Effective Fokker-Planck equation and chiral anomaly

Volume 206, number 1

PHYSICS LETTERSB

12 May 1988

EFFECTIVE F O K K E R - P L A N C K EQUATION AND CHIRAL ANOMALY Giuseppe N A R D U L L I Dipartimento di Fisica, Universit~ di Bari and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I- 70126 Bari, Italy

Received 13 January 1988

It is shownthat, even in the regularizedversion of stochasticquantization for field theories, when the processis non-markovian, there exists an effectiveFokker-Planck equation which reduces to the unregularized one when the noises become white. As an application, the chiral anomaly is computed.

1. Introduction

In the stochastic quantization (SQ) method of Parisi and Wu [ 1 ] two distinct formulations are available: either one solves the Langevin equations obeyed by dynamical fields with gaussian white noises, or one considers a Fokker-Planck equation for the probability functional at fixed (fictitious) time t. In both cases canonical results should be recovered in the limit t ~ and indeed the equivalence between SQ and canonical methods has been proved in a number of circumstances [ 2 ]. A difference between the two points of view arises, however, if one introduces the so-caUed stochastic regularization (SR) [ 3 ], which is needed to deal with divergent expressions. As a matter of fact, SR amounts to the introduction of weakly coloured noises into the Langevin equations, following the approach of Doob, Ito and Stratonovich for brownian motion [ 4 ]; in this case the process becomes non-markovian and the very Fokker-Planck formulation might be meaningless [ 5 ]. A way out has been recently found, for brownian motion, by Fox [ 6 ], who has shown that an effective Fokker-Planck hamiltonian can be obtained for weakly coloured noise, and the Stratonovich version of the stochastic calculus can be justified. As stressed by Fox, the existence of a Fokker-Planck equation does not mean that a non-markovian process becomes a markovian one, but that there exists a Markov process, described by the effective Fokker -Planck equation, as close as one desires to the original non-markovian process. The purpose of this letter is to extend Fox's results to the SQ scheme for quantum field theory: the interest of having a Fokker-Planck formulation also for the regularized version of SQ is obvious and we wish to exemplify it by calculating the chiral anomaly in this approach. Even though the chiral anomaly has been computed in the framework of SQ by a number of authors [ 7 ], all the previous calculations have adopted the Langevin point of view; only in one case [ 8 ] the attempt has been made to derive the anomaly in the Fokker-Planck formalism, but the regularization scheme adopted in ref. [ 8 ] differs by the usual SR because these authors choose to regulate the noise in the x u direction instead of adopting the usual smearing of the noise correlation in the t-direction.

2. The effective Fokker-Planck hamiltonian

To begin with, we consider the Langevin equation with weakly coloured noise; to be definite we consider fermion fields interacting with an abelian background field A ~ ' ( x ) in a 4D euclidean space: the general case as well as other examples will be discussed elsewhere; thus we have O~(x, t ) / O t = - ( I ~ - i m ) ( l ~ + i m ) ~(x, t) +~/(x, t) = W[q/] + q ( x , t),

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O~7(x, t)/Ot= - (Fp' - i m ) T ( I ~ ' +im)T~7(X, t) +g(X, t) = W[g] + r/(X, t),

(lb)

where g, ~ are fermion fields, D u = 0u- ieAu, D'u = - Ou- ieA u and {7~, 7,} = - 6u~; the two-point correlation function for the Grassmann noises r/, g is

( r/(x, t)Yl(y, t' ) ) =2K(x, y)CA ( t--t' ),

(2)

with

K(x, y) =i (I~x-im ) 8 ( x - y)

(3)

and

Ca ( t - t ' ) = (A/Z) e x p ( - A I t - t ' l ) .

(4)

In the limit A--, ~ , CA (t-- t' ) ~ 8 (t-- t' ) and the process becomes markovian; we also define the Green's function A(x, y) by the formula

f d4w K(x, w)A(w, y) =~(x--y),

(5)

as well as the function CA that satisfies

f ds CA (t--s)CA (S-- t' ) =~(t--t' ).

(6)

We wish to derive an effective Fokker-Planck equation for the probability functional defined by

P ( t ) - P [ ~ , gT;t]-- j [dr/] [d~]P[r/, ~]8[~,-~u(t) ]8[~-~t(t) ],

(7)

where ~ ( t ) - ~(x, t) and gT(t)- ~7(x, t) are solutions of eqs. (1) (they are also fuctionals of r/and ~ respectively), [... ] means functional integration. P[r/, ~] is defined as follows:

P[r/,F1]=Nexp ( - ~ l ~ d4x d4x, dtdt' CA(t--t')ff(x,t)A(x,x')r/(x',t')O(x'

, x) ) ,

(8)

where N is given by the normalization condition f [dr/] [d~]P[r/, O] = 1,

(9)

and (b(x, x' ) is a factor that enforces gauge invariance: x'

¢(x,x'):exp(iefAu(z)dzu).

