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Stochastic derivation of the chiral anomaly based on the generalized langevin equation
Volume 194, number 4
PHYSICS LETTERSB
20 August ! 987
S T O C H A S T I C DERIVATION OF T H E C H I R A L ANOMALY BASED O N T H E G E N E R A L I Z E D LANGEVIN EQUATION Mikio NAMIKI a,b, Ichiro OHBA a, Satoshi TANAKA ~ and Danilo M. YANGA ~' a Department of Physics, Waseda University, Tokyo 160, Japan b The Institute of Statistical Mathematics, Tokyo 106, Japan Received 4 March 1987
A generalized Langevin equation for fermion field is first derived within the frameworkof the stochastic quantization. Based on it, the chiral anomaly is derived directly from the stationary property of the pseudoscalardensity. This approach is convenient to observethe quantum origin of anomalies. The conservationlaw of the vector current is also derived in a similar way.
Recently, within the framework of the stochastic quantization method (SQM), several authors showed that it is possible to derive the chiral anomaly by solving the Langevin equations for fermion fields and using these solutions to calculate the divergence of the chiral current [ 1 ]. It was also argued that the chiral anomaly appears due to fluctuations in the evolution of the system to its equilibrium state [ 2 ]. In spite of these works the origin of the chiral anomaly is still not so clear. On the other hand, path-integral quaniization enables to derive the chiral anomaly from the non-invariance of the path-integral measure under the chiral transformation [ 3 ]. In this letter, we present a simple and straightforward way of deriving the chiral anomaly, within the framework of SQM, which would be more convenient to clarify its quantum origin. Our approach is based on a fermionic analogue of the generalized Langevin equation for an arbitrary observable already formulated in bosonic systems [4] ~, but it does not require to solve the Langevin equation. It is shown that the chiral anomaly comes out naturally from the stationary property of the pseudoscalar density. The conservation law of the vector current is also derived from the stationary property of the scalar density. First let us derive the generalized Langevin equation for an arbitrary functional of fermion fields, making use of the technique of Ito's stochastic differential calculus [ 6 ]. The Langevin equations for the independent fermion fields g " ( x , t) and ~ " ( x , t) are, respectively, written as [ 7] dq/"(x,t)-
~ '~~S( (xt,)t ) d t + d O ~ ( x , t ) '
d ~ ' ~ ( x ' t ) = 8 ~6S(t) - - ~ , t ) dt+dO'~(x't) '
(la,b)
where t is the fictitious time variable, and dO~ (x, t), dO'~(x, t) are Grassmann random variables satisfying (dO ~ ) = ( d 0 ~ ) = 0 ,
(d0~d0 p ) = ( d 0 ~ d 0 p ) = 0 ,
(2a,b)
(dO'~(x, t)d0P (y, t) ) = - (d0a(y, t)dO~(x, t) ) = 2 h t ~ P ~ ( x - y ) d t .
