Volume 175, number 2 PHYSICS LETTERS B 31 July 1986

STOCHASTIC EQUATIONS FOR GRAVITY

L. BAULIEU a

CERN, CH 1211 Geneva 23, Switzerland and Centro de Estudios Cientificos de Santiago, Casella 16443, Santiago 9, Chile

Received 4 February 1986

The Langevin equations governing stochastic quantization of gravity are established. These equations hold true, indepen- dently of the value of the torsion. They are covariant under the BRS symmetry associated to local diffeomorphisms and Lorentz transformations. The method determines simultaneously the Langevin equations of the diffeomorphisms and Lorentz ghosts, and of the classical vielbein and spin connection.

Stochastic quantization is a method which is for- mally equivalent to the path integral formalism or to the canonical one for building quantum scalar field theories * 1. This technique was applied first by Parisi and Wu to the case of Yang-MiUs theories [2]. In their original paper, the stochastic dynamics of the Yang-Mills field is described by means of a Langevin equation. By transposing this equation into a Fokker- Planck equation, one can prove in a rather general way, formally true for any type of gauge theory, that the stochastic quantization method and the usual methods based on a gauge-f'txed and BRS-invariant lagrangian predict the same values for any gauge- independent quantity [3]. Recently, the stochastic description of the ghost sector of Yang-Mills theories has been derived from the principle that the evolu- tion along the stochastic time must be compatible with the BRS symmetry [4,5]. The method is purely geometrical, since the BRS symmetry ,2 can be con- structed independently of the notion of a lagrangian, and it turns out that the system of Langevin equa- tions describing the whole set of classical and ghost fields of a Yang-Mills theory, including the possible gauge-restoring forces, can be determined from the

1 Permanent address: Laboratoire de Physique Th6orique, Tour 16, 4 place Jussieu, F-75005 Paris, France.

,1 For a review see ref. [1 ]. 4-2 For a review see ref. [7].

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

sole knowledge of the cohomological properties of the BRS operator [4,5]. The method has also been applied to determine the stochastic formulation of these gauge theories involving interacting p-form gauge fields [5].

The aim of this letter is to prove that stochastic quantization can be used also in curved space, when gravity is coupled. We will in fact derive the Langevin equations of gravity, with a trivial generalization to gauge systems coupled to gravity. We choose to work in the vielbein formalism. Given any quantum action coupled to gravity, the fields describing the gravita- tional sector are [8-10]

ea = e~u dxU

b u

coab=(.o ab dxu

~ab ~ab.

da b / (1)

Here a and b stand for the Lorentz indices of D- a

dimensional Lorentz symmetry, e u and CO~ b denote, respectively, the vielbein and the spin connection. ~u and ~u on the one hand, and ~ab and ~ a b on the otherhand, are, respectively, the ghosts and anti- ghosts of local diffeomorphisms and Lorentz trans- formations, b u and d ab are the corresponding Stiickelberg auxiliary fields. The latter permit one to define the BRS variations of antighosts. The action of the gauge symmetry on all the fields, i.e., that of the

133

Volume 175, number 2 PHYSICS LETTERS B 31 July 1986

BRS operator, can be geometrically determined, and any admissible action, possibly gauge-fLxed, must be invariant under s transformations. The action of s on the fields (1) is as follows [8 -10]

se a = _ ~2abeb _ ( i ~ w a b ) e b + l .~e a ,

s60ab = _ D ~ a b + i~R ab ,

s~2 ab = - ½ [~2, ~2] ab + ~ i ~ i ~ R ab ,

_ 1

s ~ a b = d a b , s ~ ta = b # ,

existence of the above-defined grading. One defines

d - - - d + s ,

a - d + ~ = [exp(-i~)] (d + s) expq~) , (5)

~oab ~ 60ab + ~2ab ,

& a b _~ 60ab + ( I2ab _ i~60ab) = [ exp (_ i0 ] ~ a b . (6)

Then one finds that eqs. (2) or (4) are equivalent to the following constraints on generalized curvatures, once expanded in ghost number [8,9]:

k - = d c o + & ~ = R - d60 + 6060,

sd ab = O , sbla = O , (2)

where D = d + co is the Lorentz.covariant derivative. R = d60 + coco is the Lorentz curvature two-form. The scalar Lorentz ghosts ~2 ab , ~2 ab and the vector field ghosts ~ and ~ are anticommuting objects, i~ is the contraction operator along the anticommuting vector field ~u, and L~ - i~d - d/~ is the graded Lie deriva- tive along ~u. The grading is defined as the sum of the Lorentz degree and ghost number. Therefore s is a graded differential operator with grading equal to one, i~ has grading O, and L~ has grading I. One can verify that s 2 = 0 on all quantities. I f one introduces the following change of variables:

g - s - L~,

~2ab ~ ~2ab _ i~60 ab = f2ab _ ~ o ) a b

~ab = d a b _ L~ ~2 ab = d ab - i~ d~2 ab , (3)

one can rewrite eqs. (2) under the equivalent and simplest form

g60ab = _ D ~ 2 a b , ~e a = _ ( 2 a b e b ,

7 ~ = d e + £oe = T = d e + 6 0 e ,

- e . (7)

