Non-perturbative Quantum Field Theory and the Geometry of Functional Spaces

TL;DR

This paper develops a non-perturbative, non-commutative geometric framework for quantum gauge theories on curved backgrounds, linking it to perturbative results and highlighting open questions.

Contribution

It introduces a novel non-commutative geometric construction over gauge connection spaces that models interacting quantum gauge and fermionic fields non-perturbatively.

Findings

Constructs a non-commutative geometry over gauge connections.

Shows the geometry yields a candidate for non-perturbative quantum gauge theory.

Recovers perturbative quantum field theory in a local limit.

Abstract

In this paper we construct a non-commutative geometry over a configuration space of gauge connections and show that it gives rise to a candidate for an interacting, non-perturbative quantum gauge theory coupled to a fermionic field on a curved background. The non-commutative geometry is given by an infinite-dimensional Bott-Dirac type operator, whose square gives the Hamilton operator, and which interacts with an algebra generated by holonomy-diffeomorphisms. The Bott-Dirac operator and the associated Hilbert space relies on a metric on the configuration space of connections, which effectively works as a covariant ultra-violet regulator. We show that the construction coincides with perturbative quantum field theory in a local limit. Questions concerning Lorentz invariance and the fermionic sector as well as the issue of existence are left open.