(6.8a)

where I have defined the regularized inverse supervielbein dyAc

E R(A)Mt;N,JbNA(#(y)) n’

I””

er

- 5’)

(6.9a) (6.9b)

which is the natural geometric object for coordinate-space regularization. Go-covariance of the process (6.8) follows from (6.9c), while the formal unregularized form of the second-order process, with N:(C) +,, a”“(q+({)) AIL, is easily read from (6.9b). The corresponding explicitly Gs-covariant matrix elements and regulator on V’ are also useful.

(Ac)M5:Nc’=(At)?Nsc(t?tf’)=(A)M5;Nt_‘ep’(t’) =~“‘(~(5))(a)p~‘Q5’GQN(~(~‘)) =G”‘(~(5))(A~),r’Q5’GQN(~(~‘)) R(A,.)M5,N,r = R(A)MC,Nt.e-l((‘) = ~“‘kw) 7

R(A),, - ‘Q5’GQ,vv(i(5’))

s,M SAL 5’),

(6.10a) (6.10b) (6.10~) (6.10d) (6.10e) (6.10f)

and R(Ac)MS;Nef may be constructed from (dC)“5CNC, by covariant matrix multiplication, as in (4.10). These objects are manifest G*-bitensors when 4” is a G,-tensor. In that frame, the superconnection is a Gc-tensor as well, and the form

$$“=

-fi$“-GMN[$],+\ljiN: <

NY(t)=s(d(‘)e([‘)E:y’;Ai’m Jm EMCiA5’ E R(A,.)Mi;N~~ENA(~(<‘))

(6.1 la) (6.11b) (6.1 lc)

shows manifest Gr-covariance of the process. A geometric interpretation of the parameter p is available in terms of the Parisi-Wu Markov time t -B-‘t, with [r] = (length)2. The resealed noise

290

M.B.HALPERN

ijA = ,/$qA maintains (fa(z, 5) fa(r’, 5’)) = 2dA8 6(z -r’) processes (6.2) and (6.8) become B-In’,=

-+

p-‘pf’=

$ijMNn,

#‘(r - <‘), while

&M + iv”,

the

(6.12a) (6.12b)

and

respectively. The barred noise terms are the forms (6.2~) and (6.8b) with rla + tjA, and prime denotes r-derivative. It follows that /I controls the deviation (of the stochastic path in r) from motion along a supergeodesic. The right side of (6.13) also suggests that the phase-space and second-order formulations will be equivalent to a simpler and more conventional first-order Parisi-Wu coordinate-space regularization at large /I. I will make this observation precise with regularized functional integration of the momenta in Section 10. Bern and I [ 191 have studied the weak-coupling expansion of (6.13) in simple cases. The unambiguous elimination of the noise-prefactor contractions (RVC ,‘s), explicitly confirming the uniqueness of the phase-space (or second-order) stochastic calculus, corresponds to the generic fact that any retarded second-order Green function vanishes at equal z.

7. REGULARIZED DEWITT MEASURE: MOMENT RELATIONS FOR ALL ,fI

This section continues the discussion of the general theory with Dewitt measure, now at the d-dimensional SD level. In particular, I begin to study here a momentexpansion technique [lo] for obtaining non-perturbative information about the momenta, and hence about the general regularized Dewitt measure for all values of the parameter p. The techniques and results of this section form the basis of the applications in Sections 9 and 10. In particular, I will complete the moment expansion at large /I in Section 10. The Dewitt-measure SD equations (5.9),

O=( {Wf}-~~(&h,-$ +Pm,F> {F, H) =I (do [rcN(BMN-&

{$~~n~(;P?)+-$}&]f’,

(7.la) (7.lb)

hold for all F[& n], and the moment expansion of F is in powers of the momenta.

COORDINATE-INVARIANT

For example, the zeroth-moment choice immediately in the zeroth-moment relation

291

REGULARIZATION

F0[4, Z] = F[d]

in

(7.1)

results

(7.2)

which states that translation-invariant averages linear in the momentum Similarly, the tirst-moment choice F, [q5, TCJ = 5 (dc) 71~3~~ SF[d]/S#” the p-independent first-moment relation

are zero. results in

(7.3a) (7.3b)

after some algebra and use of the zeroth-moment relation (7.2) to eliminate the contribution of the viscous term. Here the supercovariant derivative D/DqbM makes its first appearance, and I remark that the entire momentum dependence of (7.3) is contained quadratically in the mixed-phase/coordinate-space super-Laplacian A’. The idea behind regularized functional integration [lo] of the momenta is to find enough of these moment relations so that averages involving the momenta can be expressed in terms of coordinate averages alone. Such an integration will be carried through completely at large /I in Section 10, and the relation (7.3) will then provide a purely coordinate-space SD regularization. A taste of the integration procedure is obtained for all p with the second-moment choice F, = ~TC,%~~X,. The result is the relation

((XM~MN7tN)e)= (2q,f,;M’>-BP’

((“M’v.““$)t)~

where (7Sa) (7.5b) is the invariant Eqs. (5.7b) and integrated form, theory and the Section 6.

trace of the regularized

supermetric. The form (7Sb) follows with (6.7a). The first term on the right of (7.4) is in momentumwhile the presence of the second term indicates that the regularized momentum integration are simpler at large /I, as mentioned in

M.B.HALPERN

292

In fact, the second term does not contribute

to the global form of (7.4)

(do n,%MNn, = (d<) Y&,‘M5 (7.6) > (s ) (1 due to the zeroth-moment relation (7.2). The same result is obtained directly with the mixed-moment choice FZ = H. The fully momentum-integrated relation (7.6) is a characterization of the general regularized Dewitt measure for all j?. Moreover, the relation (7.6) is seen in Section 9 to imply a nonperturbative geometric characterization of the Weyl anomaly in the presence of the regulator for the general (d= 2)-dimensional non-linear sigma model. In this connection, it will be useful to have the trace of the supermetric in the form C4$e;MC=e((i) J (dt’) e((‘) R(A,.)Mg’N’:‘R(At.)MtcNt.,

(7.7)

where Eqs. (4.9a) and (6.10d) have been employed. With (7.7), the result (7.6) is seen as manifestly Gc-invariant in the frame where 4”” is a tensor, as it should be. I finally note that A,. is a symmetric operator in the case of the (provisional) Gs-covariant Laplacians of general relativity. The further relations R(Ac)Mt’Nt’ = R(A,.)‘?,;

