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Review

Partial Differential Equations and Quantum States in Curved Spacetimes

1
Department of Mathematics, University of California at Santa Barbara, South Hall, Santa Barbara, CA 93106, USA
2
Regional Mathematical Center, Southern Federal University, 344006 Rostov-on-Don, Russia
3
School of Mathematics, Cardiff University, Senghennydd Rd, Cardiff CF24 4AG, UK
*
Author to whom correspondence should be addressed.
Academic Editor: Emma Previato
Mathematics 2021, 9(16), 1936; https://doi.org/10.3390/math9161936
Received: 20 July 2021 / Revised: 9 August 2021 / Accepted: 9 August 2021 / Published: 13 August 2021
(This article belongs to the Special Issue Partial Differential Equations and Applications)
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. View Full-Text
Keywords: quantum field theory; partial differential equations; hyperbolic propagators; Hadamard states quantum field theory; partial differential equations; hyperbolic propagators; Hadamard states
MDPI and ACS Style

Avetisyan, Z.; Capoferri, M. Partial Differential Equations and Quantum States in Curved Spacetimes. Mathematics 2021, 9, 1936. https://doi.org/10.3390/math9161936

AMA Style

Avetisyan Z, Capoferri M. Partial Differential Equations and Quantum States in Curved Spacetimes. Mathematics. 2021; 9(16):1936. https://doi.org/10.3390/math9161936

Chicago/Turabian Style

Avetisyan, Zhirayr, and Matteo Capoferri. 2021. "Partial Differential Equations and Quantum States in Curved Spacetimes" Mathematics 9, no. 16: 1936. https://doi.org/10.3390/math9161936

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