Eqs. (7) are also equivalent to

7r~ ~ ~ ' [ p q ) ] R = +606o = ex ~ R ,

= + co e = [exp( )1 T ,

"~= e + i~e = [exp(i~)] e . (8)

The constraint that d has no component with non- zero ghost number is geometrically crucial. It enforces the inversibility of the vielbein eu, and determines the link between the diffeomorphisms and the transla- tion sector of the Poincar6 algebra [8]. The Bianchi identities DR = 0, DT = R e ensure that d 2 = "d 2 = 0, which means in fact that the constraints (7), (8) truly determine g and s with s 2 =§2 = sd + ds = gd+ d~ = 0 [8,9]. It is most convenient to introduce the follow- ing "Lorentz-covariant" operators, constructed from the "Lorentz-scalar" operators s, § and L~ = [i~, d]:

S - = s + I 2 ,

D = D + S = d + s + 6 0 + ~ 2 ,

1 ~ 2 a b = - - ½ [ f 2 , ~ 2 l a b , s'~u = - ~ {/j, ~j}u ,

~d a b = O, g / ~ u = 0 . (4)

Further, one can combine the-classical and ghost quantities into extended objects compatible with the

£ ~ = L~ + i~60 = i~D - Di~,

S - ~ + ( Z = S - £ ~ ,

i ) = D + S = d + §+6o +~2. (9)

These operators satisfy the commutat ion properties

134

Volume 175, number 2 PHYSICS LETTERS B 31 July 1986

82 1. • =~t~t~R, SD+DS = i~R ,

$ 2 = 0 , SD+DS = 0 . (10)

Eqs. (10) are trivial to verify by making use of the identities R = DD andR = I)I) t o ~ e r with the BRS equations (7) and (8). This yields DD = exp(i~R), DD = R and thus eq. (10) by expansion in ghost number.

In the stochastic approach, all fields and ghosts (1) depend on the stochastic time t in addition to the space-time variable xU [1]. Besides, one introduces a noise source for each field. We denote as rl~ b, 7 ab ,

~ab, [3ab, pa Ku -KU ' flu the noise sources of co ab _ _ ~ ' , _ _ ~ '

~2 ab , ~2 ab, dab, e~, ~t~, ~u b u, respectively. We want to determine the action of a/a t on all these fields, and also the action of the BRS operator on the noise sources. The requirement is that the stochastic evolu- tion along t must be compatible with the action of s, defined in eq. (2), or equivalently in eqs. (7) or (8). This means that a/at must commute with s

(a/a t)s = s a / a t . (11)

It is obvious that the space-time derivative d = dxU0u must also commute with a/at. Therefore one requires

(b /a t )d = "d~/3 t , (12)

or equivalently

(a /a t )d = d a / o t - L((a/at)~ ) . (13)

We shall prove that the knowledge of the BRS equa- tions of gravity, eqs. (2), i.e. of the symmetry of gravity, together with the above assumption, eq. (12), is sufficient to determine the Langevin equations of gravity. From now on, we adopt the notation • for the operator a/a t. By applying a/at to the BRS equa- tion (7) and by using eq. (13), one gets that

i

R = (d + s -- L~)cb - L{do+ [w, o5] =1~. (14)

This can be written as

R=De3 - L ~ = R. (15)

We shall expand eq. (15) in ghost number. Then as- suming that the stochastic time derivative of fields are linear functions of the noises, we shall solve the re- suiting equations, and Fred the expressions of ~ , ~ , ~, i.e., the Langevin equations of co, ~2, ~. In this deriva- tive, the nature of stochastic sources (e.g. gaussian),

can be let unspecified. We begin with the part with ghost number 2 in eq. (15):

S ~ = L{ {2 = i~ d~ = - i ~ . (16)

Since ~2 = 0, one must have S~ = 0. This implies

+K*. +m, , (17)

and the s~variation of the source KU must be

~KU = 0

¢,sKU= { ~ , K } V - ~ 3 ~ K U - K O ~ 3 c ~ u . (18)

ZU is an arbitrary vector field, with ghost number 1, which can be a function of all the fields co, e, ~, ~2. We combine eq. (17) with eq. (16) and find