9uA45))

(7.8a)

N&MC iNc’ = R(A)Ni’,Mc

(7.8b)

W)PS,Q,~~QNW’))

(7.8~)

= W)Ni’;~t

%f5 ;M5=e(~)[R2(A,.)],,-;MS=e(~)[R2(A,)]M’;,5 = [R’(A)]%,, = [R2(&JMgiM5

(7.8d) (7.8e)

are then obtained with (4.9a), (6.7a), and (6.10d) in this case. 8. REGULARIZED GENERAL NON-LINEAR SIGMA MODEL: LAPLACIANS This section is primarily a further specialization to the general d-dimensional non-linear sigma model with Dewitt measure. Although both G,- and Gc-invariance must be maintained, construction of the spacetime Laplacians are particularly simple for the sigma model because one is conventionally interested only in a particular v-frame: $“, x~, and qA are G,-scalar fields with weights 0, 1, and f, respectively. The G,-covariant and GE-invariant Hamiltonian and action of the model are (8.la) s=ij %MN

(dt) gMNgmn a,$” &,,d” -t ...

= eGmv(d),

(8.lb) (8.1~)

COORDINATE-INVARIANT

293

REGULARIZATION

where g,,(c) is an arbitrary external gravitational field (metric on the base manifold), and the G,-scalar supermetric G,,,,(4) specifies an arbitrary internal (target) manifold. Other terms of interest may be added to the action, but it should be kept in mind that the Dewitt measure agrees with the canonical measure only in the absence of higher derivatives. I also note that the superconnection (6.5) is an explicit Gs-scalar, r~J%) = fES(G), in this case, while the Einstein connection Pm,(g) is scalar under G,. To complete the regularization, I need the explicit forms of the spacetime Laplacians which enter the regulator in the formulations above. I will first construct the Laplacian (A,)$ of Eq. (4.8) on objects VMmeP1rcM, which in this case are simultaneously Gs-scalar and a G,-covariant tensor of rank one. Because V, is G,-scalar, it is not difficult to verify that the covariant spacetime derivative A, on V,,,,, A, V, = DA% V, = W,,,,

(8.2a)

involves only the superconnection. This result is a higher-dimensional generalization of standard covariant differentiation of a rank-one covariant tensor along a curve. On the other hand, since W,, is simultaneously a G; and G,-covariant tensor of rank one, its covariant spacetime derivative (8.3a)

involves the Einstein connection as well. The fully covariant spacetime Laplacian A = g”“A,,A,, so

is

(8.4) are the desired matrix elements in this case. The other matrix elements of A required above are easily constructed from this result. For example, (&,c’Nc’

= e(t)(A,)Mc;N”’

(Ac)~t ;Nr’ = (A&” are read from regularization. 595/178/Z-8

(4.8~).

This

completes

(8Sa)

S,.(<, 5’)

the Laplacians

for

the phase-space

294

M. B. HALPERN

The other

matrix

elements

(d)“c;Ncz

and (dc)Me;NC;C on GE-scalars V”= regularization of Section 6 and Section 10 below, may now be constructed from (8.5), (6.6c), and (6.10). However, equivalent and simpler forms follow from (8.2) and (8.3), GMNV ,,, N S4”, relevant to the coordinate-space

A,V”=D$$VNz

W,M

(8.6a)

A,,, W,” = D,“,;; WY D:~E

(8.6b)

a,s;+QfN

(8.6c)

D,M,; = a, s:, SF - Pm, SE + s;pfN,

because the Gs-scalar superstructures are covariantly

(8.6d)

constant,3

A,G,N(~)=a,G,N-~~,GsN-G,s~~N

=amp

[

+GMN-

The results (8.6) were stated in Ref. [12]. (‘C)!N

complete the construction linear sigma model.

I-SRMG,, - GiUSriN Finally,

M.rgR. = (gm”D,mR

1

= 0.

(8.7)

the matrix elements rN

(8.8a)

)t

(AIMS;N<,= (Ac)MN Jd(t;-t’)

(8.8b)

(Ac)“‘;N<’ = (At;)?,, fi,(C, 5’)

(8.8c)

of the fully covariant

Laplacians

for the general non-

9. PHASE-SPACE STRESS-TENSOR: GENERAL WEYL ANOMALY

A consistent advantage of the phase-space formulation is that any nontrivial coordinate-space measure has been lifted into the phase-space action H. The natural definition of the stress-tensor 19,” as the response of any matter Hamiltonian to a general deformation of the Einstein metric

(9.1) will therefore include the response of the measure as well. This phase-space stresstensor is particularly valuable in the study of anomalies, which are generally measure effects. 3 Covariant constancy of the full supermetric d,,,g,,, = aigNgRp,, = 0 follows as usual (see, e.g., (4.8b)) by definition with a:%,., = eAz&:se-‘, where A, G,, E A~~S,,GRs is given in (8.7).

COORDINATE-INVARIANT

REGULARIZATION

The phase-space stress-tensor of the general non-linear

29.5

sigma model (9.2a)

e0 = eOmngmn= -2g,,

-” kl,

+_drc gMNn 2 M N

follows from (8.1). The second terms on the right of Eqs. (9.2a) and (9.2b) are the contribution of the measure. As an example, I will apply the form (9.2b) to obtain a non-perturbative geometric characterization of the Weyl anomaly in the presence of the regulator for any (d= 2)-dimensional classically Weyl-invariant non-linear sigma model. When d = 2, the first term in (9.2b) is also the response of the classical action to a Weyl deformation, since the matter field has vanishing Weyl weight in this case. The theory is completely regularized, so averages involving the action term are truly zero and any non-zero trace of the stress-tensor e%= 7c,9MN7tN

(9.3)

comes from the measure term. The average of the right side of (9.3) was studied for the general Dewitt-measure theory in Section 7. The general non-perturbative form of the Weyl anomaly in the presence of the regulator (9.4a)

(i(d~)ee)=(1(4)1$&tiM()

=

(d<) e(<)(dt’)

e(5’)(edc’n2)Mg.Nt’(eddn2 ) Me;Ni.)

