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Construction of
Hadamard states by characteristic Cauchy problem
Christian Gérard and Michał Wrochna |
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Vol. 9 (2016), No. 1, 111–149
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DOI: 10.2140/apde.2016.9.111
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Abstract |
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We construct Hadamard states for Klein–Gordon fields in a spacetime equal to the interior of the future lightcone from a base point in a globally hyperbolic spacetime . Under some regularity conditions at the future infinity of , we identify a boundary symplectic space of functions on , which allows us to construct states for Klein–Gordon quantum fields in from states on the CCR algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on , ensuring that the bulk state it induces in satisfies the Hadamard condition. Using pseudodifferential calculus on the cone , we construct a large class of Hadamard states on the boundary with pseudodifferential covariances and characterize the pure states among them. We then show that these pure boundary states induce pure Hadamard states in . |
Keywords
Hadamard states, microlocal spectrum condition,
pseudodifferential calculus, characteristic Cauchy problem,
curved spacetimes
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Mathematical Subject Classification 2010
Primary: 35S05, 81T20
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Milestones
Received: 8 November 2014
Revised: 13 October 2015
Accepted: 16 November 2015
Published: 10 February 2016
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Authors |
| Christian Gérard | |
| Laboratoire de Mathématiques
d’Orsay Université Paris-Sud XI 91405 Orsay France |
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| Michał Wrochna | |
| Institut Joseph Fourier Université Joseph Fourier (Grenoble 1) UMR 5582 CNRS 38402 Grenoble France |
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