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Schwinger-Dyson formulation of coordinate-invariant regularization
Volume 185, number 1,2
PHYSICS LETTERSB
12 February 1987
SCHWINGER-DYSON FORMULATION OF COORDINATE-INVARIANT REGULARIZATION "~ M.B. HALPERN LawrenceBerkeley Laboratoryand Department of Physics, Universityof California, Berkeley, CA 94720, USA Received22 September1986
Geometricinterpretationof the continuumregularizationprogramis found in a Schwinger-Dysonformulationof coordinateinvariant regularization.Introductoryapplication of the generalformulationis givenfor the regularizednon-linearsigma model and regularizedeuclideangravity.A Schwinger-Dysonbonus, conceptuallyindependentof the regularization,is noted in the case of euclideangravity:Due to the differentialcharacterof the formulation,difficultiesusually associated with unbounded action are bypassed,at least in weakcoupling.
Principles for invariant non-perturbative continuum regularization of any quantum field theory have recently been given [ 1-3 ], with explicit application s to the scalar prototype [ 2 ], gauge theory [ 1,3-5 ] and gauge theory with fermions [ 6,7 ]. An introductory discussion of the program may be found in ref. [ 8 ]. In this letter I report a general framework for coordinate-invariant regularization, with introductory applications to the non-linear sigma model and Einstein gravity. Further details, explicit computations and alternate formulations will be given elsewhere [9-11 ]. For a field theory in d dimensions, the regularization scheme generally ad.rnits two equivalent formulations in terms of a regulator R(A) which is a function of an appropriate covariant spacetime laplacian A. At the level of ( d + 1 )-dimensional regularized Langevin [ 12] systems, R(A) is a covariant displacement of the noise, while at the level of d-dimensional regularized Schwinger-Dyson (SD) equations R (A)enters in a covariant regularizedfunctional laplacian. Heat kernel regularization [ 5 ] R = exp (A/A 2) is guaranteed to regularize any euclidean theory to all orders, and for safety in new theories, I will assume such regularization here. It should be born in mind however that dimension-dependent power-law regularization [2,3], which presumably allows euclidean-Minkowski rotation at finite cutoff, may also succeed as in previous cases, even for non-renormalizable theories. Advantages of the SD equations include formulation in physical spacetime, without regard for stochastic equilibration or choice of stochastic calculus. In particular, the SD equations are applicable in cases where naive action formulations are unbounded, such as euclidean gravity [ 13 ] below. The SD scheme reported here is a regularization of De Witt's supermetric prescription [ 14] for the measure, which is equivalent to the canonical measure for the non-linear sigma model, but controversial for gravity [ 15 ]. Although the De Witt measure is particularly elegant and easy to control, other gravitational measures may be regularized as well. In particular, the coordinate-space SD framework given here may be viewed as a special case in a more general phase-space regularization [ 9 ], after integration [ 7 ] of the regularized momenta. The general scheme also allows the direct regularization of Minkowski-space canonical formulations if desired. Moreover, other gravitational measures may be treated [ 10 ] by tinkering with the coordinate-space SD equations presented here. The large-cutoff unitarity of various regularized gravitational measures is therefore open ~r This work was supported by the Director, Officeof EnergyResearch, Officeof High Energyand Nuclear Physics, Division of High Energy Physics of the US Department of Energyunder contract DE-AC03-76SF00098and the National ScienceFoundation under Research Grant No. PHY-85-15857. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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12 February 1987
for future study. Coordinate-covariant SD equations are best introduced in a finite-dimensional context, where regularization is not needed. In this case, the SD equations are equivalent to more familiar action formulations, which I review first. Consider a connected riemannian manifold without boundary and coordinates x m. Coordinate deformations 8 x m are contravariant vectors under the group Gx of coordinate diffeomorphisms 2"~ =fm (x), while the metric gm, (x ) on deformations
118xll2 = g m . ( X ) S X m S X ~
(1)
is a covariant tensor. A G~-invariant measure d w = d w is constructed as do)-= and finally a partition function
(dx)e(x)
Z = J (do)) e x p [ - S ( x ) ]
=-detl/2[gmn(X)]~[pdXP ,
(2)
is given in terms of an action S(x), defined as a Gx-scalar S(2) In general, one is interested in computing averages
=S(x).
(F(x) ) =Z -1 j (do~) exp[ -S(x)]F(x),
(3)
for any F(x) in coordinate system x m, and a simple statement of reparametrization invariance is that (9°(0e) ) = (5P(x) ) for any Gx-scalar b °. Statements of this kind, trivially verifiable by change of variable in a finite-dimensional context, require coordinate-invariant regularization in a general quantum field theory. The averages of the theory are equivalently described ,i by the set of Gx-covariant SD equations 0=(LF(x)),
L=--gmnOnSOm+A,
A=gmnD~O,~=-g'~[On~-Frn,,,(g)]Or,
(4)
in which the SD operator L involves the inverse metric gm,, the Gx-covariant derivative Dn, the Christoffel connection F~m of gmn, and the Gx-covariant (scalar) laplacian A. These equations may be obtained directly from the identities 0=
f (do))e-lO,n[exp(-S)egmnO,,F]
(5)
at the action level, or as the adjoint of the covariant Fokker-Planck equation [ 16] for the Gx-scalar Fokker-Planck density at equilibrium. Further study of covariant stochastic formulations is found in refs. [ 17,18 ]. Except that regularization is generally required, the same prescription is easily followed in quantum field theory. On a d-dimensional spacetime ~m, let #u(~) be generic fields with possible spacetime (Einstein) tensor indices included in the index M. The formal partition function of interest is
Z= f ~exp(-S[q)]),
~O')N~)~[~)]~ITI~(detll2[~RX(O(~))]Hd~(~)m'
(6)
in terms of the action functional S[~] and an ultralocal supermetric [ 14] N:~N(~({)) on deformations of the field coordinates
1180112=f
(d{)~M,V(O({))Sg~M(g)8~({).
