Note on the Anomalies from Stochastic Quantization

Abstract

Based on the stochastic quantization method it is shown that different kernels in fermion Langevin equations with external gauge field lead to different types of anomalies. The proof is given in arbitrary even dimensions for vector coupling. Examples are considered in two dimensions where vector and axial-vector couplings are dual. . 1. The stochastic quantization method of Parisi and Wu 1 ) has been applied 2 ) to boson as well as fermion fields. It introduces the extra time variable r with a dimension of (lengthY Since the Dirac and Klein-Gordon operators have different dimensions, the fermionic formalism 3 )-7) differs from the bosonic one ip that the fermion Langevin equation is defined not only by the Dirac operator r but also by an additional kernel K of mass dimension 1. The simplest c~se K = [-I, where [ is a constant with a dimension of length, was originally proposed,3) leading to the fermion Langevin equation which is gauge but not chiral invariant. Hence we expect for K = [-1 the absence of the vector anomaly and the appearance of the axial anomaly8) as verified in Ref. 9). In general, K is an arbitrary operator as far as the r-oo limit exists to obtain the usual Green's functions (at least in the perturbational sense). It is pointed out lO ) that this arbitrariness is reflected in the ambiguity of the types ll ),12) of anomalies for chiral gauge couplings in perturbation theory. The purpose of this note is to obtain a necessary condition on K for the absence of the vector anomaly so that gauge invariance is kept during the quantization. For simplicity we shall consider the vector gauge theory in arbitrary even dimensions where the gauge field is regarded as external. Examples are given in two dimensions where vector and axial-vector couplings are dual. l3 ) 2. The Dirac operator of the 2 n-dimensional, Euclidean Dirac theory with the prescribed external gauge field Ap is given by r=r-(a+A)+m, where we choose Dirac matrices to be Hermitian and satisfy {rp, rv}=2opv, fl., ))=1, ... , 2n, and m is the fermion mass. Without repeating the detailed derivation we recall 10 ) that the vector and axial anomalies in this theory are given in the stochastic quantization method by