The prototype of a relativistic differential operator is often a normally hyperbolic operator. It is a classic fact that there exist unique advanced and retarded propagators for such an operator on a globally hyperbolic spacetime. With the advent of quantum field theory, a new type of propagator emerges --- the Feynman propagator which propagates positive (negative) energy solutions to the future (past) and thus combines causality with the notion of positivity of energy. These propagators are an essential ingredient of perturbative quantum field theory and are intimately connected with quantum states. Moreover, they are not unique in contrast to advanced and retarded propagators. Apart from their applications in quantum physics, these propagators also play an enormously important role in global analysis and relativistic spectral geometry.
During my stay at the Erwin Schrödinger Institute, the Feynman propagators for a wave-type operator on the Schwarzschild black hole will be investigated. The explicit formulae of Feynman propagators on Minkowski spacetime are available in the standard textbooks of quantum field theory, which employs the Fourier transform. However, on a generic globally hyperbolic spacetime there is neither a Fourier transform nor any reasonable notion of energy in the absence of a global timelike Killing vector field. A priori, it is, therefore, unclear what a Feynman propagator should be. Thanks to the classic work of Duistermaat and Hörmander who gave the elegant definition of a Feynman propagator in terms of the wavefront set condition and constructed such propagators (modulo smoothing terms) in great generality. On the other hand, there are interesting constructions of advanced and retarded propagates on the exterior region of a Schwarzschild black hole based on spectral analysis. The aim of this project is to combine microlocal techniques with spectral theoretic ideas in order to construct Feynman propagators on a Schwarzschild spacetime.
Duration of stay: 24th June to 14th July 2023
|Onirban Islam||University of Potsdam|