(7)
As in the finite-dimensional case, the measure is formally invariant and the action a formal scalar under the group G o of local field diffeomorphisms ~M(~) = FM(~(~)), n Weak coupling SD expansions are unique when supplementedwith the usual permutation symmetryof the Green functions as a boundary condition [2,3]. 112
Volume 185, number 1,2
PHYSICS LETTERS B
0~ ~6M(O=O--7 ( ~ ) ~ ( 0 ,
12 February 1987
~,, O0 R , . , 0 ~ s ~M~(~.~) =o--$-zt¢~ ~ (0 c¢~(~(O).
(8)
Ideally, the regularization should respect G¢-covariance, as well as other invariances which may be present in the theory, such as gauge invariance, or invariance under the group G¢ of Einstein diffeomorphisms ~ f m(~). For this purpose, I propose the regularized covariant SD equations 0= (LF[~]),
8S 5~M(~) 8 ~-A, L = - f (d~) ffMN(~(~)) 6#v(~)
D 6 A--- f (d~X)(d~') ~A Me; N¢' _DON(~, _ _ ) _5¢M(~) ---
(d~)(d~') NAMe;re'
~_~a(~_~,)r~e
(N(~)))
50R(~) '
in which F[~] is any field-functional, (¢tu is the inverse supermetric, and the super-Christoffel connection F~M( f~)=½ f~RL
(oo
o
o
f~ML+0--~ f~UL- - ~ -
~NM
)
(10)
appears in the functional covariant derivative D / D # v. The regularization, which is contained entirely in the regularized functional laplacian A, appears as a regularized (inverse) supermetric
~A M¢;N¢'-~ J ( d ~ " ) R ( A ) M C ; p e " R ( A ) N ¢ ' ; a¢" ffeQ(O(~")),
(11)
in which the regulator R (A) is a function of an appropriate spacetime laplacian a on tensors VM which transform like 5 OM. Ideally, A should be chosen to be covariant with respect to G o, G¢ and so on. With such a A, the matrix construction (A{) vM(~) ~-. (A¢).~VN(~),
AM¢;u,'~' ~ ( ~ ¢ ) . ~ d ( ~ _ ~t),
(12)
then guarantees that fgaMe; U¢' is a bitensor and A is a scalar under all covariances of A. It follows that the regularization is coordinate-invariant. In the formal unregularized limit RM¢;N¢' --* 3~ad(¢--¢'), A
s M¢;N¢' ~ f~MN(0(¢))ad(¢__¢,),
(13)
A
and the SD equations (9) correspond to the formal identities 0= f ~ o ) ~-1 f ( d , ) ~
8
(exp(-S)¢
8F f(/v~v(0(,))~-~)
(14)
at the action level, in analogy with eq. (5). It would be misleading however to leave the impression that I guessed the regularized form of A in eq. (9), or that its form is unique, although I believe it is the simplest possible, since it is in the y = 0 (Ito) form [ 3,5,19 ], with no field-derivatives of the regulator. In fact, I originally obtained the SD equations (9) directly from the regularized phase-space formulation [9] at equilibrium, by regularized integration [ 7 ] of the momenta. Remarkably, there is no ambiguity in the phase-space stochastic calculus [ 9,11 ], and the resulting coordinate-space SD equations are always in the form usually associated with Ito calculus in conventional coordinate-space stochastic processes. For completeness, I also give the associated markovian-regularized Ito stochastic process 113
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12 Februaxy 1987
f 8S ~jM(~) + ~am; seF~s(f¢(q~(~))) = _ ~MN(¢(~)) ~---7~7~, O9~ tg) ~ + J (d~')ga Me; a¢, qa(~'), (r/a(~, t)qs( ~', t') ) =2~asOa( ~-~')J( t-t'),
(15)
in which ga Me;A¢' is a regularized (inverse) supervielbein, ~,M~_ gMa gNA gA Me;ae' ~R(A)Me;Ne'gNA(~(~')) , ~a Me;Ne'-'-~ J (d~U)ga Me;A¢" ga Ne';A+e"
(16)
This system is a regularized form of an equation mentioned by Rumpf [ 18 ], and is equivalent to the SD equations (9) under assumption of equilibration. The covariant regularized SD equations (9) are a more fundamental result, however, since they may be applied to any theory with formal partition function eq. (6), including Minkowski theories, without regard for equilibration or boundedness of the action. In what follows, I briefly discuss the application of the general regularized formulations eqs. (9) and (15 ) in a number of cases of interest. In the general d-dimensional non-linear sigma model, ¢W(~) is a set of G~-scalar fields with G¢- and Ge-scalar action
S= 2 f ( d~) (~MNgmn(grnOMOnON+ ....
