Annales Henri Poincaré

, Volume 19, Issue 5, pp 1529–1586 | Cite as

Quantum Fields from Global Propagators on Asymptotically Minkowski and Extended de Sitter Spacetimes

  • András Vasy
  • Michał Wrochna


We consider the wave equation on asymptotically Minkowski spacetimes and the Klein–Gordon equation on even asymptotically de Sitter spaces. In both cases, we show that the extreme difference of propagators (i.e., retarded propagator minus advanced, or Feynman minus anti-Feynman), defined as Fredholm inverses, induces a symplectic form on the space of solutions with wave front set confined to the radial sets. Furthermore, we construct isomorphisms between the solution spaces and symplectic spaces of asymptotic data. As an application of this result, we obtain distinguished Hadamard two-point functions from asymptotic data. Ultimately, we prove that non-interacting Quantum Field Theory on asymptotically de Sitter spacetimes extends across the future and past conformal boundary, i.e., to a region represented by two even asymptotically hyperbolic spaces. Specifically, we show this to be true both at the level of symplectic spaces of solutions and at the level of Hadamard two-point functions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allen, B.: Vacuum states in de Sitter space. Phys. Rev. D 32, 3136 (1985)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Araki, H., Shiraishi, M.: On quasi-free states of canonical commutation relations. I Publ. RIMS Kyoto Univ. 7, 105–120 (1971)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bär, C., Ginoux, N., Pfäffle, F.: Wave Equation on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics, EMS, Zurich (2007)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bär, C., Strohmaier, A.: An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary. To appear in Am. J. Math. preprint arXiv:1506.00959, (2017)
  5. 5.
    Bär, C., Strohmaier, A.: A rigorous geometric derivation of the chiral anomaly in curved backgrounds. Commun. Math. Phys. 347(3), 703–721 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baskin, D., Vasy, A., Wunsch, J.: Asymptotics of radiation fields in asymptotically Minkowski space. Am. J. Math. 137(5), 1293–1364 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baskin, D., Wang, F.: Radiation fields on Schwarzschild spacetime. Commun. Math. Phys. 331, 477–506 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bertola, M., Gorini, V., Moschella, U., Schaeffer, R.: Correspondence between Minkowski and de Sitter quantum field theory. Phys. Lett. B 462, 249–253 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bros, J., Moschella, U.: Two point functions and quantum fields in de Sitter universe. Rev. Math. Phys. 8, 327 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bros, J., Moschella, U., Gazeau, J.P.: Quantum field theory in the de Sitter universe. Phys. Rev. Lett. 73, 1746 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brum, M., Jorás, S.E.: Hadamard state in Schwarzschild-de Sitter spacetime. Class. Quantum Grav. 32(1), 015013 (2014)ADSMathSciNetCrossRefzbMATHGoogle 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.
    Dappiaggi, C., Drago, N.: Constructing Hadamard states via an extended Møller operator. Lett. Math. Phys. 106(11), 1587–1615 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    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
  16. 16.
    Dereziński, J., Gérard, C.: Mathematics of Quantization and Quantum Fields. Cambridge Monographs in Mathematical Physics. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  17. 17.
    Dereziński, J., Siemssen, D.: Feynman propagators on static spacetimes. To appear in Rev. Math. Phys., preprint arXiv:1609.00192, (2018)
  18. 18.
    de Boer, J., Solodukhin, S.N.: A holographic reduction of Minkowski space–time. Nucl. Phys. B 665, 545 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Duistermaat, J.J., Hörmander, L.: Fourier integral operators II. Acta Math. 128, 183–269 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dyatlov, S.: Asymptotics of linear waves and resonances with applications to black holes. Commun. Math. Phys. 335, 1445–1485 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fewster, C.J., Verch, R.: A quantum weak energy inequality for Dirac fields in curved spacetime. Commun. Math. Phys. 225, 331 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Friedlander, F.G.: Radiation fields and hyperbolic scattering theory. Math. Proc. Camb. Philos. Soc. 88, 483–515 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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
  24. 24.
    Gérard, C., Wrochna, M.: Construction of Hadamard states by characteristic Cauchy problem. Anal. PDE 9(1), 111–149 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gérard, C., Wrochna, M.: Construction of Hadamard states by pseudo-differential calculus. Commun. Math. Phys. 325(2), 713–755 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gérard, C., Wrochna, M.: Hadamard states for the linearized Yang–Mills equation on curved spacetime. Commun. Math. Phys. 337(1), 253–320 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gérard, C., Wrochna, M.: The massive Feynman propagator on asymptotically Minkowski spacetimes. To appear in Am. J. Math. preprint arXiv:1609.00192, (2016)