(2c)
S(t) is the action which is obtained by replacing the arguments g ( x ) and ~(x) with ~t(x, t) and q~(x, t), respectively, in the classical action S. The operators 6 / 8 ~ and ~/~q7 are left derivatives, a and fl are spinor indices Permanent address: National Institute of Physics, College of Science, University of the Philippines Diliman, Quezon City 3004, Philippines. ~ Similar equations were also derived in ref. [ 5 ]. 530
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 194, n u m b e r 4
PHYSICS LETTERS B
20 August 1987
( o t h e r indices being suppressed, if any). Remember that, according to Ito's calculus, we have (f(¢,~v;t)dO"(x,t)) = (f(~,~u;t)dO'~(x,t)) = 0 for an arbitrary function of ~v and ~,f(~,~v;t), at time t. Note that eq. (1) is a straightforward extension of the classical field equations 8S/8~v=0 and ~ S / 8 ~ = 0 see ref. [ 8 ]. It is natural to use such Langevin equations to analyze the anomaly, since the classical Noether's theorem is derived using the field equations and the anomaly is a breaking-down of Noether's theorem. (The Langevin equation (1) necessitates introduction of a fermionic mass term to allow for convergence to equilibrium state [ 8 ]. On the other hand, the Langevin equation with a kernel [ 9] needs not the mass term for the convergence, but still needs the mass term to give the correct chiral anomaly [ 1 ].) Also note that the Planck constant h is introduced in eq. (2) as the diffusion constant of the hypothetical stochastic process, and that dO and dO are of the order of dt ~/2 The Langevin equation for an arbitrary functional, F(t)=F[~,~v;t], is obtained by making use of eq. (1) in the Taylor expansion of F(t + dt). Neglecting higher order terms involving O (dt3/2), we get the generalized Langevin equation dF( t)
f dy F ( d L\
8S(t)
8F(/)
~iS(t)
8F(t)
5~(y,t) 6p,"(y,t) +8~v,~(y,t) - - ~ , ]
+ ~ dy i dz(dO'~(y't)dOB(z't) +½dO.(y,t)dOp(z,t)
8/F(t)
8~(z,t)8~"(y,t)
"]
8F(t)
dt+dO~(y't) -8~v'~(y,t) +dO"(y't)
~iF(t) ] 8~(y,t)]
~-½dO'~(Y't)dOP(z't) 82F(t) 8p,P(z,t)6~u'~(y,t)
6ZF(t) "] 8~p(z,t)8~,~(y,t).l ,
(3)
where the order of the Grassmann variables is decided according to the prescription of the left derivatives. It is convenient to introduce a sort of partial differentials including higher order effects of de/and d~, which are defined through
~l~.r(t ) =.]f dyd~"(Y't)
1(8F(t) ~ 8F(s) "] 2\8~-(-y-~-t) 8~(y,s) . . . . ~+dtJ'
(4a)
dcV(t) = Jf dyd¢'~(Y't)
1(8F(t) 2\8~t)
(4b)
k 8F(s) 8¢"(y,s) s=t+dt]'
By expanding the second terms in the RHS ofeqs. (4a) and (4b), we can prove that the stochastic differential dF(t) is rewritten as dF(t) = d~,F(t) + aCF(t)
(5)
in terms of the partial differentials, apart from higher order terms involving O(dt3/2). We can also prove that expectation values of the partial differentials d~,F and a ~ F are expressed as
(7:l~,F(t))=dtf[d~] [d~]F[~,u/] dy[ (~lcF(t))=dt f
[dp,] [d~]F[~,~v]
f d y [F~ 8
8
8
{h ~ 8
8S
+ ~ -8S ~ ) ] "X-]p[~,~',t] ,
(6a)
(6b)
where P[~,~v,t] is the Fokker-Planck distribution at time t. Since P[~,~v,t] tends to e x p ( - S / h ) as t~oo [8], eq. (6) means that ~2 ~2 This statement on partial differentials is also valid for each component of ~u and ~, and for the bosonic field, if any.
531
Volume 194, number 4
(~l~,F(t) ) ' ~ , oo ,
PHYSICS LETTERSB
(~¢F(t) ) '~,
20 August 1987
oo.
(7)
Therefore any linear combination composed of d~,F(t) and dCF(t) has a vanishing expectation value at the limit t--,oo. Using the generalized Langevin equation (3), we can formulate a simple prescription to derive the chiral anomaly. We consider a system described by the action functional
S(t) - - S [ ¢ , ~ ; t ] = J dx ~t(x,t)(-ilO+m)~/(x,t),
(8)
where Ib = 7u [ 0# + A u ( x ) ], A u ( x ) = - igA ~( x ) T a being the SU (N) gauge field ~3 with generators T", and the 7-matrices obeying {Tu, 7~} =2gu~, g ~ = ( - 1, - 1, - 1, - 1), 7+ = -Tu, and 75 = -71727374 = -7~-. Then the Langevin equations ( l a ) and ( l b ) for ~ and q/become explicitly dq/(x,t) = ( i ~ + i A - m ) ~ ( x , t ) d t + d O ( x , t ) ,
d~(x,t) = ~ ( x , t ) ( - i ~ + i A - m ) d t + d O ( x , t ) .
