Applications of Microlocal Analysis to Quantum Field Theory on Curved Spacetime
Is the spacetime we are living in the boundary of some higher-dimensional geometric structure? This question, broadly known as the holographic principle, has its quantum counterpart: what physical content is encoded in the asymptotic structures of a spacetime? Two different models are provided by asymptotically de Sitter and anti-de Sitter spacetimes, and one can ask how a quantum theory can be described in terms of data on the horizon. It turns out that within the framework of Quantum Field Theory on curved spacetime, the problem is reduced to the study of asymptotic properties of solutions of partial differential equations that describe the classical theory, and the structure of their singularities. The goal of the project is to apply methods of singular microlocal analysis, developed by Andras´ Vasy and collaborators in the considered geometrical setting, in order to provide rigorous constructions of quantum field theories on curved spacetimes. In the de Sitter case, the critical issue is the behavior of solutions as one approaches the horizon. The Anti-de Sitter case is particularly challenging: the well-posedness of an initial value problem requires specifying data on the horizon, which poses difficulties unsolved so far in the field theoretical context.