We report on the well-posedness of the Feynman problem for the Klein–Gordon equation on asymptotically Minkowski spacetimes. The main result is the invertibility of the Klein–Gordon operator with Feynman conditions at infinite times. Furthermore, the inverse is shown to coincide with the Duistermaat–Hörmander Feynman parametrix modulo smoothing terms.
@article{SLSEDP_2019-2020____A3_0,
author = {G\'erard, Christian and Wrochna, Micha{\l}},
title = {The {Feynman} problem for the {Klein{\textendash}Gordon} equation},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:4},
pages = {1--10},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2019-2020},
doi = {10.5802/slsedp.140},
language = {en},
url = {http://www.numdam.org/articles/10.5802/slsedp.140/}
}
TY - JOUR AU - Gérard, Christian AU - Wrochna, Michał TI - The Feynman problem for the Klein–Gordon equation JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:4 PY - 2019-2020 SP - 1 EP - 10 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/articles/10.5802/slsedp.140/ DO - 10.5802/slsedp.140 LA - en ID - SLSEDP_2019-2020____A3_0 ER -
%0 Journal Article %A Gérard, Christian %A Wrochna, Michał %T The Feynman problem for the Klein–Gordon equation %J Séminaire Laurent Schwartz — EDP et applications %Z talk:4 %D 2019-2020 %P 1-10 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/articles/10.5802/slsedp.140/ %R 10.5802/slsedp.140 %G en %F SLSEDP_2019-2020____A3_0
Gérard, Christian; Wrochna, Michał. The Feynman problem for the Klein–Gordon equation. Séminaire Laurent Schwartz — EDP et applications (2019-2020), Exposé no. 4, 10 p. doi : 10.5802/slsedp.140. http://www.numdam.org/articles/10.5802/slsedp.140/
[BS16] Bär, C., Strohmaier, A.: A rigorous geometric derivation of the chiral anomaly in curved backgrounds, Commun. Math. Phys. 347, 703–721 (2016). | DOI | MR | Zbl
[BS19] Bär, C., Strohmaier, A.: An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary, Am. J. Math. 141 (9), (2019), 1421–1455. | DOI | MR | Zbl
[BVW15] Baskin, D., Vasy, A., Wunsch, J.: Asymptotics of radiation fields in asymptotically Minkowski space, Am. J. Math., 137 (5), (2015). | DOI | MR | Zbl
[DH72] Duistermaat, J.J., Hörmander, L.: Fourier integral operators II, Acta Math. 128 (1972), 183–269. | DOI | MR | Zbl
[DS18] Dereziński, J., Siemssen, D.: Feynman propagators on static spacetimes, Rev. Math. Phys. 30, (2018), 1850006. | DOI | MR | Zbl
[DS19a] Dereziński, J., Siemssen, D.: An evolution equation approach to the Klein-Gordon operator on curved spacetime, Pure Appl. Anal. 1, 215–261, (2019). | DOI | Zbl
[DS19b] Dereziński, J., Siemssen, D.: An evolution equation approach to linear Quantum Field Theory, preprint , (2019). | arXiv
[GHSZ19] Gell-Redman, J., Hassell, A., Shapiro, J., Zhang, J.: Existence and asymptotics of nonlinear Helmholtz eigenfunctions, preprint , (2019). | arXiv
[GHV16] Gell-Redman, J., Haber, N., Vasy, A.: The Feynman propagator on perturbations of Minkowski space, Commun. Math. Phys., 342, 1, (2016), 333–384. | DOI | MR | Zbl
[GOW17] Gérard, C., Oulghazi, O., Wrochna, M.: Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry, Commun. Math. Phys., 352 (2), (2017), 519-583. | DOI | MR | Zbl
[GW14] Gérard, C., Wrochna, M.: Construction of Hadamard states by pseudo-differential calculus, Commun. Math. Phys. 325 (2) (2014), 713–755. | DOI | MR | Zbl
[GW17] Gérard, C., Wrochna, M.: Hadamard property of the in and out states for Klein-Gordon fields on asymptotically static spacetimes, Ann. Henri Poincaré 18 (2017), 2715–2756. | DOI | MR | Zbl
[GW19a] Gérard, C., Wrochna, M.: The massive Feynman propagator on asymptotically Minkowski spacetimes, Am. J. Math. 141 (6), 1501–1546 (2019). | DOI | MR | Zbl
[GW19b] Gérard, C., Wrochna, M.: The massive Feynman propagator on asymptotically Minkowski spacetimes II, Int. Math. Res. Notices, doi.org/10.1093/imrn/rnz007, (2019). | DOI | MR
[Ju96] Junker, W.: Hadamard States, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime, Rev. Math. Phys. 8, (1996), 1091–1159. | DOI | Zbl
[MT15] Mizutani, H., Tzvetkov, N.: Strichartz estimates for non-elliptic Schrödinger equations on compact manifolds, Comm. PDE, Vo. 40, no. 6, 1182–1195 (2015). | DOI | Zbl
[NT19] Nakamura, S., Taira, K.: Essential self-adjointness of real principal type operators, preprint , (2019). | arXiv
[Ta17] Taira, K.: Strichartz estimates for non-degenerate Schrödinger equations, preprint , (2017). | arXiv
[Va13] Vasy, A.: Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces, (With an appendix by S. Dyatlov), Inventiones Math., 194:381–513, (2013), 194.2: 381-513. | DOI | MR | Zbl
[Va17] Vasy, A.: On the positivity of propagator differences, Annales Henri Poincaré, 18 (3), (2017), 983–1007, (2017). | DOI | MR | Zbl
[Va19] Vasy, A.: Essential self-adjointness of the wave operator and the limiting absorption principle on Lorentzian scattering spaces, to appear in J. Spectr. Theor., , (2019) | arXiv | DOI | MR | Zbl
[VW18] Vasy, A., Wrochna, M.: Quantum fields from global propagators on asymptotically Minkowski and extended de Sitter spacetimes, Ann. Henri Poincaré, 19 (5), (2018), 1529–1586. | DOI | MR | Zbl
[Wr19] Wrochna, M: Conformal extension of the Bunch-Davies state across the de Sitter boundary, J. Math. Phys. 60 (2), 022301, (2019). | DOI | MR | Zbl
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