A topological interpretation of stochastic quantization

Volume 212, number 3 PHYSICS LETTERS B 29 September 1988

A T O P O L O G I C A L I N T E R P R E T A T I O N OF S T O C H A S T I C Q U A N T I Z A T I O N

L. BAULIEU 1 and B. G R O S S M A N 2 Rockefeller University, Physics Department, 1230 York Avenue, NY 10021, USA

Received 27 June 1988

We analyze stochastic quantization in the framework of topological field theory. We consider an action which is a pure derivative in stochastic time. The fields needed to gauge fix this topological action are provided by the noise of the Langevin equation and the fermions arising from the exponentiation of a determinant. Interaction terms can be introduced by conjugation of the free BRST operator by a Morse function. The known supersymmetric action for stochastic quantization is then recovered. The for- malism that we develop has applications in Yang-Mills theory.

1. Introduction

A new set o f field theories, topological in origin, have been recently constructed by Witten [ 1,2 ], following some work of Atiyah [3 ]. The construction is motivated by an at tempt to relate topological invar- iants o f three- and four-dimensional manifolds to a relativistic quantum field theory. In addition to mathematical applications, Witten has suggested that there may be a new phase in these theories, possibly related to string theory, in which all observables are topological. This phase would be something like a hy- pothetical QCD completely determined by instanton like configurations. The states o f this phase should be determined by the zero modes o f the hamiltonian, just in the Wheeler-De Wit equation for quantum gravity. Further development in regard to these topological theories have been their recent construction from the gauge fixing of a topological term such as the Chern-Simon classes for Yang-Mills systems or o f an analogue for a-models [4 ]. The twisted supersymmetry that Witten uncovered is actually a BRST symmetry. The fields needed to form a multiplet along with the gauge fields, are just the required ghosts o f an enlarged BRST symmetry.

Our new viewpoint is a generalization o f this re- suit: namely, given a system which only depends on data at the boundaries, one can construct an action which is a pure derivative, i.e. topological, and cal- culate by a consistent BRST gauge fixing procedure a partit ion function that depends purely on the boundary conditions. We may call such a theory a topological field theory (TFT) .

Based upon the ideas on Morse theory [ 5,6,8 ] ~1, we wish to develop hamiltonian mechanics for topological field theories. In this formulation we shall obtain some intuition for the meaning of topological invariance through the introduction of symplectic 2- forms in the relevant phase spaces, and through the observation that the dynamics of these topological models are characterized by the existence o f a flow in loop space.

As an example of TFT's we will obtain stochasti- cally quantized field theories [9 -12] . Then our methods have applications for the Yang-Mills theory in d = 4 dimensions. In this case we show the existence of a flow along ghost direction which has some similarities with the one found in the general framework of stochastic quantization.

Permanent adress: LPTHE Paris VI, Universit6 Pierre et Marie Curie, 4 Place Jussieu, F-75230 Paris, France.

2 Work supported in part by the US Department of Energy, Con- tract Grant Number DE-AC02-87ER40325.B000.

~ For a discussion relating the mathematical and physical points of view, see ref. [7].

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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2. Topological interpretation of stochastic quantization

Let S [ ~ ] be a given local action, and suppose for simplicity that qb [x] is a real scalar field, and x is the ordinary spacetime position. One introduces noise so that there is a fictitious time as a function of which the system evolves toward an equilibrium distribution. We therefore assume that the action S[ ~ ] has the requisite positiveness properties in order to en- sure the consistency of the stochastic process [9 ]. At a finite stochastic time T, the stochastic partition function can be defined as [ 10,11 ]

f [ d ~ ] [ d ~ ] [ d ~ ° ] e x p ( - f dtI~m~), (1) O ~ t ~ T

with

&,oc= f dx[½(O~lOt)2+½(SSIS~) 2

+ ~(O/Ot+ 82s/8q ~2) 7 ' ] . (2)

(x, t) is the original scalar field which now depends on stochastic time t, while ~P(x, t) andkU(x, t) are scalar anticommuting fields. In the functional inte- gral ~P and ~ have periodic boundary conditions at t= 0 and t= T. With this prescription, the action I~toc is supersymmetric for each value of t [ 5-7,10 ]. Our aim is to prove the topological origin of the action (2).

