Quantizing Yang-Mills theory: From Parisi-Wu stochastic quantization to a global path integral

Abstract

Based on a generalization of the stochastic quantization scheme we recently proposed a generalized, globally defined Faddeev-Popov path integral density for the quantization of Yang-Mills theory. In this talk first our approach on the whole space of gauge potentials is shortly reviewed; in the following we discuss the corresponding global path integral on the gauge orbit space relating it to the original Parisi-Wu stochastic quantization scheme. MATHEMATICAL SETTINGIt is our aim to discuss a globally valid path integral procedure for the quantization of YangMills theory based on a recently introduced generalization [1-4] of the Parisi-Wu stochastic quantization scheme [5][6][7]; for different globally valid stochastic interpretations of the Faddeev-Popov procedure [8] see [9,10].Let P (M, G) be a principal fiber bundle with compact structure group G over the compact Euclidean space time M . Let A denote the space of all irreducible connections on P and let G denote the gauge group, which is given by all vertical automorphisms on P reduced by the centre of G. Then G acts freely on A and defines a principal Gfibration A π −→ A/G =: M over the paracompact [11] space M of all inequivalent gauge potentials with projection π. Due to the Gribov ambiguity [12] the principal G-bundle A → M is not globally trivializable.From [11] it follows that there exists a locally finite open cover U = {U α } of M together with a set of background gauge fields {A (α) 0 ∈ A} such that Γ α = {B ∈ π −1 (U α )|D *