Representation Theory in Homotopy Algebraic Quantum Field Theory

Algebraic quantum field theory (AQFT) is an axiomatic approach aiming to provide a rigorous framework for the description of quantum field theories. Although it put forward several new insights and allowed for the development of quantum field theory on curved spacetimes, it became apparent that its axioms were too strict for the description of quantum gauge field theories. This motivated the initiation of the program of homotopy AQFT. Homotopy AQFT attempts to combine the framework of the Batalin--Vilkovisky (BV) formalism with the insights gained from AQFT, with the goal to provide an axiomatic framework for quantum gauge theories. The main goal of this dissertation is to study representations of homotopical AQFTs. An important aspect of this framework is that homotopical AQFTs come equipped with a notion of weak equivalences. Two weakly equivalent homotopical AQFTs are understood to be physically equivalent. Therefore, our main challenge is to introduce a concept of representations of homotopical AQFTs such that it is consistent with the weak equivalences between homotopical AQFTs. To this end, after defining representations of homotopical AQFTs by generalizing the usual concept of representations in AQFT, we introduce a notion of weak equivalences between them. These weak equivalences determine a homotopy category of representations (i.e. the category obtained by formally inverting these weak equivalences) for each homotopical AQFT. Then, we utilize the theory of model categories to prove that, given two weakly equivalent homotopical AQFTs, the associated homotopy categories of representations are equivalent, demonstrating this way the compatibility of the proposed concept of representations with the homotopy AQFT framework. Having established the concept of representations for homotopical AQFTs, we aim next at constructing explicit representations for the homotopy AQFT associated with Maxwell $p$-forms. Assuming an ultrastatic spacetime with compact Cauchy surface, we illustrate that one can construct representations in this framework mimicking the usual procedure of constructing an appropriate two-point function for the global (differential graded) algebra of observables, and then producing a representation of the global algebra of observables in a manner similar to the Gelfand--Naimark--Segal construction. Then, a representation for the homotopical AQFT is obtained by simply restricting the representation of the global algebra of observables to the local ones.