From the idea of stochasticity, given a stochastic field q~(x, t), it is natural to consider an action which only depends on the fields at initial and final values of the stochastic time t. Such an action must corre- spond to a lagrangian which is a pure derivative with respect to t, i.e. a d = dtO/Ot exact lagrangian and must thus be of the form

Jtop=S[T]-S[O]= f dt(O~/Ot)(SS/Sq~) . O<~t<~ T

(3)

We call this action a topological action since it is in- dependent of any variation of the field

@(x, t)--*~(x, t) +e(x, t). (4)

e(x, t) is arbitrary except at the boundaries where e(x, O) = e(x, t) = 0. The BRST operator s corresponding to the gauge symmetry (4) is defined as follows:

The fields ~, T, ~ and b depend on x and t. T is the anticommuting ghost associated to e, ~P the corresponding anticommuting antighost and b the field which will be used as a Lagrange multiplier for the gauge fixing of the symmetry (4). One assumes [0/ 0t, s] =0 and one can check s2=0. Therefore we can gauge fix the topological action (3) in a BRST invariant way by adding to it the following s-exact term

J~F = J- dxdt s [ ~(O~lOt+ ~S/Sqb+ b/2 ) ]

= j dxdt[b2/2+b(O~/Ot+ 8S/8~)

- ~(O/Ot+ 82S/8cI92) ~],

~ o p ~ J = ~op - J ~ F . (6)

Eliminating the field b from J b y gaussian integration, one recovers the stochastic action Istoc written in (2), since (OdP/Ot)SS/8~ cancels against the crossed term coming from (O~/Ot+ 8S/8~) 2.

Istoch is therefore a gauge fixed BRST invariant version of the topological classical action Ic~ in (3) with the following choice of the gauge function:

0¢/0t + 8SI6~. (7)

Notice that only with this choice of a gauge function and after elimination of the auxiliary field b by its algebraic equation of motion, the BRST symmetry turns out to be identical to ordinary supersymmetry. We will not consider here the issue of concerning other gauge choices.

An anti BRST symmetry acting on the fields qb, ~u, ~' and b can also be constructed. Its generator g sat- isfies [0/0t, g]=[g, s ]=g2=0 and is defined as follows:

gO=~, g ~ = 0 ,

~u= - b, ~b=0. (8)

We can understand the existence of the anti BRST operator as resulting from the invariance of the stochastic process under reversal of the stochastic time, as can be inferred from the minus sign occurring in the equation g ~ = - b , the action (6) is also anti BRST invariant, go¢ = 0. One has indeed the following equivalent expressions for JoE:

sqb=~, sT=0 , s~P=b, s b = 0 . (5)

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(" JGF = J dxdt s [ ~O(8@/at+ 8S/8@+ b/2) ]

= f dxdt ~ [ 7'( - O@/Ot- aS~8@- b/2 ) ]

= f dxdt{sg(S+ ktqv/2) +s[ ~"(O@/Ot)]}

= j" dxdt{sg(S+ ~tF/2)-~[~(O@/Ot)]}. (9)

Rather than eliminating the field b and getting the action (2) from (4), we can express h la Nicolai [ 13 ] the field @ in terms of b. To do so, we write the partition function (6) as:

f [ d @ ] [db]det(O/Ot+ 82S/aqP)

= 8 ( b + 8@/8r +,3S/6@ )

× e x p ( f dtdx[(O@/Ot)(aS/a@)+b2/2]). 0 < t < T

(I0)

Then, we find @ as a function of b by formally in- verting the equation

-b--- 8@/8t+ aS~a@. ( 1 1 )

We can thus express the stochastic partition function as :

f [ d b ] e x p ( f dtdx[b2/2 0 < t < T

+ (O@/Ot) (as/a@) I.=~,,]), (12)

where @b is the solution of eq. (7). To get eq. ( 11 ) we have used the identity

de t (ab /a@)= f [d~U] [~u]

× exp (<Io ~ t ~ d t cLv ~P( 0/Ot + a2S/8 ~ ) ~u), (13)

which makes sense provided there is a unique solution to eq. (1 1 ). The non-uniqueness or non-existence of solutions may be related to "supersymmetry" breaking, i.e. non-conservation of ghost number due to the possible existence of zero modes. Notice that the simplistic aspect of the partition function in (12)

is of course misleading, since the last term in the action is a non-local term in function of b.

