Nonperturbative Quantum Field Theory and Noncommutative Geometry

TL;DR

This paper develops a non-perturbative framework for quantum field theories on curved backgrounds using noncommutative geometry, introducing a universal Bott-Dirac operator that defines Hamiltonians and interactions.

Contribution

It presents a novel non-perturbative approach to quantum field theory on curved spaces via noncommutative geometry and Bott-Dirac operators, extending the mathematical foundation.

Findings

Constructed a Hilbert space embedding for quantum fields on curved backgrounds.

Identified a universal Bott-Dirac operator generating free Hamiltonians.

Proved existence of non-perturbative quantum field theories for scalar and Yang-Mills cases.

Abstract

A general framework of non-perturbative quantum field theory on a curved background is presented. A quantum field theory is in this setting characterised by an embedding of a space of field configurations into a Hilbert space over $\mathbb{R}^\infty$. This embedding, which is only local up to a scale that we interpret as the Planck scale, coincides in the local and flat limit with the plane wave expansion known from canonical quantisation. We identify a universal Bott-Dirac operator acting in the Hilbert space over $\mathbb{R}^\infty$ and show that it gives rise to the free Hamiltonian both in the case of a scalar field theory and in the case of a Yang-Mills theory. These theories come with a canonical fermionic sector for which the Bott-Dirac operator also provides the Hamiltonian. We prove that these quantum field theories exist non-perturbatively for an interacting real scalar theory and for a general Yang-Mills theory, both with or without the fermionic sectors, and show that the free theories are given by semi-finite spectral triples over the respective configuration spaces. Finally, we propose a class of quantum field theories whose interactions are generated by inner fluctuations of the Bott-Dirac operator.