Ward identities for the stochastic quantization of gauge fields

Ward identities of BRS or nil-potent type are derived which express the content of local gauge invariance for stochastically quantized gauge theories. First the Langevin equation for stochastically quantized non-abelian gauge theories with gauge fixing is shown to be invariant under a restricted class of local, classical gauge transformations. Next the solution of the Langevin equation is expressed as a functional integral with a local action which is shown to be invariant under the restricted classical gauge transformations. Grassmann variables or ghosts are introduced which make the action invariant under a nil-potent transformation. They play no dynamical role, but only serve as a book-keeping device to calculate the BRS transform of non-ghost amplitudes. (All closed ghost loops vanish.) Finally the Ward identities are solved to obtain the renormalized action and restrictions on renormalization constants.