t9.4c)

then follows as a special case of the more general relations (7.5)-(7.7). The first form (9.4a) shows an entirely geometric interpretation of the general anomaly as the invariant trace of the regularized supermetric, while I have chosen the heatkernel form of the regulator in (9.4b) and (9.4~) to facilitate comparison with conventional results. The local form of the anomaly (e(5) 0(g)) = ( %$e;Mt) will be obtained at large b in Section 10. The non-perturbative form (9.4) is consistent with known results in conventional heat-kernel regularization of (one-loop) background-field problems. As a simple check, consider the flat D-dimensional internal (target) manifold ‘S,,N = e6,,, Then

296

M. B. J-JALPERN

the spacetime Laplacians of Section 8 reduce to ordinary Gr-covariant d(g) on scalars and scalar densities, and the simple forms4 (AyTNg

= (AJNp

(A,),,,=

Laplacians

= 6fj(A,.),,f

(9Sa) (9Sb)

A, S,.(t, t’) = (A,)ty

A, z [e-‘a,(egmn WA, I,+,<:Nt’ = R(AJN?,,,<

a,,)]<

(9.k)

= 6f,(eAdn2)55,

may be used directly in (7.8d). In this case, both the form of the anomaly presence of the regulator and the large ,4 result

(9Sd) in the

(9.6a)

D(e

A’D D 2AJnz)CS= g, + 24n R(g) + U( A - ‘)

(R(g) is Einstein curvature scalar) are those obtained by Alvarez [33] in a discussion of the Polyakov string. The general non-perturbative characterization (9.4) (or its local form in Section 10) invites further analysis on arbitrary target manifolds with heat-kernel and background-held methods. Further applications in the non-linear sigma model are discussed in Section 11.

10. INTEGRATION

OF THE MOMENTA:

COORDINATE-SPACE

REGULARIZAT~ON

In this section, I study regularized integration [lo] of the momenta for the general theory with De Witt measure, beginning with the SD equations (7.1) and, in particular, the implied first-moment relation (7.3). The technique is a completion at large fl of the moment expansion begun in Section 7, and variants of the technique should suffice to integrate any non-singular H[&, n] which is no more than quadratic in the momenta. As in the case of regularized Grassmann integration [lo], integration at finite b seems prohibitively complex. The first step in the integration is to recognize the large-p relation (10.1) denoted by sub-zero, which follows from the regularized SD Eqs. (7.1) under the assumption that averages do not grow at large fi. The assumed “no-growth 4 The other Laplacians are (d)“c,NS, = 6: A,,. and (d”)MC;N” = age(c) dCi.em’(c’) = 6: A,,,, where = d, 8’(5 -r’) = (A,)li,e(l’). In fact, the heat-kernel expansion (9.6b) is performed more easily on the alternate form [R2(A)]“tzMI = D(ez“ln’)SC (= De(t)(&‘““’ )& of (7.8e), which is constructed by ordinary matrix multiplication from A,,.. A<(,

COORDINATE-INVARIANT

REGULARIZATION

297

theorem” is expected to hold as in [lo], since the formal unregularized theory is independent of 8. Indeed, (10.1) is a regularized version of the formal statement (3.8b), which is true in the unregularized theory. Moreover, Bern and I [19] have explicitly verified the no-growth theorem to all orders in the phase-space regularized scalar prototype and gauge theory. The desired large-b relations expressing the momenta in terms of the coordinates are now obtained by a systematic moment expansion of (10.1). For example, the first two moments F, = n,,,,(c) F”[d] and F, = ran ~~(5~) F”“[$] result in the relations (10.2a)

(~MM(OF”Cdl >Kl,=o (x,+,(5,) nd52) F”“C41 >co,= CC!,,,; ,<,F”“Cdl >to,.

(102b)

It should be noted that the large-a result (10.2a) is stronger than the finite-~ relation (7.2), while the large-/3 result (10.2b) is consistent with the finite-/3 relation (7.4). In particular, the second-moment relation (10.2b) gives ((~M~MNTd;)(“)=

wf;.MihO,.

(10.3)

so the local form of the Weyl anomaly in Section 9 (40

@(l)),o, = IO)

(10.4)

follows at large j3 as promised. The general moment

F,,=n,,(5,)...n,~(5,)F”’

~~“““Cdl

(10.5)

in (10.1) yields the family of relations (10.6a)

(10.6b) in which the contraction symbol is defined to be the regularized supermetric. These relations are easily solved by iteration from (10.2a) and (10.2b), and the result packaged as the large-p generating functional (e~(d~)JM(y)n~(C)F[~])(0)=

(e(1/2)5(d;)(d5’,JM(~)~~~,~g.JN(5’)F[~])10,

(10.7)

for arbitrary F[c$] and source J”. The result (10.7) states that the momenta are Gaussian at large & with the regularized supermetric as two-point contraction. The last step in the derivation of a pure coordinate-space SD regularization is the application of the large-p results (10.2b) or (10.7) to eliminate the quadratic

298

M. B. HALPERN

momentum dependence in the a-independent first-moment relation (7.3), as promised in Section 7. The mixed-phase/coordinate-space super-Laplacian A ’ in (7.3b) is then naturally expressed in terms of the regularized inverse supermetric cqf~;N~’ = 9?M’(fj(<))

(10.8a)

Y;zt,Qcr3QN(q5(y))

(10.8b) (10.8~) (10.8d) where (10.8b) and (10.8~) are verified from (5.7b), (6.7a), and (6.9a). It also follows with (6.9~) and (6.10d) that

%~5’Nc’ =

i

(dt”)

e({“)

(10.9b)

R(d,.)“‘,,,,,R(d,.)Nr’;Qt;,,GPQ(~(S”)),

so the regularized inverse supermetric is a G,-bitensor and a CC-bitensor when 4” is a tensor. The final form of Eq. (7.3) at large b is the invariant regularized coordinate-space SD system (lO.lOa) (lO.lOb) (lO.lOc) which was announced in Ref. [12]. G4-covariance and CC-invariance of the regularized SD operator L, and in particular of the regularized coordinate-space super-Laplacian A in (lO.tOc), are immediate consequences of (10.9). The coordinate-space SD equations (10.10) are a regularized version of the formal identities

O=@P~(dO&exp(-S)

asMN(d(t;))

6F ~~~(5)

1 ?

