Annales Henri Poincaré

, Volume 18, Issue 8, pp 2715–2756 | Cite as

Hadamard Property of the in and out States for Klein–Gordon Fields on Asymptotically Static Spacetimes

  • Christian GérardEmail author
  • Michał Wrochna


We consider the massive Klein–Gordon equation on a class of asymptotically static spacetimes (in the long-range sense) with Cauchy surface of bounded geometry. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods.


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  1. 1.
    Ammann, B., Lauter, R., Nistor, V., Vasy, A.: Complex powers and non-compact manifolds. Comm. PDE 29, 671–705 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baskin, D., Vasy, A., Wunsch, J.: Asymptotics of radiation fields in asymptotically Minkowski space. Am. J. Math. 137(5), 1293–1364 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bär (ed.), C., Fredenhagen, K. (ed.): Quantum field theory on curved spacetimes. Lect. Notes Phys. 786, 39–58 (2009)Google Scholar
  4. 4.
    Brouder, C., Dang, N.V., Hélein, F.: A smooth introduction to the wavefront set. J. Phys. A: Math. Theor. 47(44), 443001 (2014)Google Scholar
  5. 5.
    Brum, M., Jorás, S. E.: Hadamard state in Schwarzschild–de Sitter spacetime. Class. Quantum Grav. 32(1), 015013 (2014)Google Scholar
  6. 6.
    Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Comm. Math. Phys. 208, 623–661 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Choquet-Bruhat, Y., Cotsakis, Y.: Global hyperbolicity and completeness. J. Geom. Phys. 43, 345–350 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dafermos, M., Rodnianski, I., Shlapentokh-Rothman, Y.: A scattering theory for the wave equation on Kerr black hole exteriors. J. Differ. Geom. (to appear) (2014). arXiv:1412.8379
  9. 9.
    Dappiaggi, C., Drago, N.: Constructing Hadamard states via an extended Møller operator. Lett. Math. Phys. 106(11), 1587–1615 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dappiaggi, C., Moretti, V., Pinamonti, N.: Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property. J. Math. Phys. 50, 062304 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dappiaggi, C., Moretti, V., Pinamonti, N.: Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime. Adv. Theor. Math. Phys. 15, 355 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dang, N.V.: Renormalization of Quantum Field Theory on Curved Spacetimes, a Causal Approach. Ph.D. thesis, Paris Diderot University (2013)Google Scholar
  13. 13.
    Dereziński, J., Gérard, C.: Mathematics of Quantization and Quantum Fields. Cambridge Monographs in Mathematical Physics. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  14. 14.
    Dimock, J., Kay, B.S.: Classical wave operators and asymptotic quantum field operators on curved space-times. Annales de l’I.H.P. A 37(2), 93–114 (1982)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dimock, J., Kay, B.S.: Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric. I. Ann. Phys. 175(2), 366–426 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dimock, J., Kay, B.S.: Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric. II. J. Math. Phys. 27, 2520 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Drouot, A.: A quantitative version of Hawking radiation. Ann. Henri Poincaré 18(3), 757–806 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fewster, C.J., Verch, R.: Algebraic quantum field theory in curved spacetimes. In: Advances in Algebraic Quantum Field Theory. Springer, Berlin (2015)Google Scholar
  19. 19.
    Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved space-time. II. Ann. Phys. 136, 243–272 (1981)ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Georgescu, V., Gérard, C., Häfner, D.: Asymptotic Completeness for Superradiant Klein–Gordon Equations and Applications to the De Sitter Kerr Metric (preprint) (2014). arXiv:1405.5304
  21. 21.
    Gérard, C., Oulghazi, O., Wrochna, M.: Hadamard states for the Klein–Gordon equation on Lorentzian manifolds of bounded geometry. Comm. Math. Phys. (to appear) (2016). arXiv:1602.00930
  22. 22.
    Gérard, C., Wrochna, M.: Construction of Hadamard states by pseudo-differential calculus. Comm. Math. Phys. 325(2), 713–755 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gérard, C., Wrochna, M.: Hadamard states for the linearized Yang–Mills equation on curved spacetime. Comm. Math. Phys. 337(1), 253–320 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gérard, C., Wrochna, M.: Construction of Hadamard states by characteristic Cauchy problem. Anal. PDE 9(1), 111–149 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hintz, P., Vasy, A.: Semilinear wave equations on asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes. Anal. PDE 8(8), 1807–1890 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Comm. Math. Phys. 231(2), 309–345 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hollands, S., Wald, R.M.: Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17(3), 277–311 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hollands, S., Wald, R.M.: Quantum fields in curved spacetime. In: General Relativity and Gravitation: A Centennial Perspective. Cambridge University Press, Cambridge (2015)Google Scholar
