A Yang-Mills-Dirac Quantum Field Theory Emerging From a Dirac Operator on a Configuration Space

TL;DR

This paper introduces a Dirac operator on the configuration space of $SU(2)$ gauge connections and shows that a twisted inner fluctuation yields a square that comprises the Yang–Mills Hamiltonian coupled to an operator-valued fermionic sector. A metric on the underlying 3-manifold enables a basis change that transforms the fermionic sector from one-forms to a Dirac-type fermion content, linking noncommutative geometric methods to quantized gauge theories. The results provide a geometric unification framework at the level of quantized fields, with gravity entering via the configuration-space geometry and the Chern–Simons functional playing a central role in the fluctuations. The work also clarifies the role of the twist in Clifford structure and raises further questions about Hilbert-space realizations and the handling of gauge fixing and Gribov ambiguities.

Abstract

Starting with a Dirac operator on a configuration space of $SU(2)$ gauge connections we consider its fluctuations with inner automorphisms. We show that a certain type of twisted inner fluctuations leads to a Dirac operator whose square gives the Hamiltonian of Yang-Mills quantum field theory coupled to a fermionic sector that consist of one-form fermions. We then show that if a metric exists on the underlying three-dimensional manifold then there exists a change of basis of the configuration space for which the transformed fermionic sector consists of fermions that are no-longer one-forms.