(10)

X

From eq. ( 7 ) and from the Langevin equations ( 1 ) we have ( (u= O~(x, t)/Ot)

(J(t)8[~t-~(t)]+8[~-~,(t)] OP(t)ot _ _ f [dr/] [d~]P[r/, ~] 3 d4x(\ ~8[~-~'(t)] (5~(x) =-

d4x~--~W[~lP(t)

= ~8[gt-~(t)] ~(t)) (5~(x)

8~(x) W[~lP(t)

- f d4x ~ ( x ) f [dr/] [drTlq(x, t)P[q, V/]a[~-~(t)]O[~,-~(t) ] - J" d4x ~--~-~ f [dq] [drT]rT(x, t)P[q, O]a[~,-~,(t)la[~,-~,(t)].

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The last two integrals in eq. ( 11 ) can be computed as follows: by using eq. (8) one has

~P[r/, fl]/~fl(x, t) = - ½e[r/, ~] J d4yds CA( t-s)A(x, y)r/(y, s)q)(y, x),

(12)

on the other hand, from eqs. ( 5 ) and (6) one gets

r/(x,t)e[q, fl]=e[r/,fl] f d u C a ( t - u ) fd4wg(x,w) fd4ydsCA(U-s)A(w,y)r/(y,s)fb(y,w),

(13)

which, together with eq. (12), gives

q(x,t)P[rl, fl]=-2

~P[r/, O] . duCA(t-u) ~ d4wK(x,w) ~fl(w,u)

(14)

We now apply this result to the third integral on the RHS ofeq. ( 11 ); we have ~ [dr/] [d~]r/~(x, t)P[r/, ~ l ~ [ ~ - ~ ' ( t ) ] ~ [ ~ - ~(t) ] 8P[rt, r/]

=

- 2 ~ d'w du CA (t-u)Kaa(x, w) ~ [dr/] [d~16[~/-~,(t)16[¢7- ¢/(t) ] 8ffl(w, u)

=2 f d4w du CA(t-u)K.p(x, w) I

[dr/] [dg]P[r/, Jq],~[~-~(t)]

&~[~-~(t)l 6ga(w, u)

=-2 f d4wduCA(t-u)r, ,(x,w) f tdr/ltdOl t,,Ol t - (,)l jo = - 2 f d4wduCA(t-u)K,~p(x,w ) d z ~

[dr/][dfl]P[r/,fl]6[g/-g(t)]~[gt-q/(t)]~u

), (15)

where we have performed an integration by parts. In order to evaluate the functional derivative 6 ~ / ~ appearing on the RHS ofeq. (15), we observe that, if we put

~ ( z , t) /~fla(w, u) =0(z, w)Xra(z, w; t, u),

(16)

then.~a satisfies the following equation, obtained from eq. ( 1b):

O(z'w) O'A~a(z'w;t'u)ot

= fJ d'ydt' 8if'r[ ~ , qT(z, ~ t)] ~(y,w).4~(y,w;t',u)+~a~(z-w)~(t-u)

= - f d'y (I~ +m2)-CO(y-z)~)(y, w)A~ (y, w; t, u)+Srp6(z-w)6(t-u),

( 17)

where D'u = - 0 / 0z u - ieAu (z). The solution of eq. ( 17 ) is given by

A~a(z, w; t, u) = i ds~(s-u) exp{- (t-u) [I~'z2 +rn~lX}ra~(z-w), 0

so that, by inserting this result on the RHS ofeq. ( 15 ), one obtains

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f [dr/] [d#]~U(x, t)e[rl, #]5[~u-~u(t) ]~[~u-~u(t) ]

=_2 f d4wduCA(t_u)Kaa(x,w) f d4z ~ O8 ( z ,

w)

t × j ds 5 ( s - u) exp{- ( t - u) [ I ~ + rnZ]X}~pO(z-

w)P(t)

0

d z~O(z, ~--

w) duexp{-(t-u)[A+(I~]+m2)Tl}~p6(z-w)P(t) 0

dz~O(z,w)

dOexp{-(O/A)[A+(I~+m2)T]}raO(z-w)P(t),

(19)

0 where we have put At=m in the upper limit of the O integration since A, t--,m. A similar formula holds for the fourth integral appearing on the RHS ofeq. ( 11 ); in conclusion, by introducing the function Ga (x, z) = j dO exp{-

(O/A) [A+ (l~"2+mZ)T]}J(Z--X)

(20)

