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

TL;DR

This paper derives a Yang-Mills-Dirac quantum field theory from a Dirac operator on a configuration space, revealing how fluctuations lead to a coupled gauge-fermion system and how a metric influences fermion representation.

Contribution

It introduces a novel approach connecting Dirac operators on configuration spaces with Yang-Mills-Dirac theories, highlighting the role of twisted inner fluctuations and metric-dependent basis transformations.

Findings

Fluctuations of the Dirac operator produce a Yang-Mills-Dirac Hamiltonian.

Existence of a metric allows a basis change transforming fermions from one-forms to other forms.

The framework links geometric operators with quantum field theoretical models.

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.