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arXiv · 2012.09767

On microlocalisation and the construction of Feynman Propagators for normally hyperbolic operators

Onirban Islam·Alexander Strohmaier
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Abstract

This article gives global microlocalisation constructions for normally hyperbolic operators on a vector bundle over a globally hyperbolic spacetime in geometric terms. As an application, this is used to generalise the Duistermaat-H\"{o}rmander construction of Feynman propagators, therefore incorporating the most important non-scalar 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 Dirac-type operators on a globally hyperbolic spacetime. Even though the natural bundle metric on spinors is not positive-definite, in this case, we can give a direct microlocal construction of a Feynman propagator that satisfies positivity.

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BibTeXRIS

Onirban Islam, Alexander Strohmaier. 2020-12-17. On microlocalisation and the construction of Feynman Propagators for normally hyperbolic operators. https://doi.org/10.4310/cag.241204020919

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