On microlocalization and the construction of Feynman propagators for normally hyperbolic operators
Abstract
This article gives global microlocalisation constructions for normally hyperbolic operators on globally hyperbolic spacetimes on vector bundles in geometric terms. As an application, this is used to generalise the DuistermaatHörmander construction of Feynman propagators, therefore incorporating the most important nonscalar geometric operators. It is shown that for normally hyperbolic operators that are selfadjoint with respect to a hermitian bundle metric, the Feynman propagators can be constructed to satisfy a positivity property that reflects the existence of Hadamard states in quantum field theory on curved spacetimes. We also give a more direct construction of the Feynman propagators for Diractype operators on a globally hyperbolic spacetime. Even though the natural bundle metric on spinors is not positivedefinite, in this case, we can give a direct microlocal construction of a Feynman propagator that satisfies positivity.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 DOI:
 10.48550/arXiv.2012.09767
 arXiv:
 arXiv:2012.09767
 Bibcode:
 2020arXiv201209767I
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Differential Geometry;
 35S30;
 58J40;
 81T20
 EPrint:
 47 pages