Partial Differential Equations and Quantum States in Curved Spacetimes

Zhirayr Avetisyan and Matteo Capoferri
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Zhirayr Avetisyan: Department of Mathematics, University of California at Santa Barbara, South Hall, Santa Barbara, CA 93106, USA
Matteo Capoferri: School of Mathematics, Cardiff University, Senghennydd Rd, Cardiff CF24 4AG, UK

Mathematics, 2021, vol. 9, issue 16, 1-19

Abstract: In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states —on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

Keywords: quantum field theory; partial differential equations; hyperbolic propagators; Hadamard states (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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