0

we obtain from eq. ( 11 ) the following effective Fokker-Planck equation:

t]/Ot= --HAP[~u,~; t],

0P[ ~,, ~

(21)

where the effective Fokker-Planck hamiltonian is given by HA=-- f -i +i

d'x( ~ 8

d4x

([/}2+m2)(//(x)+

(I~x-im)

~8

(I~2+m2)T#(x) )

f d4x'O(x',X)GA(X,X')8~(x,)

8 )" f d 4 x ~ ( x ) ( l ~ ' - i m ) T f d4x'O(x, x' )GT (x',x) 8~(x'

(22)

In conclusion, starting with the Langevin equations (1) with weakly coloured noises (A-,m) we have obtained an effective Fokker-Planck hamiltonian that reduces to the unregularized one in the limit A ~ m , when

GA(X,X')~a(X--X'). By performing a similarity transformation

H'a=eAHae -A, A = - f d4yd4zd4w~-~(y,Z)O(W,z)GA(z, W) 8~(W)'8

(23)

with A defined by

i(I~,:+im)5(x.y) =5(x-y),

(24)

one obtains

H,A=_f d 4 x ( ~ (rp~+rnZ)~u(x)+ 8

)

( I ~2 +m2)TqT(X) ,

(25)

which, as shown in ref. [ 5], is a positive semidefinite operator. It follows that H~ is a positive semidefinite 89

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operator as well. Thus, under the usual hypothesis of the existence of a mass gap between the zero mode and the non-zero modes one obtains P[~, ~ t] = ~ c. exp( -Ent)Z. [q/, ~] n

' CoXo[~, qT],

(26)

t~oo

where Hazn=E.z., the cn are determined by the initial condition and HaZo= 0. It is easy to see that (Co= 1 ) Zo[~, qY]= ( l / Z ) exp{-SA [~/, q~]},

(27)

with Z determined by the normalization condition and

SA = --i J d4z d4E ~(z+E/2)Gx 1(z+ E/2, z-¢/2)O(z-E/2, z+ ~/2) (I~z-~/z +im)~(z-~/2).

(28)

G- ~(x. y ) is defined by ~d4x Ga (z, x)G~ ~(x, y) = ~ ( z - y ) ,

(29)

so that l i m a ~ G~ ~(x, y) = ~ ( x - y ) ; this allows to write lim SA = lira S,, A~oo

(30)

~oo

S, = - i f d4z ~(z+¢/2)(~(z-E/2, z+E/2) (I~z_,/z +irn)~(z-E/2).

(31)

We observe that the introduction of weakly coloured noises has eventually produced a separation in the product of singular operators similar to the "point splitting technique" [ 9 ].

3. Derivation of the anomaly

In order to derive the chiral anomaly in the Fokker-Planck formalism we follow the approach of ref. [ 8 ]; we begin with the normalization condition ~ [ d ~ ] [dqY]P[¥, ~, t] = 1,

(32)

and consider the chiral transformation q/(x)--,~,' (x) =exp[iot(x)?5 ]~u(x), ~7(x)--,q7'(x) = ~ ( x ) exp[ioz(x)~,5 ].

(33)

If

J=exp (i ~ &xot(x)Q(x))

(34)

is the jacobian of the transformation, then from eq. (32) one has

i ~ d4xot(x)Q(x)=

f Ida,] [d~]6e[~, if;,t]

= ~ c n f [dg] [dq7] exp(-Ent)Z. [~, ~] n

t~oo

'-ZI

[dg] [d~] exp(-Sa)6Sa,

(35)

so that

i f d ' x o t ( x ) a ( x ) = - lim (6S,) = - l i m l f [dg] [d~71 ~0

90

e~O Z

(36)

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From eqs. (31), (33) and (36) one has 8S, = 2 im ~ d4z a(z)J5 + J d4z ( ( z + e / 2 ) B ~ ( z - , / 2 ) ,

(37)

with

(38)

•15= ~(z+ ~12)O(z- ~12, z+ EI2)~s Iv(z- ~12) and

B = O ( z - el2, z+~/2 ){ - Eu (OalOz.)I~z + [ 1 - (~u/2) (O/Ozu ) l~a (z) }ys.

(39)

We now compute (8S~) in two different ways; on one hand we have (A(y, x) defined in eq. (24)) (6S,) = 2 im f d4z ot(z) ( J5 ) - i f d4z d4x t~(x- z - ~ / 2 ) Tr {Ba(z--E/2, x) }.

(40)

The last integral can be computed by introducing the following representation for A (y, x):

7~,~#(y,x)= -i(l~y + im )g~ ,~(y-x) = --i

q/~(Y)gteP (x) E+im '

(41)

where ~ve(x) are eigenfunctions of the operator 1~: rp~ve=E~VF:by using cyclic properties of the trace we obtain (to the leading order in e): (6S~) = f daz ot(z)E~F,u Yr{yu~sA(z- e/2, z+e/2)}.

