Koopman-von Neumann Formulation of Classical Yang-Mills Theories: I

TL;DR

This paper develops a Koopman-von Neumann (KvN) classical formulation of non-Abelian gauge theories, establishing a path integral approach that parallels quantum methods and introduces classical gauge structures and charges.

Contribution

It introduces the KvN formalism for classical Yang-Mills theories, connecting classical path integrals with quantum gauge concepts like gauge-fixing and Faddeev-Popov determinants.

Findings

Classical path integral formulation requires gauge-fixing and Faddeev-Popov determinants.

Constructs classical BFV formalism and gauge charges.

Establishes relation between classical and quantum gauge structures.

Abstract

In this paper we present the Koopman-von Neumann (KvN) formulation of classical non-Abelian gauge field theories. In particular we shall explore the functional (or classical path integral) counterpart of the KvN method. In the quantum path integral quantization of Yang-Mills theories concepts like gauge-fixing and Faddeev-Popov determinant appear in a quite natural way. We will prove that these same objects are needed also in this classical path integral formulation for Yang-Mills theories. We shall also explore the classical path integral counterpart of the BFV formalism and build all the associated universal and gauge charges. These last are quite different from the analog quantum ones and we shall show the relation between the two. This paper lays the foundation of this formalism which, due to the many auxiliary fields present, is rather heavy. Applications to specific topics outlined in the paper will appear in later publications.