Stochastic regularization of fermions

Stochastic quantization is applied to the U(1)-gauge theory including fermions (QED). The early formulation of stochastically quantized QED used non-gauge-invariant Langevin equations, whereas nowadays the explicitly gauge-invariant equations of Ishikawa are preferred. We will comment on the older representation, where gauge invariance can be violated by the regularization because gauge invariance is already broken by the Langevin equations from the beginning. This is explicitly demonstrated in a few examples, including anomalous Ward-Takahashi identities and the vacuum polarization tensor Pi/sub munu/. Deeper insight into the breakdown of gauge invariance is gained by the use of the Fokker-Planck equation and the construction of equilibrium probability distributions in the case of a classical electromagnetic background field. Our investigations result in the conclusion that only Ishikawa's explicitly gauge-invariant Langevin equations can be applied without contradictions