(9a,b)
Now, in order to derive the chiral anomaly, we make a generalized Langevin equation for the pseudoscalar density. By making use of the general formula (3), we obtain
dj5(x~t)=-[i~uj~(x~t)+ 2mj5(x~t)]dt-d~(x~t)d~a(x~t)7~" +¢(x~t)75d~(x~t)+d~(x~t)75~(x~t) ,
(10)
where the pseudoscalar density and the chiral current are given by
js(x,t) =~t(x,t)75g/(x,t),
jSu(x,t) =~t(x,t)Tu75~u(x,t),
(11,12)
respectively. According to Ito's calculus, the expectation values of the last two terms on the RHS of eq. (10) become zero, but those of the remaining terms are not vanishing. The latter terms describe the systematic time evolution of J5 (x, t), including the term d0" (x, t) d 0 p (x, t) 7g" which reflects effects of quantum fluctuations. Averaging both sides of eq. (10) we obtain
d ( j s ( x , t ) ) / d t = - i ( OujSu(x,t)- 2mijs(x,t) ) - 2h tr[75O(x-x)l ,
(13)
where tr stands for trace over both spinor and SU(N) indices. Since the LHS of eq. (13) becomes zero in the equilibrium state at the limit t--,~, we get the following identity:
(OujSu(x) - 2 m i j s ( x ) ) = 2 i h t r [ 7 5 ~ ( x - x ) ] •
(14)
Eq. (14) coincides with the usual (unregularized) anomalous chiral Ward-Takahashi (WT) identity. Thus we found that the anomalous chiral WT identity is nothing but the trivial stationary property ofjs(x, t) in the equilibrium state, i.e. lira d ( J s ( x , t ) ) = 0 . t ~ at
(15)
The anomaly term 2ih t r [ 7 5 O ( x - x ) ] comes from the term dO"(x,t)dOP(x,t)7 p'~ in eq. (10), which is purely made of quantum fluctuations and gives a non-zero expectation value. It is manifest that the anomaly term is proportional to h which comes directly from the diffusion constant in eq. (2c). The ill-defined anomaly term 2i h tr[ 7 5 ~ ( x - x ) ] is evaluated by regularizing the correlation relation (2c) as
(dO'~(x,t)dOP(y,t)) = lim 2h d T Z ¢f~*(y) exp[ - (2,/M)Z]O~(x), M~oo
(16)
n
where the ¢~(x) and the 2, are eigenfunctions and eigenvalues of lO, respectively; I~O,(x) =2,O,(x). Then the same manipulation as in ref. [ 3 ] leads us to the well-defined anomaly term ~3The gauge field A~ needs not be limited to an external field. For quantized Au, however, we have to set up an additional Langevin equation for it. 532
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i< dO"(x, t)dOa(x, t)Ta5'~ > = - i(h/161t2)tr[ ~w,aaFUVF *p] ,
(17)
tr standing for the trace over SU(N) indices here only. Next, let us consider how the above derivation of the chiral anomaly is related to the chiral transformation. We study the variation of the action (8) under the local infinitesimal chiral transformation
8~(x,t)=i~(x)75~u(x,t),
8#(x,t)=i¢(x)#(x,t)7~,
(18a,b)
where ~(x) is an arbitrary infinitesimal local parameter. Then we obtain 8S(t) iO~j~(x't)+ 2 m j s ( x ' t ) = ~ ( x ' t ) 7 5 8~(x,t)
8S(t) 8~(x,t) 75V(x,t) •
(19)
In the fields were not quantized, the RHS of eq. (19) should be equal to zero as a consequence of the classical field equations 8S/8 ¥ = 0 and 5S/8~p = O. In our case of stochastically quantized field theory, however, the ~, and ~ are fields subject to the Langevin equations (1), and j~ is not a Noether current. Using eq. (1) we get [RHS ofeq. (19)]dt= ¢775( - d c / + d 0 ) - ( d ~ - d 0 ) y ~ ~/ = -
[(a~,j5 -
½d¢75 dv) +
(a~j~ -
½d¢75d~,)] + ¢75d0 +d075 ~,.
