Geometry of the Space of Gauge Orbits and the Yang-Mills Dynamical System

Abstract

In these notes we shall be mainly concerned with some global aspects of classical continuum Yang-Mills theory which may be of relevance to non-perturbative considerations in the quantum theory of the continuum Yang-Mills field. In doing so we make some assumptions about which one should be clear from the outset. In classical field theory it is necessary to introduce boundary conditions in order to solve the dynamics. In order to construct a (Euclidean) quantum theory of interacting fields it is customary to put in at the outset some cutoffs: a space (space-time) volume cutoff, as well as an ultraviolet cutoff. The latter is to be removed after renormalization, and then the burden is to take the infinite volume limit. We shall adhere to this philosophy [1]. A volume cutoff is introduced in these notes by compactifying space (space-time) in some way, right from the start in the classical theory. This is because in the functional integral approach to the quantum theory one integrates over classical field configurations. Ultraviolet regularization is not mentioned because we deal mostly with the classical theory, but it should be kept in mind. The other major hypothesis is with respect to regularity of classical field configurations.

Lectures at the 1979 Cargèse Summer Institute on “Recent developments in Gauge Theories”