(10.11)

which are true at the functional-integral level for the general theory with Dewitt measure (2.13). The continuing role of SMN as a convenient covariant kernel in these relations is clear. Due to explicit divergences in (10.1 1), and the other com-

COORDINATE-INVARIANT

REGULARIZATION

299

plications sketched in the introduction, I was originally unable to guess the corresponding covariant regularized form (10.10). With the hindsight of the result, however, the coordinate-space regularization rule FP”(&C))

S”([ - 4’) -+ c!Jyf+t

S”“(qs(()) a”“(&())

P(O) + FJy;N;

ip(( - <‘) + &WA~’ A

(10.12a) (10.12b) (10.12c)

may be used to attain regularization of more general coordinate-space SD systems and stochastic processes. In particular, Chan and I [20] have used this rule to regularize gravity with an arbitrary power-law Euclidean measure. The absence of field-derivatives of regularized superstructures in the n of (10.10) is a direct consequence of the phase-space stochastic calculus, which unambiguously eliminated such terms (RVC,‘s) at all values of a. On the other hand, this y = 0 form [3, 63 of h is usually associated with Ito calculus in conventional Parisi-Wu stochastic processes. For completeness, I also give the regularized first-order Ito stochastic process

(VA<, 1)rlB(5’f)) =26/l, Ht - 5’) &I - f)

(10.13b)

associated to the SD Eqs. (10.10) under the assumption of equilibration. The resemblance of this first-order Ito process to the apparent large-/? form of the second-order process (6.13) is intriguing but elusive, since the latter dictates its own (phase-space or second-order) stochastic calculus. After this derivation from regularized phase-space, I realized that Eq. (10.13) is a regularization, according to the rule (10.12), of an explicitly divergent formal process mentioned by Rumpf [ 341. Stratonovich processes equivalent to the SD Eqs. (10.10) are given in Appendix B. 11. COORDINATE-SPACE

REGULARIZAT~ON:

SIMPLE APPLICATIONS

As in the case of regularized Grassmann integration [lo], the simplicity of the momentum integration above reflects a diagrammatic simplification of the phasespace regularization at large p [ 191. The coordinate-space regularization (( 10.10) and (10.13)) captures the large-b simplicity, and will generally be preferable for perturbative studies of regularized theories with Dewitt measure. This section is intended as a guide to simple applications in coordinate-space regularization. In the first place, the general coordinate-space regularization ((10.10) and (10.13)) contains the previous regularizations of the program as special cases in flat space and flat superspace. Consider for example the simplest non-linear sigma

300

M. B. HALPERN

model with no external gravitational field on a flat D-dimensional internal manifold with gMN= 6,,. Then rE,,,= PMN= 0 and the regularized inverse supermetric is simply g,Mr; NS’= [R*( Cl )lee, dMN, where A = 0 = ~?,a, is ordinary flat-space Laplacian. The SD operator (10.10) reduces to

which is the regularized scalar prototype [2] with D components. Similarly, the regularization of flat-space gauge theory is obtained with @“=A;, qA =qt, &MA = gMN = 6,” dab, and &~~;Ag’ = [R(A)];;.

&,

where A is gauge-covariant Laplacian (10.13) and the SD operato? (10.10)

Cc?yf; NC’= [R*(A)]$.

[S]. The resulting

hp,,,

(11.2)

forms of the process

(11.3a)

are in The qP’ in c?,,,,~=

correspondence with Ito calculus as expected [3,6]. simplest example of Gs-invariant regularization is the case of D scalar fields a d-dimensional gravitational field. This is the non-linear sigma model with & 6,, and A,=A,+

. ...

(11.4)

where the scalar Laplacian A, is given in (9.5~) and the dots indicate possible conformal terms when d # 2. The resulting forms of (10.10) and (10.13) are (11.5) and

(11.6) 5 Although no stochastic equivalent is available, first-order SD systems L(5) 5 -WW(t) +J (dS’)[R*(d)],,. S/S&t’) or 0 = (L;(5) flA] > J (dr’)[R’(d)]~~.6/64~(5’) may also be investigated.

such as 0 = (L(r) F[(] ) with with L;(5) = -6S/6A;(C) +

COORDINATE-INVARIANT

REGULARIZATION

301

where the covariant matrix elements (dc)55, are given in (9.5). Regularized backgrounddeld problems such as this are exactly soluble [9]. For example, the choice F= 4”(t) #N(t’) in (11.6) yields immediately

<4”(5) C(T)> =s”“C(--A,.)--‘R2(d,.)155,

(I 1.7)

with (AcJtEr = A, SAC, 5’1, and higher correlators follow by Wick’s theorem. It is instructive to reobtain the (d=2)-dimensional Weyl anomaly (9.6) in this coordinate-space formulation. The phase-space stress-tensor (9.2) included the response of the measure, and, following Hawking [35], I can find an analogous coordinate-space trick in this simple case. The supermetric for the variables 4” = & q5” is 6,,, so the coordinate-space stress-tensor (11.8a) (11.8b) includes the response of the measure in the second term of (11.8b). The theory with in d = 2 dimensions, so that

A, = d, is classically Weyl-invariant

(e(t)> = (d”(SN-4)

d”(5)> ==~C~2(a;~

(11.9)

is obtained with (11.7) from the measure term alone. This is the local form of the Weyl anomaly (9.6), corresponding to the general local large-8 result (10.4). I emphasize that it is not necessary to find the corresponding coordinate-space stress-tensor on non-trivial target manifolds: Independent of its derivation from the phase-space stress tensor, the general local Weyl anomaly (10.4) is completely momentum-integrated, and may be further analyzed in coordinate-space with the regularized general non-linear sigma model in the coordinate-space form (10.10). The simplest laboratory for the perturbative study of G+-invariant regularization is the special case of the non-linear sigma model with no gravitational field and gMA=a#Alapf, which describes the variable change from Cartesian 4” to curvilinear 4”. Beginning with the free theory and a simple $(4), it should not be difficult to verify (2.7) explicitly through one loop for a simple F[q5]. Explicit one-loop G,-invariance of S-matrix elements should also be studied here first. Another simple application in the sigma model is one-loop verification of vanishing Goldstone boson mass on target manifolds M,* = [SU(n), x sU(n),]/SU(n).. 12. REGULARIZED

EUCLIDEAN

GRAVITY

As a final example of invariant coordinate-space regularization, I apply the general regularized SD Eqs. (10.10) to the case of d-dimensional Euclidean gravity with De Witt measure. There are three conceptual issues to address before the analysis.