(20)
Substituting this into eq. (19) and neglecting higher order terms involving O(dt3/2), we are led to a~,j5 (x, t) + ~J5 (x, t) = dj5 (x, t)
= - [id~'j~(x, t) + 2mjs(x, t)] d t - dO'~(x, t)dOP(x, t)7~ '~ + ~9(x, t)75dO(x, t) + d 0 ( x , 075 V(x, t ) ,
(21 )
which is just the generalized Langevin equation for J5 (x, t) given by the general prescription (3). And so the chiral anomaly is derived. A similar derivation is also possible using the Langevin equation with a kernel in ref. [9]. For comparison, we derive the conservation law of the vector current. In this case, we study the variation of the action S(t) under the phase transformation
8¢/(x,t)=i~(x)~u(x,t),
8#(x,t)=-ie(x)~(x,t).
(22a,b)
~S(t) 8S(t) iO~'ju(x,t) = - ~ ) ( x , t ) 8~9(x,t~----) ~/(x,t--------~~u(x,t) ,
(23)
Then we obtain
where ju = ~97u¢' is the vector current. A similar procedure to eq. (20) gives [RHS ofeq. ( 2 3 ) ] d t = [(a~,p- ld~d~u) - ( a e p - ½d~gd~u)] -~gd0 + d0¢,,
(24)
where p(x, t) = ~(x, t)~u(x, t) is the scalar density. Note that the terms ½d~d~u in the square brackets cancel out each other, so that duplication of random noises disappears in this case. This point is essentially different from the procedure of deriving eq. (21), where the term d 0 " d 0 a T ~ remains to give the chiral anomaly. Hence, neglecting higher order terms involving O(dt3/2), we obtain
a~,p(x, t) - ~lop(x, t) = iOUju(x, t)dt + ~(x, t)dO(x, t) - dO(x, t) ~u(x, t ) .
(25)
The expectation values of both sides in eq. (25) then give
( iOujv(x, t) ) = ( a~p(x,t) ) - ( a~p(x, t) ) .
(26)
The RHS goes to zero as t--*oo because of eq. (7). Consequently, the convervation law of the vector current 533
Volume 194, number 4
(OUju(x))
=0
PHYSICS LETTERS B
20 August 1987
(27)
is obtained. We have shown that the chiral anomaly is obtained from the stationary property of the pseudoscalar density, based on the generalized Langevin equation. Our approach is very simple and straightforward. Moreover, there is no need to solve the Langevin equations as other authors have done. The anomaly term comes from the duplication of the random noises in the generalized Langevin equation for the pseudoscalar density, reflecting effects of the quantum fluctuations. On the contrary, we have shown that the vector current is conserved, because the duplications of the random noises cancel out each other. This approach can be applied to various types of anomalies. One of the authors (D.Y) thanks the Japan Society for the Promotion of Science for its financial support.
