On the Quantisation of Linear Gauge Theories on Lorentzian Manifolds: Maxwell's Theory via Complete Gauge Fixing

TL;DR

This thesis investigates the quantisation of linear gauge theories, especially Maxwell's theory, on Lorentzian manifolds, introducing a novel gauge-fixing method and establishing the existence of Hadamard states.

Contribution

It presents a new gauge-fixing procedure at the initial data level for Maxwell's theory, enabling the suppression of unphysical degrees of freedom on globally hyperbolic spacetimes.

Findings

Proved well-posedness of the Cauchy problem for symmetric hyperbolic systems with nonlocal potentials.

Provided a detailed analysis of the classical phase space for linear gauge theories.

Constructed Hadamard states for Maxwell's theory using pseudodifferential calculus.

Abstract

This thesis is devoted to the study of hyperbolic differential operators on globally hyperbolic manifolds, linear gauge theories and their quantisation. In the first part, we treat the Cauchy problem for symmetric hyperbolic systems and normally hyperbolic operators on globally hyperbolic manifolds from first principles. Although hyperbolic equations are usually studied with local interactions, there are strong motivations from several areas of mathematical physics to consider also nonlocal interactions. As an intermezzo, we therefore take a small deviation from the classical local theory and prove well-posedness of the Cauchy problem for symmetric hyperbolic systems coupled to a broad class of nonlocal potentials. The second part presents a detailed exposition of linear gauge theories in globally hyperbolic spacetimes. Linear gauge theories are yet another deviation from the concept of hyperbolicity: the corresponding equations of motion are generically non-hyperbolic; however, can always be reduced to a constrained hyperbolic dynamics once an appropriate gauge fixing procedure has been applied. We give a thorough analysis of their Cauchy problem and classical phase space, complemented by a detailed discussion of many examples of physical interest, and discuss their quantisation following the algebraic approach to quantum field theory. The final chapter is devoted to the quantisation of Maxwell's theory on globally hyperbolic spacetimes, with the goal of proving the existence of Hadamard states. The novelty of our approach lies in a new gauge-fixing procedure at the level of initial data, which allows us to suppress the unphysical degrees of freedom. This gauge is achieved by means of a new Hodge decomposition for differential k-forms in Sobolev spaces on complete Riemannian manifolds, while states are constructed using tools from pseudodifferential calculus.