(17)
in which gm,(~) is an arbitrary external gravitational field and (~MN=e(~)GMN(~J(~))specifies the internal manifold. Fully invariant regularization is obtained when the G¢- and Ge-covariant derivatives
am V M - [ Omrg + ( OmO")rfN( ~)] Vu ----W~, A n W M ~ On WMm--I'rnrn(g) WrM + (OnON)I"~R(~) WRm,
(l 8)
are employed to construct the G~- and Ge-covariant laplacian AVM=gmnA,A,~VM. In the particular case of no external gravitational field and flat supermetric (ffMN=I~MN, w e have A= [] =_Omdm, ~ ; N&= [R2(I--q)]e&C~MN,and the regularized formulations eqs. (9) and (15) reduce to those of the scalar prototype [ 2]. The case of no gravitational field and gMA = 0~)'4]0~)M is the simplest laboratory for the explicit study of reparametrization invariance under variable change from cartesian 0 a to curvilinear ~M. As an application of the regularized formulation in ref. [ 9 ], I will obtain a non-perturbative characterization of the general sigma-model Weyl anomaly in d= 2 dimensions: The general anomaly in the presence of the regulator is the average of the invariant trace of the regularized supermetric, a form which is amenable to further analysis by heat-kernel methods at large cutoff. The original regularization of gauge theory [1,3 ] is also easily obtained as a special case of eq. (9). With (~M(~)---*A~(~) and flat supermetric ~MN'-"~(~lO(~ab, the regularized inverse supermetric becomes fgaMe; N¢' __.[R 2(A) ] ~, du~, where A is a gauge-covariant laplacian [ 1,3 ]. The system eq. (9) is then recognized as the ~ =0 (Ito) form of the SD equations of refs. [3,5]. I finally turn to d-dimensional regularized gravity. Although the vielbein may also be studied, I will choose here the metric q~M(~)--*gmn(~) as field coordinate. The most general ultralocal supermetric on deformations of the metric [ 14 ] is ~mn; rs = e[ ½(gmrg,S +g~g'O +rgmng rs] (19) where positivity (existence of real supervielbein) requires y > - 1/d. The explicit form of the regularized SD equations (9) is
114
Volume 185, number 1,2
O= A=
I
=
L-
PHYSICS LETTERSB
f (d¢)
12 February 1987
mn; rs ~Sgrs,]SS'~--Sgm.8 + A , D
8
(d~)(d~') ff~m.e;r~¢. Dg~,(~') 8gmn(~)
~d(~--~')/'a~n; •gab({)
8g,7(g)Sgm.({)
(d~)(d~')~mn#;rs¢,
(20)
'
where
(19 (ffpq;rs ..~ O.~ (ffpq;rnn 19 (finn;m) I~at~;rs=½ ~ffab;pq O~mn Ogrs -ag,,---~q '
(21a)
ffa¢; ~*¢'= f (d~,,)R(A)m.¢ab¢,, (gab;m(g({,,))R(A)rs¢,;pq¢,',
(21b)
are both G¢-tensors as indicated. I have also added a G¢ gauge-fixing term ~*gm. = DmZ. + D.Zm of the Zwanziger [20,21 ] type, whose generic form is a Lie derivative in the vertical direction on the principle bundle. For exploratory purposes, the simplest choice of A is the ordinary G¢-covariant A = DmDm on covariant secondrank symmetric tensors. This Choice guarantees G¢-invariance of the regularization, but locks in the metric as the fundamental variable. With heat-kernel regulator, the SD equations (20) regularize any De Witt-measure euclidean quantum geometry with classical action S, but I speciaUize in what follows to euclidean Einstein gravity S=f(d{)eR/x z, with R the curvature scalar and x = ~ . For computational purposes, it is convenient to choose the gauge Zm = ( O.h.m - Omh/2 ) / ( 2x) where g,,,. = 8m. + x h,.. and h = hm.,. The SD equations (20) may then be analyzed to any desired order in t¢ according to standard regularized SD methods [ 2,3,5,7 ]. Chart and I have completed an explicit one-loop check of G¢- invariance [ 10], which we will report elsewhere, along with a discussion of other gravitational measures. Here I confine myself to some simple remarks. The linearized form of the SD operator equation (20) is 1 +2g /¢2L'°)=1 f (d,)([[]hmn-~-l---~(~mn(OrOshrs-[' ]h))8h,~, ~
+ f (d{)(d~')(exp(21-1/A2))¢¢, ( 1 -
lm.; r,-- ½(a.,ra., + amid.r),
Yl-~dy)
82
(22a)
,~.; rs 8h~s(~')Sh.,,,({) '
Tin.;.- am.dr,.