References [ 1 ] J. Alfaro and M.B. Gavela, Phys. Lett. B 158 (1985) 473; R. Tzani, Phys. Rev. D 33 (1986) 1146. [2] M.B. Gavela and N. Parga, Phys. Lett. B 174 (1986) 319; Nucl. Phys. B 275 (1986) 546. [3] K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195; Phys. Rev. D 21 (1980) 2848. [4] M. Namiki, I. Ohba and S. Tanaka, Phys. Lett. B 182 (1986) 66. [5] S. Caracciolo, H.-C. Ren and Y.-S. Wu, Nucl. Phys. B 260 (1985) 381. [ 6 ] E.g., see N. lkeda and S. Watanabe, Stochastic differential equations and diffusion processes (North-Holland, Amsterdam, ! 981 ). [7] T. Fukai, H. Nakazato, I. Ohba, K. Okano and Y. Yamanaka, Prog. Theor. Phys. 69 (1983) 1600. [ 8 ] B. Sakita, in: Quantum theory of many-variable systems and fields (World Scientific, Singapore, 1985) p. 173; J.D. Breit, S. Gupta and A. Zaks, Nucl. Phys. B 233 (1984) 61; P.H. Damgaard and K. Tsokos, Nucl. Phys. B 235 (1984) 75; K. Ishikawa, Nucl. Phys. B 241 (1984) 589.
534
PHYSICS LETTERSB
20 August ! 987
S T O C H A S T I C DERIVATION OF T H E C H I R A L ANOMALY BASED O N T H E G E N E R A L I Z E D LANGEVIN EQUATION Mikio NAMIKI a,b, Ichiro OHBA a, Satoshi TANAKA ~ and Danilo M. YANGA ~' a Department of Physics, Waseda University, Tokyo 160, Japan b The Institute of Statistical Mathematics, Tokyo 106, Japan Received 4 March 1987
A generalized Langevin equation for fermion field is first derived within the frameworkof the stochastic quantization. Based on it, the chiral anomaly is derived directly from the stationary property of the pseudoscalardensity. This approach is convenient to observethe quantum origin of anomalies. The conservationlaw of the vector current is also derived in a similar way.
Recently, within the framework of the stochastic quantization method (SQM), several authors showed that it is possible to derive the chiral anomaly by solving the Langevin equations for fermion fields and using these solutions to calculate the divergence of the chiral current [ 1 ]. It was also argued that the chiral anomaly appears due to fluctuations in the evolution of the system to its equilibrium state [ 2 ]. In spite of these works the origin of the chiral anomaly is still not so clear. On the other hand, path-integral quaniization enables to derive the chiral anomaly from the non-invariance of the path-integral measure under the chiral transformation [ 3 ]. In this letter, we present a simple and straightforward way of deriving the chiral anomaly, within the framework of SQM, which would be more convenient to clarify its quantum origin. Our approach is based on a fermionic analogue of the generalized Langevin equation for an arbitrary observable already formulated in bosonic systems [4] ~, but it does not require to solve the Langevin equation. It is shown that the chiral anomaly comes out naturally from the stationary property of the pseudoscalar density. The conservation law of the vector current is also derived from the stationary property of the scalar density. First let us derive the generalized Langevin equation for an arbitrary functional of fermion fields, making use of the technique of Ito's stochastic differential calculus [ 6 ]. The Langevin equations for the independent fermion fields g " ( x , t) and ~ " ( x , t) are, respectively, written as [ 7] dq/"(x,t)-
~ '~~S( (xt,)t ) d t + d O ~ ( x , t ) '
d ~ ' ~ ( x ' t ) = 8 ~6S(t) - - ~ , t ) dt+dO'~(x't) '
(la,b)
where t is the fictitious time variable, and dO~ (x, t), dO'~(x, t) are Grassmann random variables satisfying (dO ~ ) = ( d 0 ~ ) = 0 ,
(d0~d0 p ) = ( d 0 ~ d 0 p ) = 0 ,
(2a,b)
(dO'~(x, t)d0P (y, t) ) = - (d0a(y, t)dO~(x, t) ) = 2 h t ~ P ~ ( x - y ) d t .
