Dirac Operators on Configuration Spaces and Yang-Mills Quantum Field Theory

TL;DR

This paper explores the mathematical relationship between Dirac operators on configuration spaces and the Hamiltonian structure of Yang-Mills quantum field theory, revealing new connections and spectral invariants.

Contribution

It introduces a formulation of Yang-Mills Hamiltonians via Dirac operators on configuration spaces and constructs a Bott-Dirac operator linking to the quantum field theory.

Findings

Hamilton operators arise from Dirac equations on gauge connection spaces

A Bott-Dirac operator is formulated on the configuration space

Spectral invariants related to the quantum field theory are identified

Abstract

In this paper we discuss a connection between Dirac operators on configuration spaces and Yang-Mills quantum field theory. We first show that the Hamilton operators of the self-dual and anti-self-dual sectors of a Yang-Mills quantum field theory emerge from unitary transformations of a Dirac equation formulated on a configuration space of gauge connections. Secondly, we formulate a Bott-Dirac operator on the configuration space and demonstrate how the Hamilton operator of a Yang-Mills quantum field theory coupled to a fermionic sector emerges from its square. Finally, we discuss a spectral invariant that emerges in this framework.