(22b)
Choosing F= hm.(¢~)l&,(~2) in the linearized SD equations, the free regularized graviton propagator is obtained
+
T
1 ~d
1 +2y
mn;rs 3 - d + 2 y
6mn-~--+&,
[] /_1\
-D
,tee,'
(23)
and the free regularized n-point functions follow from this result by Wick's theorem. Only the first term of eq. (23) 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 choice y = - 1/2, and although positivity of the supermetric is lost, perturbative analysis is greatly simplified in this case [ 10]. In particular, as seen in eq. (22), the tensor structure of the SD solid-line factors [2,3] at 7 = - 1/2 is simply lm,; ~s. In the case of euclidean gravity, the usual functional integral eq. (6) is ill-defined due to the unstable confor115
Volume 185, number 1,2
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22 February 1987
mal m o d e [ 13 ], a n d is not the correct c o n t i n u a t i o n o f t h e M i n k o w s k i integral. I a m taking the SD equations as fundamental, a n d the result eq. (23) shows that the unregularized SD equations have p r o v i d e d the correct euclidean rotation. F o r d > 2 a n d positive supermetric (7 > - 1/d), the stochastic f o r m u l a t i o n eq. (15 ) does not equilibrate, while the successful SD t r e a t m e n t o f the unstable m o d e (trace) in eq. (22) is equivalent to the following toy model: F o r a single degree o f f r e e d o m x with action S ( x ) , the SD equations ( 4 ) are 0 = ( - S ' F ' +F" ), where p r i m e denotes x-derivative. Choosing F=x2/2 a n d ( u n b o u n d e d ) action S = - x a / 2 , one obtains i m m e d i a t e l y 0 = ( x 2 + 1 ) , which is the prescription o f G i b b o n s , H a w k i n g a n d Perry [ 13 ]. F o r d > 2 a n d non-positive supermetric (~ < - 1/d), the c o m b i n a t i o n ~¢,*n;rs5 S/Sgrs effectively stabilizes the mode, in qualitative agreement with a final r e m a r k by R u m p f [ 18 ] in a stochastic context. The p o i n t o f Gibbons, H a w k i n g a n d Perry [ 13 ] is that a correctly c o n t i n u e d euclidean functional integration for gravity m u s t involve a complex g,,n-contour in some directions, a n d related ideas have been studied in ref. [ 22 ]. As a differential formulation, the SD equations allow us to c o m p u t e while leaving the question o f euclidean integration contour to a later stage. In rough analogy with the Schr6dinger equation for scattering versus b o u n d state regions, differential equations are often used to effect analytic continuations in this way. I would like to t h a n k O. Alvarez, K. Bardakci, Z. Bern, H.S. Chan, H. Htiffel, C. Kounnas, N. Marcus, H. Neuberger, A. Niemi, A. Sagnotti, H. Sonoda, H. Stapp, P. W i n d e y a n d B. Z u m i n o for helpful discussions.
References [ 1] Z. Bern, M.B. Halpern, L. Sadun and C. Taubes, Phys. Lett. B 165 (1985) 151. [2] Z. Bern, M.B. Halpern, L. Sadun and C. Taubes, Nucl. Phys. 284 (1987) 1. [3] Z. Bern, M.B. Halpern, L. Sadun and C. Taubes, Nucl. Phys. 284 (1987) 35. [4] Z. Bern, M.B. Halpern and L. Sadun, Nuel. Phys. 284 (1987) 92. [5] Z. Bern, M.B. Halpern and N.G. Kalivas, Heat kernel regularization of gauge theory, preprint LBL-21286 UCB-PTH-86/6, to be published in Phys. Rev. D. [ 6 ] Z. Bern, H.S. Chan and M.B. Halpern, Continuum regularization of gauge theory with fermions, preprint LBL-21552 UCB-PTH86/14, to be published in Z. Phys. C. [7] Z. Bern, H.S. Chan and M.B. Halpern, Non-Grassmann formulation 0f regularized gauge theory with fermions, prerint LBL-21786 UCB-PTH-86/17. [8] M.B. Halpern, in: Proc. Symp. on Topological and geometric methods in field theory (Espoo, 1986) (World Scientific, Singapore, 1986). [ 9 ] M.B. Halpern, Coordinate-invariant regularization, in preparation. [ 10] H.S. Chan and M.B. Halpern, Regularized quantum gravity, in preparation. [ 11 ] Z. Bern and M.B. Halpern, Phase-space versus coordinate-space reguiarizations, in preparation. [ 12] P. Langevin, C.R. Acad. Sei. Paris 146 (1908) 530; G. Parisi and Y.-S. Wu, Sci. Sin. 24 (1981) 483, [13] G.W. Gibbons, S.W. Hawking and M.J. Perry, Nucl. Phys. B 138 (1978) 141; S.W. Hawking, in: General relativity, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge, 1979). [ 14] B.S. De Witt, J. Math. Phys. 3 (1962) 1073; in: General relativity, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge, 1979); in" Recent developments in gravitation, eds. M. IAvy and S. Deser (Plenum, New York, 1979). [15] C.W. Misner, Rev. Mod. Phys. 29 (1957) 497; E.S. Fradkin and G.A. Vilkovisky, Phys. Rev. D 8 (1973) 4241. [16] R. Graham, Z. Phys. B 26 (1977) 397; M. Claudson and M.B. Halpern, Phys. Rev. D 31 (1985) 3310; Ann. Phys. (NY) 166 (1986) 33. [ 17] R. Graham, Phys. Lett. A 109 (1985) 209; S. Caraceiolo, 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. [ 18] H. Rumpf, Phys. Rev. D 33 (1986) 942. [19] R.E. Mortensen, J. Star. Phys. 1 (1969) 271.
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[20] D. Zwanziger, Nucl. Phys. B 192 (1981) 259; E.G. Floratos and J. Iliopoulos, Nucl. Phys. B 214 (1983) 392; E.G. Floratos, J. Iliopoul0s and D. Zwanziger, Nucl. Phys. B 241 (1984 ) 221; H.S. Chan and M.B. Halpern, Phys. Rev. D 33 (1986) 540. [21 ] J. Sakamoto, Prog. Theor. Phys. 70 (1983) 1424; T. Fukai and K. Okano, Prog. Theor. Phys. 73 (1985) 790. [22] I. Bengtsson and H. HiJffel, Phys. Lett. B 176 (1986) 391.