Now comes the existence of a flow along stochastic time. Using t as a time we wish to introduce a hamiltonian mechanics. We define the momentum of @ as H~ and ~P is the momentum of 7'. The hamiltonian corresponding to ~¢is:

J ax[ ½H~ + ½ (asia@) 2- ~'( a2slae~) ~'] . H=

(14)

As a consequence of the BRST and anti BRST symmetry (5), (8) for the action, the hamiltonian H is invariant under the actions of the following operators:

Q= J dx~(a/acl)+fiS/6@),

Q= f dx~(8/8@+aS/a@). (15)

One has Q 2 = Q : = 0 and QQ+QQ=H and thus [Q, H] = [Q, H] =0. A natural symplectic 2-form in the phase space is:

~o= 8~a!P+ 8@8H~. (16)

The equations of motion in fictitious time t are determined by

ia/at o9 = 8H, ( 17 )

where the contractions (denoted by the symbol ia/0,) of the vector field O/Ot are defined by

ia/a,8~= ½ (a2S/a(~ 2 ) ~/,

ia/o,a~P= - ½ (a2slaq p) ~v,

ia/ala@=H~,

i o / s t a H ~ .~ (aS i la (~) ) ( a 2 S / a ( ~ 2 ) _ ~ ( a 3S/ /a(~3 ) ~r_/.

(18) We therefore see that the existence of a well de-

fined stochastic mechanics depends upon the absence of zero modes for 62S/acI~. It is now interesting to observe that the topological interpretation of a stochastic theory can be related to a Morse theory construction. We shall in fact use the action S as a Morse function, and t as a parameter in loop space. We introduce the free operators

J dx ~ala@, d =

= f dx ~-5/a@. (19) d* o

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Using that kv and ~P are conjugate, the free bosonic hamiltonian is:

Ho = d * d + * d d = ~ dx62/Stib 2 , (20)

we conjugate d and d* by e s

ds = e - Sde s ,

*ds = e -S 'de s . (21)

In this way we find

H=ds*ds +*dsd,

(22)

We recognize the hamiltonian (14) and the differ- ential operators ds and *d~. We identify them with the BRST and anti BRST charges ( 15 ) that we had built for the sake of covariantly describing the topological gauge symmetry (4) and gauge fixing the topological action (3).

The non-vanishing of the anti-commutator of Q and Q, QQ+ Q Q = H , which is currently used to interpret d~ and *d~ as "supersymmetry" generators, is in fact the consequence of the absence of the auxiliary field b which we have eliminated by gaussian integration. It is indeed true in hamiltonian mechanics that the equation H = 0 can be interpreted as an equation of motion, which explains how off-shell nilpotency is broken by the equation QQ+ Q Q = H .

The Witten index for the theory is just defined as:

Tr( - )Fexp ( -- TH)

= J O<~t<~ T

_ tp(O/Ot + ~2S/~cI9) ku] (23)

and should not depend on T. It can be calculated for T ~ + ~ , and its value is expected to determine whether or not the ghost number is concerved as a result of the possible existence of zero modes.

One can furthermore rescale S in (21 ) into rS and t by z -~, where z is a real parameter. In the limit z--.oo, which can be seen as the semi-classical limit h--.0, one finds that the index is concentrated at 6S/ 6q~=0 and that ~P(82S/8cI~2)~ determines the fer- mion number F which appears in the left hand side of (23). Again one sees that the possibility of zero

modes for the operator 82S/6qb 2 is the crux of the matter: If such zero modes exist, the right hand side of (23) vanishes. As a consequence, the Witten index may vanish and the "supersymmetry" may be broken, which means in reality that the ghost number is not conserved. In this case the model has a non- trivial topological sector and the stochastic process is ill-defined.

