Perturbative Approach to Gauge Fields

Abstract

One of the most challenging attacks using the Parisi-Wu stochastic quantization method is undoubtedly its application to gauge fields. Parisi and Wu (1981) presented an interesting plan to remarkably simplify the gauge fixing procedure, by means of their stochastic quantization (SQM), claiming that we need neither any gauge fixing procedure nor the Faddeev-Popov ghost. Before this pioneering work, we used to rely upon the Faddeev-Popov trick in order to fix the gauge within the path-integral formulation of field theory. While true that the Faddeev-Popov formula turns out to reveal a rather beautiful structure of the field theory with constraints, i e., the BRS-symmetry (Becchi, Rouet and Stora 1986; Kugo and Ojima 1978,1979), it is known that the Gribov problem (Gribov 1978) exists. Parisi and Wu first derived the Landau-gauge propagator of a gauge field without resorting to the usual gauge-fixing term, and conjectured that the Faddeev-Popov term to recover the gauge invariance and unitarity, in the case of a non-Abelian gauge field, could automatically be derived within the framework of SQM. We also expect to avoid the troublesome Gribov problem (Parisi and Wu 1981) by means of SQM.