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SCHWINGER-DYSON FORMULATION OF COORDINATE-INVARIANT REGULARIZATION "~ M.B. HALPERN LawrenceBerkeley Laboratoryand Department of Physics, Universityof California, Berkeley, CA 94720, USA Received22 September1986
Geometricinterpretationof the continuumregularizationprogramis found in a Schwinger-Dysonformulationof coordinateinvariant regularization.Introductoryapplication of the generalformulationis givenfor the regularizednon-linearsigma model and regularizedeuclideangravity.A Schwinger-Dysonbonus, conceptuallyindependentof the regularization,is noted in the case of euclideangravity:Due to the differentialcharacterof the formulation,difficultiesusually associated with unbounded action are bypassed,at least in weakcoupling.
Principles for invariant non-perturbative continuum regularization of any quantum field theory have recently been given [ 1-3 ], with explicit application s to the scalar prototype [ 2 ], gauge theory [ 1,3-5 ] and gauge theory with fermions [ 6,7 ]. An introductory discussion of the program may be found in ref. [ 8 ]. In this letter I report a general framework for coordinate-invariant regularization, with introductory applications to the non-linear sigma model and Einstein gravity. Further details, explicit computations and alternate formulations will be given elsewhere [9-11 ]. For a field theory in d dimensions, the regularization scheme generally ad.rnits two equivalent formulations in terms of a regulator R(A) which is a function of an appropriate covariant spacetime laplacian A. At the level of ( d + 1 )-dimensional regularized Langevin [ 12] systems, R(A) is a covariant displacement of the noise, while at the level of d-dimensional regularized Schwinger-Dyson (SD) equations R (A)enters in a covariant regularizedfunctional laplacian. Heat kernel regularization [ 5 ] R = exp (A/A 2) is guaranteed to regularize any euclidean theory to all orders, and for safety in new theories, I will assume such regularization here. It should be born in mind however that dimension-dependent power-law regularization [2,3], which presumably allows euclidean-Minkowski rotation at finite cutoff, may also succeed as in previous cases, even for non-renormalizable theories. Advantages of the SD equations include formulation in physical spacetime, without regard for stochastic equilibration or choice of stochastic calculus. In particular, the SD equations are applicable in cases where naive action formulations are unbounded, such as euclidean gravity [ 13 ] below. The SD scheme reported here is a regularization of De Witt's supermetric prescription [ 14] for the measure, which is equivalent to the canonical measure for the non-linear sigma model, but controversial for gravity [ 15 ]. Although the De Witt measure is particularly elegant and easy to control, other gravitational measures may be regularized as well. In particular, the coordinate-space SD framework given here may be viewed as a special case in a more general phase-space regularization [ 9 ], after integration [ 7 ] of the regularized momenta. The general scheme also allows the direct regularization of Minkowski-space canonical formulations if desired. Moreover, other gravitational measures may be treated [ 10 ] by tinkering with the coordinate-space SD equations presented here. The large-cutoff unitarity of various regularized gravitational measures is therefore open ~r This work was supported by the Director, Officeof EnergyResearch, Officeof High Energyand Nuclear Physics, Division of High Energy Physics of the US Department of Energyunder contract DE-AC03-76SF00098and the National ScienceFoundation under Research Grant No. PHY-85-15857. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
111
Volume 185, number 1,2
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for future study. Coordinate-covariant SD equations are best introduced in a finite-dimensional context, where regularization is not needed. In this case, the SD equations are equivalent to more familiar action formulations, which I review first. Consider a connected riemannian manifold without boundary and coordinates x m. Coordinate deformations 8 x m are contravariant vectors under the group Gx of coordinate diffeomorphisms 2"~ =fm (x), while the metric gm, (x ) on deformations
118xll2 = g m . ( X ) S X m S X ~
(1)
is a covariant tensor. A G~-invariant measure d w = d w is constructed as do)-= and finally a partition function
(dx)e(x)
Z = J (do)) e x p [ - S ( x ) ]
=-detl/2[gmn(X)]~[pdXP ,
(2)
is given in terms of an action S(x), defined as a Gx-scalar S(2) In general, one is interested in computing averages
=S(x).
(F(x) ) =Z -1 j (do~) exp[ -S(x)]F(x),
(3)
for any F(x) in coordinate system x m, and a simple statement of reparametrization invariance is that (9°(0e) ) = (5P(x) ) for any Gx-scalar b °. Statements of this kind, trivially verifiable by change of variable in a finite-dimensional context, require coordinate-invariant regularization in a general quantum field theory. The averages of the theory are equivalently described ,i by the set of Gx-covariant SD equations 0=(LF(x)),
L=--gmnOnSOm+A,
A=gmnD~O,~=-g'~[On~-Frn,,,(g)]Or,
(4)
in which the SD operator L involves the inverse metric gm,, the Gx-covariant derivative Dn, the Christoffel connection F~m of gmn, and the Gx-covariant (scalar) laplacian A. These equations may be obtained directly from the identities 0=
f (do))e-lO,n[exp(-S)egmnO,,F]
(5)
at the action level, or as the adjoint of the covariant Fokker-Planck equation [ 16] for the Gx-scalar Fokker-Planck density at equilibrium. Further study of covariant stochastic formulations is found in refs. [ 17,18 ]. Except that regularization is generally required, the same prescription is easily followed in quantum field theory. On a d-dimensional spacetime ~m, let #u(~) be generic fields with possible spacetime (Einstein) tensor indices included in the index M. The formal partition function of interest is
Z= f ~exp(-S[q)]),
~O')N~)~[~)]~ITI~(detll2[~RX(O(~))]Hd~(~)m'
(6)
in terms of the action functional S[~] and an ultralocal supermetric [ 14] N:~N(~({)) on deformations of the field coordinates
1180112=f
(d{)~M,V(O({))Sg~M(g)8~({).
(7)
As in the finite-dimensional case, the measure is formally invariant and the action a formal scalar under the group G o of local field diffeomorphisms ~M(~) = FM(~(~)), n Weak coupling SD expansions are unique when supplementedwith the usual permutation symmetryof the Green functions as a boundary condition [2,3]. 112
Volume 185, number 1,2
PHYSICS LETTERS B
0~ ~6M(O=O--7 ( ~ ) ~ ( 0 ,
12 February 1987
~,, O0 R , . , 0 ~ s ~M~(~.~) =o--$-zt¢~ ~ (0 c¢~(~(O).
(8)
Ideally, the regularization should respect G¢-covariance, as well as other invariances which may be present in the theory, such as gauge invariance, or invariance under the group G¢ of Einstein diffeomorphisms ~ f m(~). For this purpose, I propose the regularized covariant SD equations 0= (LF[~]),
8S 5~M(~) 8 ~-A, L = - f (d~) ffMN(~(~)) 6#v(~)
D 6 A--- f (d~X)(d~') ~A Me; N¢' _DON(~, _ _ ) _5¢M(~) ---
(d~)(d~') NAMe;re'
~_~a(~_~,)r~e
(N(~)))
50R(~) '
in which F[~] is any field-functional, (¢tu is the inverse supermetric, and the super-Christoffel connection F~M( f~)=½ f~RL
(oo
o
o
f~ML+0--~ f~UL- - ~ -
~NM
)
(10)
appears in the functional covariant derivative D / D # v. The regularization, which is contained entirely in the regularized functional laplacian A, appears as a regularized (inverse) supermetric
~A M¢;N¢'-~ J ( d ~ " ) R ( A ) M C ; p e " R ( A ) N ¢ ' ; a¢" ffeQ(O(~")),
(11)
in which the regulator R (A) is a function of an appropriate spacetime laplacian a on tensors VM which transform like 5 OM. Ideally, A should be chosen to be covariant with respect to G o, G¢ and so on. With such a A, the matrix construction (A{) vM(~) ~-. (A¢).~VN(~),
AM¢;u,'~' ~ ( ~ ¢ ) . ~ d ( ~ _ ~t),
(12)
then guarantees that fgaMe; U¢' is a bitensor and A is a scalar under all covariances of A. It follows that the regularization is coordinate-invariant. In the formal unregularized limit RM¢;N¢' --* 3~ad(¢--¢'), A
s M¢;N¢' ~ f~MN(0(¢))ad(¢__¢,),
(13)
A
and the SD equations (9) correspond to the formal identities 0= f ~ o ) ~-1 f ( d , ) ~
8
(exp(-S)¢
8F f(/v~v(0(,))~-~)
(14)
at the action level, in analogy with eq. (5). It would be misleading however to leave the impression that I guessed the regularized form of A in eq. (9), or that its form is unique, although I believe it is the simplest possible, since it is in the y = 0 (Ito) form [ 3,5,19 ], with no field-derivatives of the regulator. In fact, I originally obtained the SD equations (9) directly from the regularized phase-space formulation [9] at equilibrium, by regularized integration [ 7 ] of the momenta. Remarkably, there is no ambiguity in the phase-space stochastic calculus [ 9,11 ], and the resulting coordinate-space SD equations are always in the form usually associated with Ito calculus in conventional coordinate-space stochastic processes. For completeness, I also give the associated markovian-regularized Ito stochastic process 113
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f 8S ~jM(~) + ~am; seF~s(f¢(q~(~))) = _ ~MN(¢(~)) ~---7~7~, O9~ tg) ~ + J (d~')ga Me; a¢, qa(~'), (r/a(~, t)qs( ~', t') ) =2~asOa( ~-~')J( t-t'),
(15)
in which ga Me;A¢' is a regularized (inverse) supervielbein, ~,M~_ gMa gNA gA Me;ae' ~R(A)Me;Ne'gNA(~(~')) , ~a Me;Ne'-'-~ J (d~U)ga Me;A¢" ga Ne';A+e"
(16)
This system is a regularized form of an equation mentioned by Rumpf [ 18 ], and is equivalent to the SD equations (9) under assumption of equilibration. The covariant regularized SD equations (9) are a more fundamental result, however, since they may be applied to any theory with formal partition function eq. (6), including Minkowski theories, without regard for equilibration or boundedness of the action. In what follows, I briefly discuss the application of the general regularized formulations eqs. (9) and (15 ) in a number of cases of interest. In the general d-dimensional non-linear sigma model, ¢W(~) is a set of G~-scalar fields with G¢- and Ge-scalar action
S= 2 f ( d~) (~MNgmn(grnOMOnON+ ....
