Symmetry of the Hadamard coefficients for a formally self-adjoint, normally hyperbolic operator on a globally hyperbolic Lorentzian manifold

19.04.2018, 16:15  –  Haus 9, Raum 0.14 Forschungsseminar Differentialgeometrie

Max Lewandowski

In the case of the Klein-Gordon field on Minkowski space a certain linear combination of the 4 standard fundamental solutions (advanced, retarded, Feynman and anti-Feynman propagator) yields a Hadamard state. Inspired by the construction of the advanced and retarded fundamental solutions on globally hyperbolic spacetimes for normally hyperbolic operators acting on sections in some vector bundle, a sketch of the corresponding construction for the Feynman and anti-Feynman propagator by a Hadamard series will be given. We will see that for the operator being formally self-adjoint the corresponding Hadamard coefficients turn out to have a certain symmetry property leading to the right antisymmetric part of the corresponding linear combination of fundamental solutions in the general setting, i.e. that of a Hadamard state.

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