Advertisement

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
Article

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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

Copyright information

© 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

Personalised recommendations