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

This thesis is devoted to the study of hyperbolic differential operators on globally hyperbolic Lorentzian manifolds, linear field-theoretic models exhibiting a gauge symmetry, 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, complemented by many examples and a discussion of Green hyperbolicity. Although hyperbolic equations are usually studied in the context of 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, with applications, for instance, to the Maxwell equations in linear dispersive media. The subsequent 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. In particular, we give a thorough analysis of their Cauchy problems and construct the corresponding classical phase space. A central part of this section is the presentation of several examples of direct physical interest, some of which have not appeared yet in the literature in this context. Moreover, we explain how linear gauge theories are quantised following the algebraic approach to quantum field theory, which offers a mathematically rigorous quantisation scheme, ideally suited for field theories defined on Lorentzian backgrounds. In particular, we introduce the notion of Hadamard states, which are physically distinguished states whose two-point function has a specific singularity structure as a bidistribution, and provide a detailed bibliographic review of existence results for such states. The final chapter of this thesis is devoted to the quantisation of Maxwell’s theory on globally hyperbolic spacetimes. After a detailed discussion of the Cauchy and gauge problems for Maxwell's theory on Lorentzian manifolds, the central goal is to prove the existence of Hadamard states for Maxwell's theory on any globally hyperbolic spacetime. The novelty of our approach lies in a new gauge-fixing procedure at the level of initial data, which allows us to suppress all 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 (possibly non-compact) Riemannian manifolds. Using tools from microlocal analysis, we explicitly construct Hadamard states on ultrastatic spacetimes for the completely gauge-fixed theory and define states obtained as the pull-back thereof along the gauge-fixing projector, while ensuring that this construction preserves the Hadamard property. For general spacetimes, we employ a deformation argument. This thesis is complemented by an appendix that, alongside other supplementary materials, provides a self-contained introduction to microlocal analysis and pseudodifferential calculus.