(2c)
S(t) is the action which is obtained by replacing the arguments g ( x ) and ~(x) with ~t(x, t) and q~(x, t), respectively, in the classical action S. The operators 6 / 8 ~ and ~/~q7 are left derivatives, a and fl are spinor indices Permanent address: National Institute of Physics, College of Science, University of the Philippines Diliman, Quezon City 3004, Philippines. ~ Similar equations were also derived in ref. [ 5 ]. 530
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 194, n u m b e r 4
PHYSICS LETTERS B
20 August 1987
( o t h e r indices being suppressed, if any). Remember that, according to Ito's calculus, we have (f(¢,~v;t)dO"(x,t)) = (f(~,~u;t)dO'~(x,t)) = 0 for an arbitrary function of ~v and ~,f(~,~v;t), at time t. Note that eq. (1) is a straightforward extension of the classical field equations 8S/8~v=0 and ~ S / 8 ~ = 0 see ref. [ 8 ]. It is natural to use such Langevin equations to analyze the anomaly, since the classical Noether's theorem is derived using the field equations and the anomaly is a breaking-down of Noether's theorem. (The Langevin equation (1) necessitates introduction of a fermionic mass term to allow for convergence to equilibrium state [ 8 ]. On the other hand, the Langevin equation with a kernel [ 9] needs not the mass term for the convergence, but still needs the mass term to give the correct chiral anomaly [ 1 ].) Also note that the Planck constant h is introduced in eq. (2) as the diffusion constant of the hypothetical stochastic process, and that dO and dO are of the order of dt ~/2 The Langevin equation for an arbitrary functional, F(t)=F[~,~v;t], is obtained by making use of eq. (1) in the Taylor expansion of F(t + dt). Neglecting higher order terms involving O (dt3/2), we get the generalized Langevin equation dF( t)
f dy F ( d L\
8S(t)
8F(/)
~iS(t)
8F(t)
5~(y,t) 6p,"(y,t) +8~v,~(y,t) - - ~ , ]
+ ~ dy i dz(dO'~(y't)dOB(z't) +½dO.(y,t)dOp(z,t)
8/F(t)
8~(z,t)8~"(y,t)
"]
8F(t)
dt+dO~(y't) -8~v'~(y,t) +dO"(y't)
~iF(t) ] 8~(y,t)]
~-½dO'~(Y't)dOP(z't) 82F(t) 8p,P(z,t)6~u'~(y,t)
6ZF(t) "] 8~p(z,t)8~,~(y,t).l ,
(3)
where the order of the Grassmann variables is decided according to the prescription of the left derivatives. It is convenient to introduce a sort of partial differentials including higher order effects of de/and d~, which are defined through
~l~.r(t ) =.]f dyd~"(Y't)
1(8F(t) ~ 8F(s) "] 2\8~-(-y-~-t) 8~(y,s) . . . . ~+dtJ'
(4a)
dcV(t) = Jf dyd¢'~(Y't)
1(8F(t) 2\8~t)
(4b)
k 8F(s) 8¢"(y,s) s=t+dt]'
By expanding the second terms in the RHS ofeqs. (4a) and (4b), we can prove that the stochastic differential dF(t) is rewritten as dF(t) = d~,F(t) + aCF(t)
(5)
in terms of the partial differentials, apart from higher order terms involving O(dt3/2). We can also prove that expectation values of the partial differentials d~,F and a ~ F are expressed as
(7:l~,F(t))=dtf[d~] [d~]F[~,u/] dy[ (~lcF(t))=dt f
[dp,] [d~]F[~,~v]
f d y [F~ 8
8
8
{h ~ 8
8S
+ ~ -8S ~ ) ] "X-]p[~,~',t] ,
(6a)
(6b)
where P[~,~v,t] is the Fokker-Planck distribution at time t. Since P[~,~v,t] tends to e x p ( - S / h ) as t~oo [8], eq. (6) means that ~2 ~2 This statement on partial differentials is also valid for each component of ~u and ~, and for the bosonic field, if any.
531
Volume 194, number 4
(~l~,F(t) ) ' ~ , oo ,
PHYSICS LETTERSB
(~¢F(t) ) '~,
20 August 1987
oo.
