On Gauge Invariances in Stochastic Quantization

Abstract

We show that the symmetry of stochastically quantized gauge theories is governed by a single differential operator. The latter combines supersymmetry and ordinary gauge transformations. Quantum field theory can be defined on the basis of a parabolic differential operator $\frac{\partial }{{\partial t}} - \left( {\frac{\partial }{{\partial x^i }}} \right)^2 ,$ with a Hamiltonian of the type $H = \frac{1}{2}\left[ {Q,\overline Q } \right],$ where Q has has deep relationship with the conserved charge of a topoplogical gauge theory. We display the examples of Yang-Mills theory. We also show the relevant equations for gravity, with interesting remarks for the 2-dimensional case. For the stochastic quantization of a first order action, we present another method which defines quantum field theory from a second order supersymmetric action while maintaining gauge invariance. As an application, the theory defined by the “two dimensional Chern Simons action” ∫ _M2 Tr ϕ F is related to a four dimensional theory defined by a gauge fixed action of the second Chern class ∫ _M4 TrFF.