C*-algebraic results in the search for quantum gauge fields

TL;DR

This thesis advances the mathematical understanding of quantum gauge theories by establishing new results in noncommutative geometry's spectral action and constructing rigorous C*-algebra frameworks for lattice gauge theories.

Contribution

It proves the existence of higher-order spectral shift functions, provides a convergent series expansion of the spectral action, and constructs continuum limit C*-algebras for quantum gauge theories.

Findings

Proved higher-order spectral shift functions under Schatten class assumptions.

Derived a convergent series expansion of the spectral action involving Chern--Simons and Yang--Mills forms.

Constructed continuum limit C*-algebras demonstrating strict deformation quantization.

Abstract

This thesis consists of two parts, both situated in operator theory, and both motivated by the quest for rigorous quantizations of gauge theories. The first part is based on [Skripka,vN - JST 2022], [van Suijlekom,vN - JNCG 2021], and [van Suijlekom,vN - JHEP 2022], and concerns the spectral action of noncommutative geometry and its perturbative expansions. We prove the existence of a higher-order spectral shift function under the relative Schatten class assumption, give a converging series expansion of the spectral action in terms of Chern--Simons and Yang--Mills forms, and show one-loop renormalizability of the spectral action in a generalized sense. The second part is based on [Stienstra,vN 2020] and [vN - LMP 2022] and concerns a non-perturbative approach to quantum gauge theory by means of Hamiltonian lattice gauge theory and strict quantization. We construct C*-algebras of U(1)^n-gauge observables on the lattice, show that they are conserved under the relevant time evolutions, construct continuum limit C*-algebras, and show that the result constitutes a strict deformation quantization.