Paracausal deformations, M{\o}ller operators, and Hadamard states in CCR AQFT.

In this thesis, we address several problems related to the bosonic classical and algebraic quantum field theories in curved spacetime. In particular, the main question is: how do the theories change under finite global variations of the spacetime metric tensor? To answer this question a new deformation tool, the paracausal deformation, is developed and studied on its own as a new approach to investigate the structure of the space of globally hyperbolic metric tensors associated with a smooth manifold $\M$. Then the classical M{\o}ller maps are constructed to compare solutions of the hyperbolic PDEs defining the classical field theories and the quantum M{\o}ller $*$-isomorphisms follow to compare the CCR quantum algebras associated to the propagation of the quantum fields on the different background geometries. These maps possess the important property of preserving Hadamard states, providing a new way to implement the deformation argument used to prove the existence of such states in general globally hyperbolic spacetime. Moreover, the algebraic quantization of the Proca field, i.e the massive spin 1 field, on a general globally hyperbolic spacetime is for the first time studied in detail: by employing techniques coming from microlocal analysis and spectral theory a Hadamard state is constructed on ultrastatic spacetimes and then the M{\o}ller operator is used to prove the existence of such states in general globally hyperbolic spacetimes. A discussion about the definition of Hadamard states for the massive vector fields closes the work. The thesis is based on two works on algebraic quantization of bosonic field theories and Hadamard states: \cite{Norm}, \cite{Proca}. The papers are co-authored by my supervisor Prof. Valter Moretti (UniTN) and cosupervisor Simone Murro (UniGe). The first \cite{DefArg1} has not been included since, at the time it was written, the paracausal deformation, the construction of M{\o}ller operators, the right approach to intertwine the causal propagators and all the other tools developed in the subsequent works were still at a rough stage.