The possibility of supersymmetry breaking in stochastic quantization is more clearly determined by looking at the Fokker-Planck equation for the prob- ability distribution. It is well known that the Langevin approach is equivalent to the Fokker- Planck equation for P ( qO, t) [ 10 ]:

(d/dt)/~(q~, t) = Yf/~( q~, t ) ,

~e = _ ½82 / 8 ~ 2 + ½ ( 8 s / 8 ~ ) 2 _ ½82s/sq~ ,

/~(q~, t)=P(q~, t ) e x p [ - S ( ~ ) ] . (24)

The partition function is

Z = j dq )P (~ , t ) . (25)

The function exp [ - S ( ~ ) ] is an eigenfunction of a~ with eigenvalue 0. In the asymptotic limit t-+ + oo, this determines P so that

P ( ~ , t) ~ Z - ~ e x p [ - S ( ~ ) ] . (26)

The Fokker-Planck interpretation of stochastic quantization may suggest a precise mathematical meaning to Witten's idea of a topological phase. Namely, in quantum gravity, the principle of equiv- alence determines the Wheeler-DeWitt equation H = 0. Analogously, in topological quantum field theory, stochastic evolution indicates the evolution to a zero eigenvalue of ~ if the system is ergodic ,2

3. Topological Yang-Mills symmetry and flow in ghost number

We shall see now that the ideas explained in the

~2 The q u a n t u m mechanica l case wi th a f ini te n u m b e r of degrees of f reedom is well understood. I f S ( ~ ) = Yo~n~<u0Jib% then the par i ty of N de te rmines whether supe r symmet ry is broken. I f N is odd, all the probabi l i ty dis t r ibut ion is concentrated at x = - co as t ~ + o r . Because the poten t ia l is u n b o u n d e d f rom below, the par t ic le escapes to - oo wi th probabi l i ty one. The Langevin equa t ion will therefore not have a solut ion for mos t r/.

354


previous section can be extended in a Yang-Mills theory.

Consider the case of a topological field theory for four-dimensional Yang-Mills. The classical action is:

/ top= J" T r F ^ F . M 4

It is invariant under the t ransformat ion

A - + A + D c + ~ ,

(27)

" topological" gauge

(28)

which yields F ~ - [ c, F] + D ~ c is a scalar ghost and 7 '= ~ d x ~ is a 1-form ghost which vanishes at the boundary of M4 (if there is one) . The gauge fixing of /top can be done in a BRST invariant way [ 4 ]. By introducing the relevant set of ghosts and antighosts, one gets the following expression for a topological part i t ion function, which is a generalization of that o fWi t t en [ 1,4]

exp [ - W, op ]

= f d A d c d g d ~ d 2 ~ d O d ~ d O e x p ( J ) . (29)

J is obta ined by adding to fM4Tr F ^ F an s-exact term, which after el imination of some auxiliary fields yields [ 4 ]:

J= ~ TrF^F+ ~ d4xTr{(Fc~a+ec~&,~F'a) 2 M 4 M 4

+)?,~p D,~ ~a + 0D,~ ~ '~+ ~ D . D ' ~

+ (F~p + e~aF ~, )?~p)c+)?~a[ ~ . , ~up]

+ ~ [ ~ u , ~u] _½(O.AC~)2_eO~,(D.c+ So-)}.

(30)

;~.p is a self-dual 2-form, 0 and Oare commut ing scalar ghosts while X.p, ~ . , c, g, r~ are an t icommut ing ghosts.

Consider now the equations satisfied by the above ment ioned auxiliary fields denoted as b and b~p:

b= - O.A ~, b.p = - (F~p+e~&aF y6) . (31)

In each topological sector these equations can be in- verted with a unique solution for A as function of b and b . a. Then, assuming the following identity, which is formally true when the functional integration on A is restricted in a given topological sector:

1 = J dA dc dgd To~ dg,~p dO dCTdf/8(b- 0,~A °~)

×8(b.p-F,~a-e .&aF~a)exp(J- f T r F ^ F M 4

+ f daxTr[(F~a+ea&6Fr6)2-½(~aA~)2]), M 4

(32)

we can rewrite the part i t ion function as follows:

exp ( - W) = f [db~p] [ db ]

b ~ p - T r F A FIA=A~ b b~o) ) ×exp(Mf4 (b2+ , ) "

