Universal constructions in algebraic and locally covariant quantum field theory

The present work is concerned with the application of categorical methods in algebraic and locally covariant quantum field theory. Attention is particularly paid to colimits and left Kan extensions, understanding
K. Fredenhagen’s universal algebra, which is a
global (unital) (C)*-algebra associated with a not necessarily up-directed net of local (unital) (C)*-algebras, from the point of view of category theory. The main technical result centres on explicit expressions for the universal algebra and its non-triviality in the case that a net of local unital *-algebras is constructed from linear symplectic spaces via a functorial quantisation prescription. Non-up-directed nets of local (unital) (C)*-algebras typically arise for quantum field theories in a generic curved spacetime with an arbitrary topology. As an example the field strength tensor description of
the classical and the quantised free Maxwell field in curved spacetimes is considered. Employing colimits and left Kan extensions, a universal classical and quantum field
theory are constructed. Both fail local covariance and dynamical locality but can be reduced to locally covariant and dynamically local theories.
To understand C.J. Isham’s twisted quantum fields from the point of view of algebraic and locally covariant quantum field theory, an abstract categorical framework is introduced, which utilises recent ideas of C.J. Fewster on the automorphisms of a locally covariant theory and the group of the global gauge transformations of a theory. The general formalism allows to consider twisted variants of generic locally covariant theories, which need not refer to (quantum) fields at all, on single curved spacetimes.
It is argued that the general categorical scheme leads naturally to the classification of the twisted variants of a locally covariant theory by the isomorphism classes of flat smooth principal bundles over the fixed single curved spacetime the twisted variants are considered on. The general categorical scheme and the classification of twisted variants are illustrated by the example of twisted variants of multiple free and minimally coupled real scalar fields of the same mass.
Finally, a new family of pure and quasifree states for the quantised free massive Dirac field on 4-dimensional, oriented and globally hyperbolic ultrastatic slabs with
compact spatial section is constructed, arising from a recent description of F. Finster’s fermionic projector. These FP-states (“FP” for fermionic projector) are tested for the Hadamard property with some negative and some positive results.