302

M. B. HALPERN

Existence of quantum gravity. Perturbative renormalization-group theory suggests that non-renormalizable theories are at best large-distance approximations to some renormalizable theory. Whether or not this is true non-perturbatively, regularization of quantum gravity underscores the fact that the program has reached the point where any theory may be continuum-regularized. Measure. Dewitt’s measure is controversial for gravity, its leading competitors being the canonical measure [ 14, 361 and other power-law Euclidean measures [37]. The regularization program shows no prejudice in such matters, and Dewitt measure is studied here only as a particularly elegant example: Regularization of canonical gravity has been discussed above at the phase-space level, and general power-law Euclidean measures will be investigated in some detail along with Dewitt measure in [20]. Stability. gravity

Gibbons et al. [38] have argued that the action of Euclidean-Einstein

S= C2

s

(dt) eR,

iC=JGZ,

(12.1)

with R the curvature scalar, is bottomless due to the Weyl mode,6 and that the correct Euclidean integration for gravity must therefore involve a complex g,, contour in some directions. As differential formulations, SD systems bypass such considerations, giving directly the correct Euclidean result, at least in weak coupling, in complete agreement with the prescription of Gibbons et al. Although the vielbein 4” =e,, may also be studied, I will discuss the metric (6” = g,, (and its G, = G, transforms) as field-coordinate. The explicit form of the regularized SD system (10.10) for gravity with any S is (12.2a)

0 = (~~Cg,,l>

( 12.2b)

(12.2c) Here gde;rs

6 An explicitly

Gc-invariant

form

of the unstable

I

Weyl

a yde;mn ah, mode

-- a gmn;rs agde

can be exhibited

(12.3a)

I

in weak

coupling

[20].

COORDINATE-INVARIANT

REGULARIZATION

303

R(A),,S;Pq5”R(A),,r’;dey”~pq;de( g(5”)) (12.3b) c”r:ms=s(cl(“) = s (dt") e([") R(A,),,C(;Pqe"R(Ac),s5,dee"Gpq;de( g(5"))

(12.3~)

are respectively G,-tensor and Gs-bitensor as indicated when g,, is a Gr-tensor, and the regulator is taken as heat kernel. I have also added a Gs-gauge-fixing term z grm of the Zwanziger type [ 16, 171, discussed generically in Appendix A, whose form for tensor g,, is [39] 9; g,, = D,Z, + D,Z,. Continuing in the frame with tensor g,,, the most general ultralocal superstructures on deformations of the metric are

Qmn = e II2Emn ub ab 2

FPncrs- eGmn’ Is,

~m41,,;,.~=e~1Gmn;rs

(12.4a)

EE = i (e;e; + e;ez) + cg”” dub

(12.4b)

G”“; ” _- -; ( ,mgm + gm.TgN’)+ yg”ng”

(12.4~) (12.4d)

&‘z&‘:sg and y = c(cd+ 2). It follows that positivity of the supermetric (real supervielbein) requires y > -l/d. With (12.4), the superconnection (12.3a) is easily expressed in manifestly Gr-tensor form [ZO]. To complete the regularization, a choice of spacetime Laplacian A is required. As noted above, the ordinary spacetime Laplacians A = D”D, of general relativity provide Gs-invariant regularization for gravitational interactions in any particular v-frame. For example, with g,, a tensor, the ordinary Laplacian on second-rank symmetric Cc-tensors P’,,, where

ymn;rs

=

(D’D, ~A, = (A,),,-‘,,(~) A mnr?-StE (At),,- S”(< - c’) (A,.),,,;‘““’ = (A),,,~“~‘epl([‘) R(A,.),,,;‘“5’

(12.5a) (12.5b) (12.5~)

= R(A),,Si’SS’e-‘(~‘),

(125d)

will guarantee Gs-invariance of the SD system (12.2) in this frame. The explicit form of these matrix elements will be given in [20]. Similarly, with the Laplacian on second-rank tensors with weight 1. g mn+gmn=e-‘gmnv (A),,,;“~’

-+ (~)mn5;n~’ = e -‘(5)(AcL,,‘“C45)

dd(t - 5’11,

(12.6)

will suffice for Gs-invariant regularization in this frame. Manifestly G,- and G,covariant Laplacians for gravity have not yet been constructed. I will continue here

304

M. B. HALPERN

with the provisional G,-invariant Laplacian (12.5) and tensor g,, as the preferred variable. A convenient gauge choice for computations is (12.7) where g,, = 6,,,, + rch,, and h = h,,. The SD Eqs. (12.2a)-( 12.2c) may then be analyzed to any desired order in K according to a geometric generalization [20] of previous regularized SD diagrammatic rules [2, 3, 6, S-101. Using these rules, Chan and I will give an explicit one-loop check of Gs-invariance of the regularization in Ref. [20). Here I emphasize the viability of the system with respect to the stability issue. The linearized form of the SD operator (12.2b) with the Einstein action (12.1) is IC~L’~’ =

5

(dt;) A,,,,: ,,h,, &

mn

mn: rs %,(t’)

where 0 is ordinary flat-space Laplacian, is

hm(5)

(12’8)

and the action plus gauge-fixing matrix A

in terms of the tranverse projector T,,,,. The linearized SD equations 0 = (L(O)F) may be solved with the breakup h,, = Em, + d- ’ b,,h into the traceless part and the trace. For example, the choice F= h”,,(r) h”,,(<‘) in the linearized equations gives (12.10) immediately, and quadratic choices involving the trace give coupled equations soluble in terms of (12.10). The final result is the free regularized graviton propagator [ 121

e20/“2

x2

[ 1 -

- 0

et’)

(12.11)

COORDINATE-INVARIANT

REGULARIZATION

305

and the free regularized n-point functions may be constructed from this result by Wick’s theorem. Only the first term of ( 12.11) contributes to the transverse graviton, and this term is the standard Feynman gauge result, regularized after Euclidean rotation. The Feynman term is exact with the supermetric parameter choice y = -t, and although positivity of the supermetric is lost with this choice, perturbative analysis is greatly simplified [20]. In particular, as seen in Eqs. (12.8) and (12.9), the SD solid line factors [2, 31 are trivially obtainable when y = - 4. In spite of the Gc-invariant unstable Weyl mode, the SD system (12.2) has given a (regularized) propagator which is gauge-equivalent to the correct Euclidean perturbation theory for all values of the supermetric parameter y. It is instructive to see how the SD formulation manages to do this in a simpler example. The SD system for a single degree of freedom x with Boltzmann factor exp[ --S(x)] is O=(-S’F+F’),