S(~ + i~co) = O. (19)

This implies

~2ab = - i . ~ a o + s~ab + Tab

= _ i~zcoab +S vab + Tab _ iKcoab , (20)

provided that one has

$ 7 ab = O¢~sTab = - [ ~ , 7 ] ab • (21)

V ab , which is valued in the Lie algebra ~£ of the Lorentz group and has ghost number 0, plays a role analogous to KU in eq. (17) and can depend on all the fields. In the stochastic process, when the sources KU and T ab are specified as gaussian ones, the func- tions Z u and V ab are expected to stabilize the t-evo- lution of the unphysical modes in coab, corresponding, respectively, to the local diffeomorphisms and Lorentz invariances.

To determine the Langevin equation which ex- presses the form of &, we isolate the part with ghost number 1 in eqz(15), and we use the already-found expressions of £Z and ~. One finds

S~ + D(-i~co + S I ? + 3') - L~w = 0 . (22)

Using the commutation property SD + DS = 0, one gets from this equation

S(& - D V ) - i~R + I )7 = 0 . ( 2 3 )

Eq. (.23) is clearly consistent, since $7 = 0, SR = 0 and St = 0. We insert the expression (17) of ~ into (23) and get

S(~o - DV - i z R ) - iKR + 1)3' = 0 . (24)

135

Volume 175, number 2 PHYSICS LETTERS B 31 July 1986

The solution of this equation is

~ab = DVab + i z R a b + 6£/6coab + ~ab , (25)

where the BRS variation of the source r/ab of the spin connection is

s~ab = _DTab + i x R a b " (26)

In eq. (25), we have used the fact that the cohomology of S is non-trivial for ~£-valued quantities with ghost number 0. Indeed, s-invariant but not g-exact lagrangians exist, such as the Einstein lagrangian, the Weyl lagrangian and the cosmological constant [9,10]. BRS-invariant gauge-fixing terms which are of the type s(M) can be included in £. However, they corre- spond to mere redefinitions of V ab by terms at least quadratic in the ghosts and antighosts, and one can restrict £ in eq. (25) to be a classical gauge-invariant lagrangian.

The stochastic equation for the vielbein is obtained from the BRS equation T = T. From T= ~v, d = e, one gets

I)b + ~ e - L~e = ~, (27)

where we have already computed 65 and gu. The part with ghost number 1 yields

Se + ( - i . ~ - L.~ + S V + y )e = 0 , (28)

or

S(e + Ve) - £ } ° e + 7 = O. (29)

Here we have used Se = 0 and the definition (9) of the covariant Lie derivative £ ~o. Further, one has

£~e =£~g~° e + £~-e = S(£~e) + £ ~ e

so that one gets finally

ba = _ vabe b + £gea +8£ /8e a + pa , (30)

where the noise source pa for the vielbein undergoes the following BRS transformation law: ^

Sp a = - 7abe b + £wKea . (31)

In the same way as o3 contains ~£/3~, d contains 0£/3d as a consequence of the non-triviality of the cohomology of S for objects with ghost number 0 and valued in the fundamental representation of the Lorentz algebra.

We can now summarize the results. The Langevin equation for the classical and ghost fields of gravity

are simply

ba = 8£/8e a _ Vabe b - - £ ~ e a + pa ,

~oab = 8£/86oab + D V ab + i z R a b + r] ab ,

~ab = s v a b _ i~ztoab + 7ab _ iKwab ,

= + ( 3 2 )

Besides, the BRS variations of the sources are

Spa = _7abeb + £ ~ e a ,

srlab = _D.yab + iKR ,

SKU = O, S,./ab = 0. (33)

The derivation of these equations has been purely geometrical. The dynamics, that is to say the nature of sources, has been left free. It could be specified by imposing that all sources are gaussian ones. Other choices, which could appear as more natural ones in view of the invariance under general coordinate trans- formations, can be used. In eqs. (32), the functions V ab and Zu characterize the possible gauge-restoring forces of the stochastic process corresponding to the diffeomorphism and Lorentz invariances. Physics is expected to be independent of the choice of these functions [3]. In contrast with eq. (32), eqs. (33), which express the BRS variations of noises, are inde- pendent of the gauge-restoring forces. These features are similar to those encountered in flat space [4,5]. Notice that the stochastic equations (32) and (33) hold true for any admissible gravitational lagrangian £, including these cases where the torsion is non- vanishing.