(7)
Therefore any linear combination composed of d~,F(t) and dCF(t) has a vanishing expectation value at the limit t--,oo. Using the generalized Langevin equation (3), we can formulate a simple prescription to derive the chiral anomaly. We consider a system described by the action functional
S(t) - - S [ ¢ , ~ ; t ] = J dx ~t(x,t)(-ilO+m)~/(x,t),
(8)
where Ib = 7u [ 0# + A u ( x ) ], A u ( x ) = - igA ~( x ) T a being the SU (N) gauge field ~3 with generators T", and the 7-matrices obeying {Tu, 7~} =2gu~, g ~ = ( - 1, - 1, - 1, - 1), 7+ = -Tu, and 75 = -71727374 = -7~-. Then the Langevin equations ( l a ) and ( l b ) for ~ and q/become explicitly dq/(x,t) = ( i ~ + i A - m ) ~ ( x , t ) d t + d O ( x , t ) ,
d~(x,t) = ~ ( x , t ) ( - i ~ + i A - m ) d t + d O ( x , t ) .
(9a,b)
Now, in order to derive the chiral anomaly, we make a generalized Langevin equation for the pseudoscalar density. By making use of the general formula (3), we obtain
dj5(x~t)=-[i~uj~(x~t)+ 2mj5(x~t)]dt-d~(x~t)d~a(x~t)7~" +¢(x~t)75d~(x~t)+d~(x~t)75~(x~t) ,
(10)
where the pseudoscalar density and the chiral current are given by
js(x,t) =~t(x,t)75g/(x,t),
jSu(x,t) =~t(x,t)Tu75~u(x,t),
(11,12)
respectively. According to Ito's calculus, the expectation values of the last two terms on the RHS of eq. (10) become zero, but those of the remaining terms are not vanishing. The latter terms describe the systematic time evolution of J5 (x, t), including the term d0" (x, t) d 0 p (x, t) 7g" which reflects effects of quantum fluctuations. Averaging both sides of eq. (10) we obtain
d ( j s ( x , t ) ) / d t = - i ( OujSu(x,t)- 2mijs(x,t) ) - 2h tr[75O(x-x)l ,
(13)
where tr stands for trace over both spinor and SU(N) indices. Since the LHS of eq. (13) becomes zero in the equilibrium state at the limit t--,~, we get the following identity:
(OujSu(x) - 2 m i j s ( x ) ) = 2 i h t r [ 7 5 ~ ( x - x ) ] •
(14)
Eq. (14) coincides with the usual (unregularized) anomalous chiral Ward-Takahashi (WT) identity. Thus we found that the anomalous chiral WT identity is nothing but the trivial stationary property ofjs(x, t) in the equilibrium state, i.e. lira d ( J s ( x , t ) ) = 0 . t ~ at
(15)
The anomaly term 2ih t r [ 7 5 O ( x - x ) ] comes from the term dO"(x,t)dOP(x,t)7 p'~ in eq. (10), which is purely made of quantum fluctuations and gives a non-zero expectation value. It is manifest that the anomaly term is proportional to h which comes directly from the diffusion constant in eq. (2c). The ill-defined anomaly term 2i h tr[ 7 5 ~ ( x - x ) ] is evaluated by regularizing the correlation relation (2c) as
(dO'~(x,t)dOP(y,t)) = lim 2h d T Z ¢f~*(y) exp[ - (2,/M)Z]O~(x), M~oo
(16)
n
where the ¢~(x) and the 2, are eigenfunctions and eigenvalues of lO, respectively; I~O,(x) =2,O,(x). Then the same manipulation as in ref. [ 3 ] leads us to the well-defined anomaly term ~3The gauge field A~ needs not be limited to an external field. For quantized Au, however, we have to set up an additional Langevin equation for it. 532
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i< dO"(x, t)dOa(x, t)Ta5'~ > = - i(h/161t2)tr[ ~w,aaFUVF *p] ,
(17)
tr standing for the trace over SU(N) indices here only. Next, let us consider how the above derivation of the chiral anomaly is related to the chiral transformation. We study the variation of the action (8) under the local infinitesimal chiral transformation
8~(x,t)=i~(x)75~u(x,t),
8#(x,t)=i¢(x)#(x,t)7~,
(18a,b)
where ~(x) is an arbitrary infinitesimal local parameter. Then we obtain 8S(t) iO~j~(x't)+ 2 m j s ( x ' t ) = ~ ( x ' t ) 7 5 8~(x,t)
8S(t) 8~(x,t) 75V(x,t) •
(19)
In the fields were not quantized, the RHS of eq. (19) should be equal to zero as a consequence of the classical field equations 8S/8 ¥ = 0 and 5S/8~p = O. In our case of stochastically quantized field theory, however, the ~, and ~ are fields subject to the Langevin equations (1), and j~ is not a Noether current. Using eq. (1) we get [RHS ofeq. (19)]dt= ¢775( - d c / + d 0 ) - ( d ~ - d 0 ) y ~ ~/ = -
[(a~,j5 -
½d¢75 dv) +
(a~j~ -
½d¢75d~,)] + ¢75d0 +d075 ~,.
