Basic Ideas of Stochastic Quantization

Abstract

We describe basic ideas of the stochastic quantization which was originally proposed by Parisi and Wu. We start from a brief survey of stochastic-dynamical approaches to quantum mechanics, as a historical background, in which one can observe important characteristics of the Parisi-Wu stochastic quantization method that are different from others. Next we give an outline of the stochastic quantization, in which a neutral scalar field is quantized as a simple example. We show that this method enables us to quantize gauge fields without resorting to the conventional gauge-fixing procedure and the Faddeev-Popov trick. Furthermore, we introduce a generalized (kerneled) Langevin equation to extend the mathematical formulation of the stochastic quantization: It is illustrative application is given by a quantization of dynamical systems with bottomless actions. Finally, we develop a general formulation of stochastic quantization within the framework of a (4 + 1)-dimensional field theory.