(12.12)

where prime denotes x-derivative. These equations are exactly and uniquely soluble for S(X) = ax2/2, although the unstable case a < 0 no longer corresponds to a welldefined partition function, as in Euclidean gravity. For example, choose F= x’/2 in ( 12.12) to obtain immediately o= ( -m2+ which is the prescription

1 ),

(12.13)

of Gibbons et al. [38] when a -C0. The general solution T[J]

= (eJr) = e”/‘”

(12.14)

is easily obtained from the Schwinger form [Z, 3] of (12.12). In fact, this is the mechanism by which the SD Eqs. (12.2) produce the correct result (12.11) for most values of y (including, for example, d> 2 and positive supermetric y > -l/d), although the gauge and basis-fixed form of the stochastic formulation (10.13) fails to equilibrate. There is however, a range of y corresponding to negative supermetric for which the correct result (12.11) may also be obtained in another way. The mechanism is easiest seen from the value 1mn,Jl/2 of the action plus gauge-fixing matrix (12.9) at y = - $. The non-positive eigenvalues of the matrix in this case means that the negative supermetric kernel in the drift term -5+& Ts&S/6g, has effectively stabilized the Weyl mode, and the corresponding form of the process (10.13) exhibits weak-coupling equilibration. Explicit construction of the Zwanziger gaugefixed Langevin-Green function G = (a, - A) ~ ‘, G,,;,,(k,

t - t’) = @(t - 2’) eek2(‘-“)i2

],,;,,

6 +e

T

[e-“*L(d- ‘)(r-f’W - 11 , (12.15)

306

M. B. HALPERN

shows effective stabilization window

and stochastic equilibration

il(d-l)+l>O-i

d>2: -l
in the negative supermetric

-f.

(12.16)

I emphasize however that the SD Eqs. (12.2) give the correct result (12.11) for all y, independent of such considerations. After completion of this work, I learned that Rumpf [34] had previously observed the stabilizing effect of negative supermetric in a stochastic context. His result for the allowed range of y in d = 4 dimensions, computed without gauge-fixing, is in agreement with (12.16). In summary, it seems that the Euclidean functional integral formulation of gravity is down by two strikes: We do not yet know the non-perturbative integration contour [40], and the regularization program does not regularize the functional integral formulation [2]. This suggests that the simultaneous SD stabilization and regularization of gravity may be more than a coincidence. In any case, the SD Eqs. (12.2a)-(12.2c) are viable, and will be studied elsewhere [20] in some detail.

APPENDIX

A: GENERIC ZWANZIGER

GAUGE-FIXING

Zwanziger’s non-perturbative ghostless gauge-fixing [ 163 in d dimensions is equivalent to a more or less ordinary Faddeev-Popov “flow” [17] gauge-fixing’ in d + 1 dimensions. As a result, conventional (d + 1)-dimensional flow-gauge Slavnov-Taylor identities [43] and BRST structure are readily available for Zwanziger gauge-fixing. Derivation of non-perturbative’ Ward identities directly at the regularized stochastic or SD levels is an important open problem. The generic d-dimensional form of the gauge-fixing is the addition at the stochastic or SD level of a Lie derivative in the vertical direction on the principal bundle, that is a gauge transformation. I will briefly follow such generic gaugefixing through the developments of the text. When the Hamiltonian H[& n] has a local symmetry (such as G,, G,-, or Yang-Mills gauge-invariance), the phase-space process (3.1) or its regularized forms (4.1) and (6.2) may be gauge-fixed by the addition of the gauge transformations fi-‘9=7rM and fi-‘9=#” to the right side of the 7i., and $” equations, respectively. Here dc: denotes Lie derivative in the Z-direction, as in the Gr-gauge’ The class of flow gauges is ghostless, infrared-soft, and, with proper boundary conditions, Gribovcopy free. Particular flow gauges have been independently rediscovered by a number of authors: The usual Minkowski rotate of the Euclidean flow gauge a = -V*/i, * q*/1,5 = co (sharp) is the Minkowskispace ghostless gauge of Cheng and Tsai [41], while Steiner’s gauge [42] is the Minkowski flow gauge a + -V’/a. * See, however, Refs. [44,2, 3, 5, 7-10) for explicit d-dimensional Ward identities at the one-loop level.

COORDINATE-INVARIANT

307

REGULARIZATION

fixing example 9: g,, = D,Z, + D,Z, for the metric tensor of gravity. The result at the level of the SD equations (3.7) or the regularized forms (5.9) and (7.1) is the addition of the term

1

to the SD operator, where G generates the gauge transformation with parameter Z. The gauge-fixing term (A.l) annihilates on gauge-invariant quantities and effectively commutes with the rest of the regularized SD operator, so gauge-invariant quantities will be independent of the gauge-fixing [ l&2]. In general, Z may be chosen as a function of nM and &“‘, but, in order to simplify the elimination of the momenta in Section 6 and the regularized functional integration of Section 7, I will assume that Z(d) is a function of the coordinates alone. The extra gauge-fixing terms in the phase-space process (6.2) and in particular the extra term in the canonical momenta n,=!YMN(~N-Ij-

‘2yN),

(A.21

straightforwardly generate a number of extra O(B-‘) terms in the second-order processes of Section 6. The example of gauge theory will be given in [ 191. With the same moment choices in Section 7, there are additional gauge-fixing terms in some of the moment relations. The zeroth-moment relation (7.2) becomes

which corresponds canonical momenta

to the additional gauge-fixing term in the definition (6.2b) or (A.2), and the extra terms

6 (&)(Z 4”) -+p-‘G sdM

s

(dt)q,CciMN7

* FCd1 w I >

of the

(A.4)

are found on the right side of the first-moment relation (7.3a). I call attention to the first term in (A.4), which is order fi”. This (/?I/?) term arises from the (p) viscous term of (7.1) in combination with the (fl-‘) second term of (A.3a), and will provide the expected p-independent coordinate-space gauge-fixing at large /I. The right side of the local second-moment relation (7.4) picks up the extra gaugefixing terms

(A.51

M. B.HALPERN

308

which vanish on (d{) integration because H is gauge-invariant. As a result, the global gauge-invariant statement (7.6) requires no corrections. Finally, the Zwanziger terms are not involved in the large-8 results (lO.l)-( 10.4) and (10.7). It follows that the coordinate-space SD operator (lO.lOb) picks up only the expected coordinate-space gauge-fixing term ~(&)(5?#“)~/8&“‘, which is the first term in (A.4).