  29. 29.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Springer, Berlin (1985)zbMATHGoogle Scholar
  30. 30.
    Idelon–Riton, G.: Scattering Theory for the Dirac Equation in Schwarzschild–Anti-de Sitter Space-Time (preprint) (2014). arXiv:1412.0869
  31. 31.
    Isozaki, H.: QFT for scalar particles in external fields on Riemannian manifolds. Rev. Math. Phys. 13(6), 767–798 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Junker, W.: Adiabatic Vacua and Hadamard States for Scalar Quantum Fields on Curved Spacetime. PhD thesis, University of Hamburg (1995)Google Scholar
  33. 33.
    Junker, W., Schrohe, E.: Adiabatic vacuum states on general space-time manifolds: definition, construction, and physical properties. Ann. Henri Poincaré 3, 1113–1181 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep. 207, 49 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Khavkine, I., Moretti, V.: Algebraic QFT in curved spacetime and quasifree Hadamard states: an introduction. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds.) Advances in Algebraic Quantum Field Theory. Springer, Berlin (2015)Google Scholar
  36. 36.
    Kordyukov, Y.: \(L^{p}\)-Theory of elliptic differential operators on manifolds of bounded geometry. Acta Appl. Math. 23, 223–260 (1991)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Lundberg, L.-E.: Relativistic quantum theory for charged spinless particles in external vector fields. Comm. Math. Phys. 31, 295–316 (1973)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Moretti, V.: Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence. Comm. Math. Phys. 268, 727–756 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Moretti, V.: Quantum out-states holographically induced by asymptotic flatness: invariance under space-time symmetries, energy positivity and Hadamard property. Comm. Math. Phys. 279, 31–75 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Nicolas, J.-P.: Conformal scattering on the Schwarzschild metric. Ann. Inst. Fourier 66(3), 1175–1216 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Radzikowski, M.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Comm. Math. Phys. 179, 529–553 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, London (1975)zbMATHGoogle Scholar
  43. 43.
    Ruijsenaars, S.N.M.: Charged particles in external fields I. Classical theory. J. Math. Phys. 18(4), 720–737 (1977)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Ruzhansky, M., Wirth, J.: Dispersive estimates for \(t\)-dependent hyperbolic systems. J. Differ. Equ. 251, 941–969 (2011)ADSCrossRefzbMATHGoogle Scholar
  45. 45.
    Sahlmann, H., Verch, R.: Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13(10), 1203–1246 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Sanders, K.: Equivalence of the (generalized) Hadamard and microlocal spectrum condition for (generalized) free fields in curved space-time. Comm. Math. Phys. 295, 485–501 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Seiler, R.: Quantum theory of particles with spin zero and one half in external fields. Comm. Math. Phys. 25, 127–151 (1972)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    Shubin, M.A.: Spectral theory of elliptic operators on non-compact manifolds. Astérisque 207, 37–108 (1992)zbMATHGoogle Scholar
  49. 49.
    Shubin, M.A.: Pseudo-Differential Operators and Spectral Theory. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  50. 50.
    Vasy, A.: Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces, (With an appendix by S. Dyatlov). Invent. Math. 194(2), 381–513 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Vasy, A.: On the positivity of propagator differences. Ann. Henri Poincaré 18(3), 983–1007 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Vasy, A., Wrochna, M.: Quantum Fields from Global Propagators on Asymptotically Minkowski and Extended de Sitter Spacetimes (Preprint) (2015). arXiv:1512.08052
  53. 53.
    Wald, R.M.: Existence of the S-matrix in quantum field theory in curved space-time. Ann. Phys. (N.Y.) 118, 490–510 (1979)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)CrossRefzbMATHGoogle Scholar
  55. 55.
    Wirth, J.: On \(t\)-Dependent Hyperbolic Systems. Part 2 (preprint) (2015). arXiv:1508.02635
  56. 56.
    Wrochna, M.: Singularities of Two-point Functions in Quantum Field Theory. PhD thesis, University of Göttingen (2013)Google Scholar

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité Paris-Sud XIOrsay CedexFrance
  2. 2.Institut Fourier, UMR 5582 CNRSUniversité Grenoble AlpesGrenoble Cedex 09France

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