(42)

On the other hand, by integrating by parts in eq. (37), one obtains (~S~) = J d4z or(z) (2 ira J5 -OuJsu),

(43)

with

Jsu = ~(z+ ~12 )?u 75~v(z- ~12 )q)(z- e/2, z + e/2 ),

(44)

so that, by comparison, we get

Ou( Jsu ) = 2 im( J5 ) --E,F~u Tr{yuy57~(z--E/2, z+E/2)}.

(45)

The trace can be computed as follows: lira Tr{yu?5~ ( z - e/2, z+ e/2)} = lira Tr{yu?5~ (z, y)} e~O

=

y~z

-ilimy~=Tr{r~r5 [rp~+im]-~}8(z-y)=

~- --~e Fav Tr{yuY,7~Y~Yv}lim ---~-0 y~z 0z~

f d4k -ilimy~ J (--~n)4 Tr(7~75 [ l ~ + i m ] - ~ e x p [ - i k ( z - y ) ] }

d4k e x p [ - i k ( z - y ) ] e _ (2n) 4 (k2+m2) 2 = --

e~ ~-~n2/~#E,,~# ~-i"

(46)

Taking into account the prescription [9 ] l i m , ~ o : E ' / d =g"~'/4, one finally gets 0r (Jsu) = 2 ira(J5 ) + (e/8n 2) Fu,,F~,

(47)

which is the result we wanted to prove. We can also obtain the jacobian of the transformation (33); as a matter of fact, from eqs. ( 3 3 ) - ( 3 6 ) , (42) and ( 4 6 ) - ( 4 7 ) one obtains J-~l+i

daxot(x)Q(x)=l+

e:

d4xa(x)Fu~F~,,

(48)

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this is Fujikawa's result [ 10]: whereas Fujikawa computes the jacobian directly, we have obtained it from a comparison with (~S,).

4. Conclusion In conclusion we have been able to obtain an effective Fokker-Planck equation in a stochastic regularized quantum field theory, when the gaussian noises appearing in the Langevin equations are weakly coloured; as an example we have computed from the effective Fokker-Planck hamiltonian the chiral anomaly, recovering the usual results of refs. [ 9 ], [ 10 ].

Acknowledgement We thank P. Cea for very useful discussions.

References [ 1 ] G. Parisi and Y. Wu, Sci. Sin. 24 ( 1981 ) 483. [2] D. Zwanziger, Nucl. Phys. B 192 ( 1981 ) 259; E. Floratos and J. Iliopoulos, Nucl. Phys. B 214 (1983) 392; W. Grimus and H. Hueffel, Z. Phys. C 18 ( 1983 ) 129; W. Grimus and G. Nardulli, Nuovo Cimento 91 A (1986) 384. [ 3 ] J.D. Breit, S. Gupta and A. Zaks, Nucl. Phys. B 233 (1984) 61. [4] J.L. Doob, Ann. Math. 43 (1942) 351; K. Ito, Proc. Imp. Acad. Tokyo 20 (1944) 519; Mem. Am. Math. Soc. 4 ( 1951 ) 1; R.L. Stratonovich, SIAM J. Control 4 (1966) 362. [ 5 ] B. Sakita, 7th Johns Hopkins Workshop, eds. G. Domokos and S. Kovesi-Domokos (World Scientific, Singapore, 1983 ). [6] R.F. Fox, Phys. Rev. A 33 (1986) 467; Phys. Rev. A 34 (1986) 4525; J. Star. Phys. 46 (1987) 1145. [7] J. Alfaro and M.B. Gavela, Phys. Lett. B 158 (1985) 473; E.R. Nissimov and S.J. Pacheva, Phys. Lett. B 171 (1986) 267; M.B. Gavela and N. Parga, Phys. Lett. B 174 (1986) 319; R. Kirschner, E.R. Nissimov and S.J. Pacheva, Phys. Lett. B 174 (1986) 324; E.S. Egorian, E.R. Nissimov and S.J. Pacheva, Lett. Math. Phys. 11 ( 1986 ) 209; J.P. Ader and J.C. Wallet, Z. Phys. C 32 (1986) 575; Z. Bern, H.S. Chan and M.B. Halpern, Z. Phys. C 33 (1986) 77; R. Tsani, Phys. Rev. D 33 (1986) 1146. [8] J.A. Magpantay and M. Reuter, preprint DESY 87-004 (1987). [9 ] R. Jackiw, in: Current algebra and anomalies, eds. S.B. Treiman, R. Jackiw, B. Zumino and E. Witten (World Scientific, Singapore, 1985) p. 81. [ 10] K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195; Phys. Rev. D 21 (1980) 2848; D 22 (1980) 1499 (E).

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