(17)
in which gm,(~) is an arbitrary external gravitational field and (~MN=e(~)GMN(~J(~))specifies the internal manifold. Fully invariant regularization is obtained when the G¢- and Ge-covariant derivatives
am V M - [ Omrg + ( OmO")rfN( ~)] Vu ----W~, A n W M ~ On WMm--I'rnrn(g) WrM + (OnON)I"~R(~) WRm,
(l 8)
are employed to construct the G~- and Ge-covariant laplacian AVM=gmnA,A,~VM. In the particular case of no external gravitational field and flat supermetric (ffMN=I~MN, w e have A= [] =_Omdm, ~ ; N&= [R2(I--q)]e&C~MN,and the regularized formulations eqs. (9) and (15) reduce to those of the scalar prototype [ 2]. The case of no gravitational field and gMA = 0~)'4]0~)M is the simplest laboratory for the explicit study of reparametrization invariance under variable change from cartesian 0 a to curvilinear ~M. As an application of the regularized formulation in ref. [ 9 ], I will obtain a non-perturbative characterization of the general sigma-model Weyl anomaly in d= 2 dimensions: The general anomaly in the presence of the regulator is the average of the invariant trace of the regularized supermetric, a form which is amenable to further analysis by heat-kernel methods at large cutoff. The original regularization of gauge theory [1,3 ] is also easily obtained as a special case of eq. (9). With (~M(~)---*A~(~) and flat supermetric ~MN'-"~(~lO(~ab, the regularized inverse supermetric becomes fgaMe; N¢' __.[R 2(A) ] ~, du~, where A is a gauge-covariant laplacian [ 1,3 ]. The system eq. (9) is then recognized as the ~ =0 (Ito) form of the SD equations of refs. [3,5]. I finally turn to d-dimensional regularized gravity. Although the vielbein may also be studied, I will choose here the metric q~M(~)--*gmn(~) as field coordinate. The most general ultralocal supermetric on deformations of the metric [ 14 ] is ~mn; rs = e[ ½(gmrg,S +g~g'O +rgmng rs] (19) where positivity (existence of real supervielbein) requires y > - 1/d. The explicit form of the regularized SD equations (9) is
114
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O=
I
=
L-
PHYSICS LETTERSB
f (d¢)
12 February 1987
mn; rs ~Sgrs,]SS'~--Sgm.8 + A , D
8
(d~)(d~') ff~m.e;r~¢. Dg~,(~') 8gmn(~)
~d(~--~')/'a~n; •gab({)
8g,7(g)Sgm.({)
(d~)(d~')~mn#;rs¢,
(20)
'
where
(19 (ffpq;rs ..~ O.~ (ffpq;rnn 19 (finn;m) I~at~;rs=½ ~ffab;pq O~mn Ogrs -ag,,---~q '
(21a)
ffa¢; ~*¢'= f (d~,,)R(A)m.¢ab¢,, (gab;m(g({,,))R(A)rs¢,;pq¢,',
(21b)
are both G¢-tensors as indicated. I have also added a G¢ gauge-fixing term ~*gm. = DmZ. + D.Zm of the Zwanziger [20,21 ] type, whose generic form is a Lie derivative in the vertical direction on the principle bundle. For exploratory purposes, the simplest choice of A is the ordinary G¢-covariant A = DmDm on covariant secondrank symmetric tensors. This Choice guarantees G¢-invariance of the regularization, but locks in the metric as the fundamental variable. With heat-kernel regulator, the SD equations (20) regularize any De Witt-measure euclidean quantum geometry with classical action S, but I speciaUize in what follows to euclidean Einstein gravity S=f(d{)eR/x z, with R the curvature scalar and x = ~ . For computational purposes, it is convenient to choose the gauge Zm = ( O.h.m - Omh/2 ) / ( 2x) where g,,,. = 8m. + x h,.. and h = hm.,. The SD equations (20) may then be analyzed to any desired order in t¢ according to standard regularized SD methods [ 2,3,5,7 ]. Chart and I have completed an explicit one-loop check of G¢- invariance [ 10], which we will report elsewhere, along with a discussion of other gravitational measures. Here I confine myself to some simple remarks. The linearized form of the SD operator equation (20) is 1 +2g /¢2L'°)=1 f (d,)([[]hmn-~-l---~(~mn(OrOshrs-[' ]h))8h,~, ~
+ f (d{)(d~')(exp(21-1/A2))¢¢, ( 1 -
lm.; r,-- ½(a.,ra., + amid.r),
Yl-~dy)
82
(22a)
,~.; rs 8h~s(~')Sh.,,,({) '
Tin.;.- am.dr,.
