The Metric Nature of Matter

TL;DR

This paper develops a geometric framework using a metric on gauge connection space, leading to a non-perturbative quantum field theory that unifies Yang-Mills and Dirac fields on curved backgrounds.

Contribution

It introduces an infinite-dimensional Bott-Dirac operator and a non-commutative algebra to encode quantum field relations in a geometric, metric-based setting.

Findings

Constructs a metric structure on gauge connections.

Derives Yang-Mills and Dirac Hamiltonians from the Bott-Dirac operator.

Establishes a geometric, non-perturbative approach to quantum field theory.

Abstract

We construct a metric structure on a configuration space of gauge connections and show that it naturally produces a candidate for a non-perturbative, 3+1 dimensional Yang-Mills-Dirac quantum field theory on a curved background. The metric structure is an infinite-dimensional Bott-Dirac operator and the fermionic sector of the emerging quantum field theory is generated by the infinite-dimensional Clifford algebra required to construct this operator. The Bott-Dirac operator interacts with the $\mathbf{HD}(M)$ algebra, which is a non-commutative algebra generated by holonomy-diffeomorphisms on the underlying manifold, i.e. parallel-transforms along flows of vector fields. This algebra combined with the Bott-Dirac operator encode the canonical commutation and anti-commutation relations of the quantised bosonic and fermionic fields. The square of the Bott-Dirac operator produces both the Yang-Mills Hamilton operator and the Dirac Hamilton operator as well as a topological Yang-Mills term alongside higher-derivative terms and a metric invariant.