(20)
Substituting this into eq. (19) and neglecting higher order terms involving O(dt3/2), we are led to a~,j5 (x, t) + ~J5 (x, t) = dj5 (x, t)
= - [id~'j~(x, t) + 2mjs(x, t)] d t - dO'~(x, t)dOP(x, t)7~ '~ + ~9(x, t)75dO(x, t) + d 0 ( x , 075 V(x, t ) ,
(21 )
which is just the generalized Langevin equation for J5 (x, t) given by the general prescription (3). And so the chiral anomaly is derived. A similar derivation is also possible using the Langevin equation with a kernel in ref. [9]. For comparison, we derive the conservation law of the vector current. In this case, we study the variation of the action S(t) under the phase transformation
8¢/(x,t)=i~(x)~u(x,t),
8#(x,t)=-ie(x)~(x,t).
(22a,b)
~S(t) 8S(t) iO~'ju(x,t) = - ~ ) ( x , t ) 8~9(x,t~----) ~/(x,t--------~~u(x,t) ,
(23)
Then we obtain
where ju = ~97u¢' is the vector current. A similar procedure to eq. (20) gives [RHS ofeq. ( 2 3 ) ] d t = [(a~,p- ld~d~u) - ( a e p - ½d~gd~u)] -~gd0 + d0¢,,
(24)
where p(x, t) = ~(x, t)~u(x, t) is the scalar density. Note that the terms ½d~d~u in the square brackets cancel out each other, so that duplication of random noises disappears in this case. This point is essentially different from the procedure of deriving eq. (21), where the term d 0 " d 0 a T ~ remains to give the chiral anomaly. Hence, neglecting higher order terms involving O(dt3/2), we obtain
a~,p(x, t) - ~lop(x, t) = iOUju(x, t)dt + ~(x, t)dO(x, t) - dO(x, t) ~u(x, t ) .
(25)
The expectation values of both sides in eq. (25) then give
( iOujv(x, t) ) = ( a~p(x,t) ) - ( a~p(x, t) ) .
(26)
The RHS goes to zero as t--*oo because of eq. (7). Consequently, the convervation law of the vector current 533
Volume 194, number 4
(OUju(x))
=0
PHYSICS LETTERS B
20 August 1987
(27)
is obtained. We have shown that the chiral anomaly is obtained from the stationary property of the pseudoscalar density, based on the generalized Langevin equation. Our approach is very simple and straightforward. Moreover, there is no need to solve the Langevin equations as other authors have done. The anomaly term comes from the duplication of the random noises in the generalized Langevin equation for the pseudoscalar density, reflecting effects of the quantum fluctuations. On the contrary, we have shown that the vector current is conserved, because the duplications of the random noises cancel out each other. This approach can be applied to various types of anomalies. One of the authors (D.Y) thanks the Japan Society for the Promotion of Science for its financial support.
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