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Analytic Hadamard States, Calderón Projectors and Wick Rotation Near Analytic Cauchy Surfaces

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Abstract

We consider the Klein–Gordon equation on analytic spacetimes with an analytic Cauchy surface. In this setting, we prove the existence of pure analytic Hadamard states. The proof is based on considering an elliptic operator obtained by Wick rotating the Klein–Gordon operator in a neighborhood of a Cauchy hypersurface. The Cauchy data of Hadamard two-point functions are constructed as Calderón projectors (suitably generalized if the hypersurface is non-compact) for the elliptic operator.

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Acknowledgements

The authors would like to thank Pierre Schapira for all the useful discussions. Support from the Grants ANR-12-BS01-012-01 and ANR-16-CE40-0012-01 is gratefully acknowledged.

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris-Sud XI, 91405, Orsay Cedex, France

    Christian Gérard

  2. CNRS, Institut Fourier, Université Grenoble Alpes, 38000, Grenoble, France

    Michał Wrochna

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  1. Christian Gérard
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  2. Michał Wrochna
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Correspondence to Michał Wrochna.

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Communicated by Y. Kawahigashi

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Gérard, C., Wrochna, M. Analytic Hadamard States, Calderón Projectors and Wick Rotation Near Analytic Cauchy Surfaces. Commun. Math. Phys. 366, 29–65 (2019). https://doi.org/10.1007/s00220-019-03349-z

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