Vol. 9, No. 1, 2016

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Construction of Hadamard states by characteristic Cauchy problem

Christian Gérard and Michał Wrochna

Vol. 9 (2016), No. 1, 111–149
DOI: 10.2140/apde.2016.9.111
Abstract

We construct Hadamard states for Klein–Gordon fields in a spacetime M0 equal to the interior of the future lightcone C from a base point p in a globally hyperbolic spacetime (M,g).

Under some regularity conditions at the future infinity of C, we identify a boundary symplectic space of functions on C, which allows us to construct states for Klein–Gordon quantum fields in M0 from states on the CCR algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on C, ensuring that the bulk state it induces in M0 satisfies the Hadamard condition.

Using pseudodifferential calculus on the cone C, 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 M0.

Keywords
Hadamard states, microlocal spectrum condition, pseudodifferential calculus, characteristic Cauchy problem, curved spacetimes
Mathematical Subject Classification 2010
Primary: 35S05, 81T20
Milestones
Received: 8 November 2014
Revised: 13 October 2015
Accepted: 16 November 2015
Published: 10 February 2016
Authors
Christian Gérard
Laboratoire de Mathématiques d’Orsay
Université Paris-Sud XI
91405 Orsay
France
Michał Wrochna
Institut Joseph Fourier
Université Joseph Fourier (Grenoble 1) UMR 5582 CNRS
38402 Grenoble
France