APPENDIX

B: COORDINATE-SPACE STRATONOVKH EQUIVALENTS

The regularized super-Laplacian

m of the coordinate-space

SD Eqs. (10.10) (B.la)

is in y = 0 [3,6] form, with no derivatives of the regularized superstructures, so the Ito process (10.13) is the most economical stochastic equivalent. The system also has Stratonovich equivalents which are regularized field-theoretic versions of the processes constructed in Refs. [28, 291. Additionally, the Stratonovich approach naturally generates the coordinate-covariant generalizations of the regularized (y = 1) Stratonovich structures originally reported for gauge theory Cl, 31. The simplest Stratonovich equivalent of (B.1) is the G,- and G,-covariant but “basis-dependent” [28, 291 process PC0 FM(()

+ FM(t) = -~MN(4(r))

= 9?~~;QT$,(~(q@)))

z+ s4”(r)

+ j (d<‘)(d<“)

j (dt’) cY,M’;~~‘~/,(~‘)

(B.2a)

El(~‘;A~“&&~-~”

(B.2b)

in which the extra second term of FM (relative to the Ito process (10.13)) generates RVC, counterterms [6] in the Langevin trees to cancel the Stratonovich RVC, loops. I also note that the structure F M involves a regularized version of the superspin-connection WMAB, since FM(c)

2 hd(0)

W-Q~&(2?)

&MA + 4fNa a4N

L =

WMAB=

-ad(O) -&‘A

as expected in analogy with [28, 291.

5

(B.3a) (B.3b)

IMAbNBW~~,

a 7 a; + r;,(9) [ a4

1

sp

1

(B.3c)

COORDINATE-INVARIANT

REGULARIZATION

309

A completely covariant and basis-independent Stratonovich equivalent can also be constructed. Begin with the regularized field-theoretic version of the covariant process of [29], (B.4)

where

(B.5) is the super-parallel-transport matrix in terms of the super-spin-connection matrices The path in (8.5) runs from a fixed reference field 4,,(l) to an (WMU)AB= WMAB. initial condition d( <, t, ) = 4 ,( <), and then from #1(() along the actual stochastic path. The anti-path ordering symbol PP indicates that the factors are ordered from left to right along the path. Following [29], useful identities are (B.6a)

where the retarded property of the process (B.4) has been used in both steps. Then standard methods yield the SD Eqs. (B.la) and (B.1 b) but with L& replaced by the Stratonovich (y = 1) form (B.7a) =

where

I

(dt’)(di;“)

R(LI)~“‘.~,,

$&

[ +“h%‘))

310

M. B. HALPERN

are respectively the forms of supercovariant derivative on super-tangent-space vectors TB and on G,-covariant vectors VQ. The regularized Stratonovich process (B.4) and the associated regularized y = 1 super-Laplacian m s in (B.7) are the coordinate-covariant generalizations of the original Stratonovich structures reported for gauge theory [ 1, 33. I remark also that the process (B.4) is the only fully covariant coordinate-space process I have discovered with no divergence in its formal structure, so this process also is regularized according to the geometric generalization of the first principles of the program. As in the case of gauge theory [3,6], the right side of the covariant process (B.4) may be modified with a covariant RVCl counterterm

(B.9a)

- S”(t’- t”) R(d)MS;Q<*rg.dd(S”)) to obtain correspondence

(B.9b)

with the m of (B.lb).

ACKNOWLEDGMENTS I thank 0. Alvarez, K. Bardakci, Z. Bern, H. S. Chan, A. Gonzalez-Arroyo, H. Hiiffel, C. Kounnas, N. Marcus, A. Mufioz-Sudupe, H. Neuberger, A. Niemi, D. Roekaerts, A. Sagnotti, J. Sakamoto, H. Sonoda, H. Stapp, P. Windey, and B. Zumino for helpful discussions.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Z. BERN, M. B. HALPERN, L. SADUN, AND C. TAUBES, Phys. Left. B 165 (1985), 151. Z. BERN, M. B. HALPERN, L. SADUN, AND C. TAUBES, Nuci. Phys. B 284 (1987), 1. Z. BERN, M. B. HALPERN, L. SAWN, AND C. TAUBES, Nucl. Phys B 284 (1987), 35. M. B. HALPERN, in “Proceedings of the Symposium on Topological and Geometric Methods in Field Theory, Espoo, 1986,” World, Cleveland, 1986. Z. BERN, M. B. HALPERN, AND L. SADUN, Nucl. Phys. B 284 (1987), 92. Z. BERN, M. B. HALPERN, AND N. G. KALIVAS, Phys. Reu. D 35 (1987), 753. Z. BERN, M. B. HALPERN, AND L. SADUN, “Continuum Regularization of Quantum Field Theory. IV. Langevin Renormalization,” LBL-21960, UCB-PTH-86/26, Z. Phys. C, in press. L. SADUN, “Continuum Regularization of Quantum Field Theory. V. Schwinger-Dyson Renormalization,” Harvard University preprint, December 1986. Z. Phys. C., in press. Z. BERN, H. S. CHAN, AND M. B. HALPERN, Z. Phys. C 33 (1986). 77. 2. BERN, H. S. CHAN, AND M. B. HALPERN, Z. Phys. C 34 (1987), 267.

