Supersymmetry and Gauge Invariance in Stochastic Quantization

Abstract

Stochastic quantization is an alternative to the Feynman path integral for quantizing a theory. In Ref. 1, Parisi and Wu have suggested applying stochastic quantization to gauge theories. Numerous works have followed.² One of the motivations of Parisi and Wu was that no gauge fixing is necessary to compute gauge-invariant quantities in stochastic quantization, since the stochastic evolution can be consistently defined from a drift force equal to minus the gradient of the classical action with respect to the gauge field, with no reference to the ghosts that occur in the ordinary path integral formalism. However, it has been realized that it is useful to introduce a kind of gauge fixing in stochastic quantization: a drift force can be defined along gauge orbits.³ This permits a consistent renormalizability of the stochastically quantized gauge theory. Moreover, with a particular choice of this drift force, it seems that the gauge field is confined within the first Gribov horizon, and so one naturally escapes the Gribov problem.^3,4 The freedom in the Langevin equation of a gauge theory, which permits the introduction of the gauge-dependent drift force, follows in fact from the simple geometrical principle that stochastic evolution be compatible with the gauge symmetry.⁵