Green operators in low regularity spacetimes and quantum field theory
Green operators in low regularity spacetimes and quantum field theory
In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field φ on a globally hyperbolic spacetime M with C1,1 metric g. This first entails showing that the (classical) Cauchy problem for the wave equation is well-posed for initial data and sources in Sobolev spaces and then constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying the relevant function spaces we need to control the norms of both φ and gφ in order to ensure that
g ◦ G± and G± ◦ g are the identity maps on those spaces. The causal propagator G = G+ − G− is then used to define a symplectic form ω on a normed space V(M) which is shown to be isomorphic to ker(g). This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local C∗-algebras.
low regularity, weak solutions, Green operators, quantum field theory
Hoermann, G.
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Sanchez Sanchez, Yafet Erasmo
72589503-da03-4d66-9429-f3598ce7681e
Spreitzer, C
3f995747-2758-40c6-83e4-3ef6bb240359
Vickers, James
719cd73f-c462-417d-a341-0b042db88634
3 September 2020
Hoermann, G.
6bc24b6d-8dc4-4a7a-a7c1-0d45882f1eec
Sanchez Sanchez, Yafet Erasmo
72589503-da03-4d66-9429-f3598ce7681e
Spreitzer, C
3f995747-2758-40c6-83e4-3ef6bb240359
Vickers, James
719cd73f-c462-417d-a341-0b042db88634
Hoermann, G., Sanchez Sanchez, Yafet Erasmo, Spreitzer, C and Vickers, James
(2020)
Green operators in low regularity spacetimes and quantum field theory.
Classical and Quantum Gravity, 37 (17), [175009].
(doi:10.1088/1361-6382/ab839a).
Abstract
In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field φ on a globally hyperbolic spacetime M with C1,1 metric g. This first entails showing that the (classical) Cauchy problem for the wave equation is well-posed for initial data and sources in Sobolev spaces and then constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying the relevant function spaces we need to control the norms of both φ and gφ in order to ensure that
g ◦ G± and G± ◦ g are the identity maps on those spaces. The causal propagator G = G+ − G− is then used to define a symplectic form ω on a normed space V(M) which is shown to be isomorphic to ker(g). This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local C∗-algebras.
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2020_Class._Quantum_Grav._37_175009
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Accepted/In Press date: 26 March 2020
e-pub ahead of print date: 3 August 2020
Published date: 3 September 2020
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© 2020 The Author(s). Published by IOP Publishing Ltd
Keywords:
low regularity, weak solutions, Green operators, quantum field theory
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Local EPrints ID: 443217
URI: http://eprints.soton.ac.uk/id/eprint/443217
ISSN: 0264-9381
PURE UUID: f6252fb2-8233-4dde-a17f-482931ff76c1
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Date deposited: 17 Aug 2020 16:30
Last modified: 17 Mar 2024 02:32
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Author:
G. Hoermann
Author:
Yafet Erasmo Sanchez Sanchez
Author:
C Spreitzer
Author:
James Vickers
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