(33)

(33) is a generalization of (12) . The notat ion I A =A(b,b.p) means that the A dependence must be ex- pressed as a function of the b and b.p solution of (31 ). The non-triviali ty of the part i t ion function is a consequence of the fact that A (b, b .p) is not a local func- t ion of b and b.p. Furthermore, the Morse theory for the topological field theory should detect the degen- eracy in the solutions of eq. (31 ) as zero modes in the presence of classical solutions, i.e. instantons. The Morse theory version of the Yang-Mills topological field theory is displayed in ref. [ 1 ] ~3.

To complete the analogy with what we have done in the stochastic quantizat ion we wish to obtain a hamiltonian interpretation. We shall introduce a flow along the variable 0, formally defined as s = dO (0/00).

We consider the space ~¢/f¢ of connections defined modulo usual gauge transformations. One de- fines the following symplectic 2-form in this space

In Witten's work the Morse function is recognized for the hamiltonian three-dimensional system as the three-dimensional Chern-Simons form 3 (A) = fTr (AdA + 2AAA ). We can now make the following simple observation. The critical point dA=0 corresponds to flat connections, i.e. dA+AA=0. Asso- ciated with these flat connections on three dimensions are instantons in the time evoluting four-dimensional theory. Thus, if we consider for instance an SU (2) gauge group broken down to U(1 ), the Chern-Simons form can be associated with a linking number of circles in three dimensions [ 14]. The instanton changes this linking number by one unit.

355


f TrF^SA 8A (34) A

M 4

and the contract ion opera tor ± ( that we formally

unders tand as i0/0o) by

± 8 A = 7 j ' , (35)

where ~ ' s tands for ~ ' = ~u+DC and thus s ~ ' =0 . The act ion of Z on all other fields c, ~ , , ¢ is zero. The first cocycle of the topological field theory is ~M3 Tr(FkU' ) where F = d A + A A and M3 is a 3-cycle in M4. The existence of a flow along 0 follows from the equali ty

± ~ T r S A D 8 A = 8 ~ Tr ~ ' F . (36) M 3 M 3

T r ( T ' F ) can be considered as a hami l ton ian which generates t ranslat ion along the ghost direct ion 0.

I f one considers ordinary gauge symmetry, then the contract ion opera tor is s imply defined as:

± S A = D c . (37)

In this case one finds the remarkable relat ion be- tween the covar iant anomaly fMaTr(cFF) (which is equivalent to the consistent anomaly fM4Tr C(8/ 8AF)Q5 with dQs=Tr(FFF) and the degenerate symplect ic 2- form fM4Tr FSASA [ 14]:

± I- Tr F ^ 8/1 ^ 6A = 8 ~ T r ( c F F ) . (38) M 4 M 4

This expression indicates that the first cocycle i, related to a flow along ghost number [ 15,16 ].

Finally, we note that the degenerate points of the presymplect ic 2-form fM4Tr(FA 8/1 ^ 8A) are interesting in themselves. These flat connect ions corre- spond to bundles which are tr ivial on a two- d imensional space in ~¢/f# de te rmined by the 2-form fM4TrF^ 8/1 A 8.4. I f this space is a self-dual or antiself-dual plane, the degenerate crit ical points corre- spond to antiself-dual fields respectively.

4. Conclusion

We have shown in this article the topological origin of stochastic quant izat ion, as s temming from a BRST

invar iant gauge fixing of an act ion that is a pure de- r ivat ive in stochastic t ime. In hami l ton ian formal- ism, and with a par t icular choice of the gauge function, the BRST invariance can be ident i f ied as supersymmetry , with a Morse theory interpretat ion. By introducing symplectic 2-forms in the space & f i e l d configurations, and generalizing these ideas to the Yang-Mil l s theory, we have shown the existence of a flow in ghost number generated by the cocycles of the gauge group.

Acknowledgement

One of us (L.B.) wishes to thank for hospi ta l i ty the Physics Depar tment of Rockefeller Univers i ty where this work was done, and I.M. Singer for most useful discussions.

References

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A topological interpretation of stochastic quantization