There is no difficulty in obtaining the other sto- chastic equations for the antighost sector of the theory, using the anti-BRS operator of gravity [8]. The method is detailed in refs. [4,5] ir/the case of flat-space gauge symmetries. One gets~u =~ZU + K'u, and bU is obtained from the computation o{bU = s~ -u, using the chain rule. In turn, one finds ~2 = S V a b -

i p z w + ~ - iR w, Finally, t ab = s~ ab can also be com- puted from the chain rule.

It is now rather simple to derive the stochastic equations for a scalar field ¢ and a Yang-MiUs system coupled to gravity, with gauge field AudXU, valued in the Lie algebra q of a given Lie group, and ghost e.

136

Volume 175, number 2 PHYSICS LETTERS B 31 July 1986

One can apply the same method as we have dope for gravity, starting from the BRS equations F = dA + A A = F and G = D0 = G, or equivalently/~ = exp(- i~F) and G = exp(i~G) [8,9]. One gets the fol- lowing Langevin equation for A, c, ~:

= D V ~ + i z F + 72~ +8£16A ,

c = S V q - i~zA + 7 ~ - iKA ,

;~ = - v q ¢ + 7t + 8£ /80 ,

D = d + c o + A , S - s + ~ + c . (34)

V q - " e cnaracterizes the gauge-r storing force in the Yang-MilIs sector, ZU and KU have been defined above, and ~ q , 3' ~, X are, respectively, the noise sources for the Yang-Mills field A. the ghost c, and the scalar field X. The BRS transformation laws of sources are

S~?q = -D') ,q + i K F ,

kx = -~,qp + zA+~¢. (35)

To conclude, we have shown that the Langevin equations describing stochastic quantization can be deduced very straightforwardly in the cases of gravity and of gauge systems coupled to gravity. Our deriva- tion has been based on the hypothesis that gauge (i.e. BRS) transformations commute with stochastic time evolution. We find it interesting that the derivation of the stochastic equations of gravity is directly linked to the cohomological properties of the BRS operator of the corresponding symmetries. As in the case of gauge symmetries in fiat space, one retains at once the Langevin equations of classical and ghost fields. The dynamics of the latter fields is therefore defined in a non-perturbative way, independently o f the usual field theory formalisms. Besides, the method provides in a systematic way the most general form of the pos- sible gauge-restoring forces which stabilize the stochastic evolution of unphysical modes of gauge fields, while preserving the gauge independence of

physical quantities. The knowledge of stochastic equations of the ghosts of gravity might be useful for entering into the study of gravitational anomalies within the framework of stochastic quantization. It is an open question whether one could use the above stochastic equations to explore non-perturbative as- pects of gravity. Using an Einstein-Hilbert lagrangian for £ will certainly produce difficulties, since the cor- responding action is not positive deffmite. These prob- lems have been tackled in ref. [11].

Part of this work was done in Erice, during the 1985 Mathematical Physics School. It is a pleasure to thank the organizers of the school, G. Velo and A. Wightman, for their kind invitation. The work was completed in Santiago, with support from a grant of the Tinker foundation to the Centro de Estudios Scientificos de Santiago. I am grateful to Claudio Teitelboim for having made possible may stay here. Finally, I have enjoyed numerous and stimulating dis- cussions with my colleague Larry Blum.

References

[1] P.K. Mitter, Gift Lectures (1985); D. Zwanziger, Erice Lectures (1985); S. Chaturvedi, A.K. Kapoor and V. Srinivasan, Hyderabad preprint HUTP 85/30 (1985), and references therein.

[2] G. Parisi and Y.S. Wu, Sci. Sinica 24 (1981) 259. [3] D. Zwanziger, Nucl. Phys. B192 (1981) 259;

L. Baulieu and D. Zwanziger, Nucl. Phys. B193 (1982) 163.

[4] L. Balieu, Phys. Lett. B 171 (1986) 396. [5] L. Baulieu, LPTHE preprint 85/42 (1985), Nucl. Phys.

B, to be published. [6] G. Becchi, A. Rouet and R. Stora, Phys. Lett. B 52

(1974) 344. [7] L. Baulieu, Phys. Rep. 129 (1985) 1. [8] L. Baulieu and M. Bellon, Phys. Lett, B 161 (1985) 96;

Nucl. Phys. B266 (1986) 75; L. Baulieu, A. Georges and S. Ouvry, LPTENS preprint 85/36 (1985), submitted to Nucl. Phys. B.

[9] L. Baulieu and J. Thierry-Mieg, Phys, Lett. B 145 (1984) 53.

[10] F. Langouche, T. Schficker and R. Stora, Phys. Lett. B 145 (1984) 342.

[11] H. Hfiffel and H. Rumpf, Z. Phys. C 29 (1985) 319; Phys. Lett. B 148 (1984) 104; H. Rumpf, Vienna preprint UWTH PH 15 (1985).

137