(22b)
Choosing F= hm.(¢~)l&,(~2) in the linearized SD equations, the free regularized graviton propagator is obtained
+
T
1 ~d
1 +2y
mn;rs 3 - d + 2 y
6mn-~--+&,
[] /_1\
-D
,tee,'
(23)
and the free regularized n-point functions follow from this result by Wick's theorem. Only the first term of eq. (23) 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 choice y = - 1/2, and although positivity of the supermetric is lost, perturbative analysis is greatly simplified in this case [ 10]. In particular, as seen in eq. (22), the tensor structure of the SD solid-line factors [2,3] at 7 = - 1/2 is simply lm,; ~s. In the case of euclidean gravity, the usual functional integral eq. (6) is ill-defined due to the unstable confor115
Volume 185, number 1,2
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22 February 1987
mal m o d e [ 13 ], a n d is not the correct c o n t i n u a t i o n o f t h e M i n k o w s k i integral. I a m taking the SD equations as fundamental, a n d the result eq. (23) shows that the unregularized SD equations have p r o v i d e d the correct euclidean rotation. F o r d > 2 a n d positive supermetric (7 > - 1/d), the stochastic f o r m u l a t i o n eq. (15 ) does not equilibrate, while the successful SD t r e a t m e n t o f the unstable m o d e (trace) in eq. (22) is equivalent to the following toy model: F o r a single degree o f f r e e d o m x with action S ( x ) , the SD equations ( 4 ) are 0 = ( - S ' F ' +F" ), where p r i m e denotes x-derivative. Choosing F=x2/2 a n d ( u n b o u n d e d ) action S = - x a / 2 , one obtains i m m e d i a t e l y 0 = ( x 2 + 1 ) , which is the prescription o f G i b b o n s , H a w k i n g a n d Perry [ 13 ]. F o r d > 2 a n d non-positive supermetric (~ < - 1/d), the c o m b i n a t i o n ~¢,*n;rs5 S/Sgrs effectively stabilizes the mode, in qualitative agreement with a final r e m a r k by R u m p f [ 18 ] in a stochastic context. The p o i n t o f Gibbons, H a w k i n g a n d Perry [ 13 ] is that a correctly c o n t i n u e d euclidean functional integration for gravity m u s t involve a complex g,,n-contour in some directions, a n d related ideas have been studied in ref. [ 22 ]. As a differential formulation, the SD equations allow us to c o m p u t e while leaving the question o f euclidean integration contour to a later stage. In rough analogy with the Schr6dinger equation for scattering versus b o u n d state regions, differential equations are often used to effect analytic continuations in this way. I would like to t h a n k O. Alvarez, K. Bardakci, Z. Bern, H.S. Chan, H. Htiffel, C. Kounnas, N. Marcus, H. Neuberger, A. Niemi, A. Sagnotti, H. Sonoda, H. Stapp, P. W i n d e y a n d B. Z u m i n o for helpful discussions.
References [ 1] Z. Bern, M.B. Halpern, L. Sadun and C. Taubes, Phys. Lett. B 165 (1985) 151. [2] Z. Bern, M.B. Halpern, L. Sadun and C. Taubes, Nucl. Phys. 284 (1987) 1. [3] Z. Bern, M.B. Halpern, L. Sadun and C. Taubes, Nucl. Phys. 284 (1987) 35. [4] Z. Bern, M.B. Halpern and L. Sadun, Nuel. Phys. 284 (1987) 92. [5] Z. Bern, M.B. Halpern and N.G. Kalivas, Heat kernel regularization of gauge theory, preprint LBL-21286 UCB-PTH-86/6, to be published in Phys. Rev. D. [ 6 ] Z. Bern, H.S. Chan and M.B. Halpern, Continuum regularization of gauge theory with fermions, preprint LBL-21552 UCB-PTH86/14, to be published in Z. Phys. C. [7] Z. Bern, H.S. Chan and M.B. Halpern, Non-Grassmann formulation 0f regularized gauge theory with fermions, prerint LBL-21786 UCB-PTH-86/17. [8] M.B. Halpern, in: Proc. Symp. on Topological and geometric methods in field theory (Espoo, 1986) (World Scientific, Singapore, 1986). [ 9 ] M.B. Halpern, Coordinate-invariant regularization, in preparation. [ 10] H.S. Chan and M.B. Halpern, Regularized quantum gravity, in preparation. [ 11 ] Z. Bern and M.B. Halpern, Phase-space versus coordinate-space reguiarizations, in preparation. [ 12] P. Langevin, C.R. Acad. Sei. Paris 146 (1908) 530; G. Parisi and Y.-S. Wu, Sci. Sin. 24 (1981) 483, [13] G.W. Gibbons, S.W. Hawking and M.J. Perry, Nucl. Phys. B 138 (1978) 141; S.W. Hawking, in: General relativity, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge, 1979). [ 14] B.S. De Witt, J. Math. Phys. 3 (1962) 1073; in: General relativity, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge, 1979); in" Recent developments in gravitation, eds. M. IAvy and S. Deser (Plenum, New York, 1979). [15] C.W. Misner, Rev. Mod. Phys. 29 (1957) 497; E.S. Fradkin and G.A. Vilkovisky, Phys. Rev. D 8 (1973) 4241. [16] R. Graham, Z. Phys. B 26 (1977) 397; M. Claudson and M.B. Halpern, Phys. Rev. D 31 (1985) 3310; Ann. Phys. (NY) 166 (1986) 33. [ 17] R. Graham, Phys. Lett. A 109 (1985) 209; S. Caraceiolo, 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. [ 18] H. Rumpf, Phys. Rev. D 33 (1986) 942. [19] R.E. Mortensen, J. Star. Phys. 1 (1969) 271.
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[20] D. Zwanziger, Nucl. Phys. B 192 (1981) 259; E.G. Floratos and J. Iliopoulos, Nucl. Phys. B 214 (1983) 392; E.G. Floratos, J. Iliopoul0s and D. Zwanziger, Nucl. Phys. B 241 (1984 ) 221; H.S. Chan and M.B. Halpern, Phys. Rev. D 33 (1986) 540. [21 ] J. Sakamoto, Prog. Theor. Phys. 70 (1983) 1424; T. Fukai and K. Okano, Prog. Theor. Phys. 73 (1985) 790. [22] I. Bengtsson and H. HiJffel, Phys. Lett. B 176 (1986) 391.
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