Gribov Problem and Stochastic Quantization

Abstract: The standard procedure for quantizing gauge fields is the Faddeev-Popov quantization, which performs gauge fixing in the path integral formulation and introduces additional ghost fields. This approach provides the foundation for calculations in quantum Yang-Mills theory. However, in 1978, Vladimir Gribov showed that the gauge-fixing procedure was incomplete, with residual gauge copies (called Gribov copies) still entering the path integral even after gauge fixing. These copies impact the infrared behavior of the theory and modify gauge-dependent quantities, such as gluon and ghost propagators, as they represent redundant integrations over gauge-equivalent configurations. Furthermore, their existence breaks down the Faddeev-Popov prescription at a fundamental level. To partially resolve this, Gribov proposed restricting the path integral to the Gribov region, which alters the gluon propagator semiclassically in a way that points to gluon confinement in the Yang-Mills theory. In this thesis, we comprehensively study the Gribov problem analytically. After reviewing Faddeev-Popov quantization, the BRST symmetry of the complete Lagrangian and the Gribov problem in depth, we detail Gribov's semi-classical resolution involving restriction of the path integral to the Gribov region, outlining its effects on the theory. Further, we elucidate stochastic quantization prescription for quantizing the gauge fields. This alternate quantization prescription hints towards a formalism devoid of the Gribov problem, making it an interesting candidate for quantizing and studying the non-perturbative regime of gauge theories.