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arXiv · 2212.06533

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

Teun D. H. van Nuland
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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.

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Teun D. H. van Nuland. 2022-12-13. C*-algebraic results in the search for quantum gauge fields. https://arxiv.org/abs/2212.06533

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