COORDINATE-INVARIANT

REGULARIZATION

311

KALIVAS, “Continuum Regularization of Supertield Supersymmetry,” Phys. Rev. D 36 (1987), 1210. 12. M. B. HALPERN, Phys. Lett. B 185 (1987) 111. 13. B. S. DE WITT, J. Math. Phys. 3 (1962), 1073; B. S. DE WITT, in “General Relativity” (S. W. Hawking and W. Israel, Eds.), Cambridge Univ. Press, Cambridge, 1979; B. S. DE WITT, in “Recent Developments in Gravitation” (M. Levy and S. Deser, Eds.), Plenum, New York, 1979. 14. R. ARNOWITT. S. DESER, AND C. W. MISNER, in “Gravitation. an Introduction to Current Research” (L. Witten, Ed.), Wiley, New York, 1962; K. KUCHAR, in “Quantum Gravity. 2. A Second Oxford Symposium” (C. J. Isham, R. Penrose, and D. W. Sciama Eds.), Oxford Univ, Press (Clarendon), London/New York, 198 1. 15. R. E. MORTENSEN. J. Stat. Ph-vs. 1, No. 2 (1969), 271. 16. D. ZWANZIGER. Nucl. Phys. B 192 (1981), 259; L. BAULIEU AND D. ZWANZIGER, Nucl. Phys. B 193 (1981). 163; D. ZWANZIGER, Phys. Left. B 114 (1982), 337; D. ZWANZIGER. Nucl. Phys. B 209 (1982). 336; E. G. FLORATOS AND J. ILIOPOULOS, Nucl. Phys. B 214 (1983) 392; E. G. FLORATOS, J. ILIOPOULOS, AND D. ZWANZIGER, Nucl. Phys. B 241 (1984). 221; E. SEILER.MPI-PAE/PTh 20/84, p. 259, published in Schladming School, 1984. 17. H. S. CHAN AND M. B. HALPERN, Phvs. Rev. D 33 ( 1986). 540. 18. C. BECCHI. A. ROUET. AND R. STORA, Phys. Lett. B 52 (1974). 344; E. S. FRADKIN AND G. A. VILKOVISKY, Phys. Lett. B 55 (1975), 224; Preprint CERN-TH-2332 (1977); Lett. Nuovo Cimento 13 (1975) 187; 1. A. BATALIN AND G. A. VILKOVISKY. Phys. Lett. B 69 (1977) 309; E. S. FRAUKIN AND T. E. FRADKINA, Phys. Lea. B 72 (1978), 343; M. HENNEAUX, Phys. Rep. 126 (1985), 1. 19. Z. BERN AND M. B. HALPERN, “Diagrammatic Methods in Phase-Space Regularization,” in preparation. Quantum Gravity,” LBL-22986, 20. H.S. CHAN AND M. B. HALPERN, “Continuum-Regularized UCB-PTH-87/8. Zeitschrift Phys. C, in press. 21. B. SAKITA, in “The Seventh Johns Hopkins Workshop” G. Domokos and S. Kovesi-Domokos. Eds.), World. Cleveland, 1983; K. ISHIKAWA, Nucl. Phys. B 241 (1984) 589. 22. V. DE ALFARO. S. FUBINI, AND G. FURLAN, Nuovo Cimerzto A 74 (1983). 365. 23. M. B. HALPERN. Nucl. Phys. B 254 (1985), 603. 24. P. LANGEVIN. C. R. Acad. Sci. Paris 146 (1908), 530. 25. G. PARISI AND Wu YONG-SHI, Sci. Sin. 24 (1981) 483. 26. S. RYANG. T. SAITO, AND K. SHIGEMOTO, Prog. Theor. Phys. 73 (1985). 1295; A. M. HOROWITZ, Phys. Lett. B 156 (1985). 89. 27. D. J. E. CALLAWAY AND A. RAHMAN, Phys. Rev. Lett. 49 (1982), 613; Phys. Rev. D 28 (1983), 1506. 28. R. GRAHAM, Z. Phys. B 26 (1977). 397. 29. M. CLAUDSON AND M. B. HALPERN, Phys. Rev. D 31 (1985) 3310; Ann. Phys (N.Y.) 166 (1986), 33. 30. R. GRAHAM, Phys. Lett. A 109 (1985), 209; S. CARACCIOLO, H.-C. REN, AND Y.-S. Wu, Nucl. Phys. B 260 (1985). 381; G. G. BATROUNI, H. KAWAI, AND P. ROSSI, J. Math. Phys. 27 (1986). 1646. 31. K. FUJIKAWA, Phvs. Rev. Lett. 42 (1979) 1195; Phys. Rev. D 21 (1980), 28481 32. R. JACKIW AND R. RAJARAMAN, Phys. Rev. Lett. 54 (1985), 1219; A. J. NIEMI AND G. W. SEMENOFF, Phys. Rev. Lett. 55 (1985), 927, 2627; L. D. FADDEEV AND S. L. SHATASHVILI, Phys. Lett. B 167 (1986) 225. 33. 0. ALVAREZ, Nucl. Phys. B 216 (1983), 125. 34. H. RUMPF, Phyx Rev. D 33 (1986). 942. 35. S. W. HAWKING, Commun. Math. Phys. 55 (1977) 133. 36. E. S. FRADKIN AND I. V. TYUTIN, Phys. Rev. D 2 ( 1970), 2841; E. S. FRADKIN AND G. A. VILKOVISKY, Phys. Rev. D 8 (1973). 4241. 37. C. W. MISNER, Rev. Mod. Phys. 29 (1957) 497: L. D. FADDEEVAND V. N. POPOV, Soviet Phys. Usp. 16 (1974) 777. 38. G. W. GIBBONS, S. W. HAWKING, AND M. J. PERRY, Nucl. Phys. B 138 (1978), 141; S. W. HAWKING, in “General Relativity,” (S. W. Hawking and W. Israel, Eds.), Cambridge Univ. Press, Cambridge, 1979. 11. N.

312

M. B. HALPERN

39. J. SAKAMOTO,

Prog.

Theor.

Phys. 70 (1983),

1424; T. FUKAI

AND K. OKANO,

Prog.

Theor.

Phys. 73

(1985), 790. 40. I. BENGTSWN AND H. H~~FFEL, Phys. Left. B 176 (1986), 391. 41. H. CHENG AND E. C. TSAI, Phys. Reo. D 34 (1986), 3858. 42. F. STEINER, Phys. Left. B 173 (1986), 321. 43. A. MuAoz SUDUPE AND R. F. ALVAREZ-ESTRADA, “Flow-Gauge

Zwanziger’s Gauge-Fixing,” 44. A. Mufioz

Slavnov-Taylor

Identities for

LBL preprint, April 1987.

SUDUPE AND R. F. ALVAREZ-ESTRADA,

Phys. Left.

B

164 (1985), 102; 166 (1986), 186.