Global wave parametrices on globally hyperbolic spacetimes

Abstract

In a recent work the first named author, Levitin and Vassiliev have constructed the wave propagator on a closed Riemannian manifold $M$ as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. In this paper, first we give a natural reinterpretation of the underlying algorithmic construction in the language of ultrastatic Lorentzian manifolds. Subsequently we show that the construction carries over to the case of static backgrounds thanks to a suitable reduction to the ultrastatic scenario. Finally we prove that the overall procedure can be generalised to any globally hyperbolic spacetime with compact Cauchy surfaces. As an application, we discuss how, from our procedure, one can recover the local Hadamard expansion which plays a key role in all applications in quantum field theory on curved backgrounds.

arXiv comment28 pages, final version accepted for publication

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Cited by in corpus (8)

• Invariant subspaces of elliptic systems II: spectral theory
• Invariant subspaces of elliptic systems I: pseudodifferential projections
• Global propagator for the massless Dirac operator and spectral asymptotics
• Diagonalization of elliptic systems via pseudodifferential projections
• Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds
• On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary
• Propagators in curved spacetimes from operator theory
• On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods