Stochastic diagrams and Feynman diagrams

Abstract

We study the relationship between ordinary perturbation theory and perturbation theory obtained from stochastic quantization. We give a simple proof that, except in gauge theories, the several stochastic diagrams of a given topology are together equivalent to the corresponding Feynman diagram. Our analysis is presented in Minkowski space, but most of it may readily be adapted to euclidean space. The field propagator may be a non-diagonal matrix, such as is the case in real-time thermal field theory. We present a new version of the Langevin equation which directly reproduces the usual axial-gauge perturbation theory. Otherwise, we find that for gauge theories the relationship between ordinary and stochastic perturbation theory is not simple, and we present a recursive method of reconstructing Feynman diagrams from stochastic diagrams, without the need explicitly to introduce ghost fields. We consider both the original Parisi-Wu version of the Langevin equation, and Zwanziger's modified version with its stochastic gauge-fixing term.