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

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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.

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References

  1. Ammann, B., Lauter, R., Nistor, V., Vasy, A.: Complex powers and non-compact manifolds. Comm. PDE 29, 671–705 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baskin, D., Vasy, A., Wunsch, J.: Asymptotics of radiation fields in asymptotically Minkowski space. Am. J. Math. 137(5), 1293–1364 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bär (ed.), C., Fredenhagen, K. (ed.): Quantum field theory on curved spacetimes. Lect. Notes Phys. 786, 39–58 (2009)

  4. Brouder, C., Dang, N.V., Hélein, F.: A smooth introduction to the wavefront set. J. Phys. A: Math. Theor. 47(44), 443001 (2014)

  5. Brum, M., Jorás, S. E.: Hadamard state in Schwarzschild–de Sitter spacetime. Class. Quantum Grav. 32(1), 015013 (2014)

  6. Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Comm. Math. Phys. 208, 623–661 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Choquet-Bruhat, Y., Cotsakis, Y.: Global hyperbolicity and completeness. J. Geom. Phys. 43, 345–350 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. Dappiaggi, C., Drago, N.: Constructing Hadamard states via an extended Møller operator. Lett. Math. Phys. 106(11), 1587–1615 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dang, N.V.: Renormalization of Quantum Field Theory on Curved Spacetimes, a Causal Approach. Ph.D. thesis, Paris Diderot University (2013)

  13. Dereziński, J., Gérard, C.: Mathematics of Quantization and Quantum Fields. Cambridge Monographs in Mathematical Physics. Cambridge University Press, Cambridge (2012)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Drouot, A.: A quantitative version of Hawking radiation. Ann. Henri Poincaré 18(3), 757–806 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Fewster, C.J., Verch, R.: Algebraic quantum field theory in curved spacetimes. In: Advances in Algebraic Quantum Field Theory. Springer, Berlin (2015)

  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)

    Article  ADS  MATH  Google Scholar 

  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. 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. Gérard, C., Wrochna, M.: Construction of Hadamard states by pseudo-differential calculus. Comm. Math. Phys. 325(2), 713–755 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Gérard, C., Wrochna, M.: Hadamard states for the linearized Yang–Mills equation on curved spacetime. Comm. Math. Phys. 337(1), 253–320 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Gérard, C., Wrochna, M.: Construction of Hadamard states by characteristic Cauchy problem. Anal. PDE 9(1), 111–149 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hollands, S., Wald, R.M.: Quantum fields in curved spacetime. In: General Relativity and Gravitation: A Centennial Perspective. Cambridge University Press, Cambridge (2015)

  29. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Springer, Berlin (1985)

    MATH  Google Scholar 

  30. Idelon–Riton, G.: Scattering Theory for the Dirac Equation in Schwarzschild–Anti-de Sitter Space-Time (preprint) (2014). arXiv:1412.0869

  31. Isozaki, H.: QFT for scalar particles in external fields on Riemannian manifolds. Rev. Math. Phys. 13(6), 767–798 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Junker, W.: Adiabatic Vacua and Hadamard States for Scalar Quantum Fields on Curved Spacetime. PhD thesis, University of Hamburg (1995)

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

  36. Kordyukov, Y.: \(L^{p}\)-Theory of elliptic differential operators on manifolds of bounded geometry. Acta Appl. Math. 23, 223–260 (1991)

    MathSciNet  MATH  Google Scholar 

  37. Lundberg, L.-E.: Relativistic quantum theory for charged spinless particles in external vector fields. Comm. Math. Phys. 31, 295–316 (1973)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Nicolas, J.-P.: Conformal scattering on the Schwarzschild metric. Ann. Inst. Fourier 66(3), 1175–1216 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, London (1975)

    MATH  Google Scholar 

  43. Ruijsenaars, S.N.M.: Charged particles in external fields I. Classical theory. J. Math. Phys. 18(4), 720–737 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  44. Ruzhansky, M., Wirth, J.: Dispersive estimates for \(t\)-dependent hyperbolic systems. J. Differ. Equ. 251, 941–969 (2011)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Seiler, R.: Quantum theory of particles with spin zero and one half in external fields. Comm. Math. Phys. 25, 127–151 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  48. Shubin, M.A.: Spectral theory of elliptic operators on non-compact manifolds. Astérisque 207, 37–108 (1992)

    MATH  Google Scholar 

  49. Shubin, M.A.: Pseudo-Differential Operators and Spectral Theory. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Vasy, A.: On the positivity of propagator differences. Ann. Henri Poincaré 18(3), 983–1007 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Vasy, A., Wrochna, M.: Quantum Fields from Global Propagators on Asymptotically Minkowski and Extended de Sitter Spacetimes (Preprint) (2015). arXiv:1512.08052

  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)

    Article  ADS  MathSciNet  Google Scholar 

  54. Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)

    Book  MATH  Google Scholar 

  55. Wirth, J.: On \(t\)-Dependent Hyperbolic Systems. Part 2 (preprint) (2015). arXiv:1508.02635

  56. Wrochna, M.: Singularities of Two-point Functions in Quantum Field Theory. PhD thesis, University of Göttingen (2013)

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Correspondence to Christian Gérard.

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Communicated by Karl-Henning Rehren.

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Gérard, C., Wrochna, M. Hadamard Property of the in and out States for Klein–Gordon Fields on Asymptotically Static Spacetimes. Ann. Henri Poincaré 18, 2715–2756 (2017). https://doi.org/10.1007/s00023-017-0573-2

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  • DOI: https://doi.org/10.1007/s00023-017-0573-2

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