Functorial formulation of Algebraic Quantum Field Theory in curved spacetimes and the Reeh-Schlieder Theorem - Estêves, Ana Camila Costa (2021)
In this project we studied how Category Theory can be used in the formulation of Algebraic Quantum Field Theory in curved spacetimes and how the Reeh-Schlieder property translates to general curved spacetimes. Category Theory concepts such as functors, natural transformations and natural equivalences are used in the definition of a Locally Covariant Quantum Field Theory, that arose in a context in which it was of interest to generalize Axiomatic Quantum Field Theory to curved spacetimes taking into consideration the ideas of locality and covariance. In fact, a Locally Covariant Quantum Field Theory is defined as a covariant functor, which can be related to another Locally Covariant Quantum Field Theory by a natural transformation. The equivalence between theories then becomes clear if this natural transformation is an isomorphism. Furthermore, the Reeh-Schlieder theorem is of great significance in the realm of Quantum Field Theory, since it provides a great deal of properties for the vacuum state and it has relevance in justifying applications of Tomita-Takesaki modular theory in Quantum Field Theories. It has already been proven that states with a weak form of the Reeh-Schlieder property always exist in general curved spacetimes. This was accomplished using the spacetime deformation technique and assuming the time-slice axiom in a Locally Covariant Quantum Field Theory.