  28. 28.
    Gérard, C., Wrochna, M.: Hadamard property of the in and out states for Klein–Gordon fields on asymptotically static spacetimes. Ann. Henri Poincaré 18(8), 2715–2756 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gell-Redman, J., Haber, N., Vasy, A.: The Feynman propagator on perturbations of Minkowski space. Commun. Math. Phys. 342(1), 333–384 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87(2), 186–225 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Guillarmou, C.: Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds. Duke Math. J. 129(1), 1–37 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Haag, R.: Local Qantum Physics: Fields, Particles, Algebras. Texts and Monographs in Physics. Springer, Berlin (1992)CrossRefGoogle Scholar
  33. 33.
    Haber, N., Vasy, A.: Propagation of Singularities Around a Lagrangian Submanifold of Radial Points Microlocal Methods in Mathematical Physics and Global Analysis, pp. 113–116. Springer, Basel (2013)CrossRefzbMATHGoogle Scholar
  34. 34.
    Hintz, P.: Global analysis of linear and nonlinear wave equations on cosmological spacetimes. Ph.D. Thesis, Stanford University, (2015)Google Scholar
  35. 35.
    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
  36. 36.
    Hollands, S., Wald, R.M.: Quantum fields in curved spacetime. In: General Relativity and Gravitation: A Centennial Perspective, Cambridge University Press, (2015)Google Scholar
  37. 37.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I–IV. Classics in Mathematics. Springer, Berlin (2007)CrossRefGoogle Scholar
  38. 38.
    Jaffe, A., Jäkel, C., Martinez, R.E.: Complex classical fields: a framework for reflection positivity. Commun. Math. Phys. 329, 1–28 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Jaffe, A., Ritter, G.: Quantum field theory on curved backgrounds. I. The Euclidean functional integral. Commun. Math. Phys. 270, 545–572 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Jaffe, A., Ritter, G.: Reflection positivity and monotonicity. J. Math. Phys. 49(052301), 1–10 (2008)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Joshi, M., Sá Barreto, A.: Inverse scattering on asymptotically hyperbolic manifolds. Acta Math. 184(1), 41–86 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kay, B.S.: The principle of locality and quantum field theory on (non globally hyperbolic) curved spacetimes. Rev. Math. Phys. 4(Special Issue), 167–195 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kay, B.S., Lupo, U.: Non-existence of isometry-invariant Hadamard states for a Kruskal black hole in a box and for massless fields on 1+1 Minkowski spacetime with a uniformly accelerating mirror. Class. Quantum Grav. 33(21), 215001 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    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
  45. 45.
    Khavkine, I., Moretti, V.: Algebraic QFT in curved spacetime and quasifree Hadamard states: an introduction. In: Advances in Algebraic Quantum Field Theory, Springer (2015)Google Scholar
  46. 46.
    Mazzeo, R., Melrose, R.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75(2), 260–310 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Melrose, R.: The Atiyah-Patodi-Singer Index Theorem, vol. 4. AK Peters, Wellesley (1993)zbMATHGoogle Scholar
  48. 48.
    Melrose, R.: Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces. Lecture Notes in Pure and Appl. Math., vol. 161, Dekker, New York, 85–130 (1994)Google Scholar
  49. 49.
    Melrose, R.: Geometric Scattering Theory, vol. 1. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  50. 50.
    Melrose, R.: Lecture notes for ‘18.157: Introduction to microlocal analysis’. Available at (2009)
  51. 51.
    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. Commun. Math. Phys. 268, 727–756 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Moretti, V.: Quantum out-states holographically induced by asymptotic flatness: invariance under space-time symmetries, energy positivity and Hadamard property. Commun. Math. Phys. 279, 31–75 (2008)ADSCrossRefzbMATHGoogle Scholar
  53. 53.
    Moschella, U., Schaeffer, R.: Quantum theory on Lobatchevski spaces. Class. Quant. Grav. 24, 3571–3602 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Radzikowski, M.: Micro-local approach to the Hadamard condition in quantum field theory on curved space–time. Commun. Math. Phys. 179, 529–553 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Radzikowski, M.: A Local to global singularity theorem for quantum field theory on curved space–time. Commun. Math. Phys. 180, 1 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Rehren, K.-H.: Boundaries in relativistic quantum field theory. To appear in the proceedings of the XVIII International Congress on Mathematical Physics, Santiago de Chile, July 2015. preprint arXiv:1601.00826 (2016)
  57. 57.
    Sanders, K.: Equivalence of the (generalized) Hadamard and microlocal spectrum condition for (generalized) free fields in curved space–time. Commun. Math. Phys. 295, 485–501 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Sanders, K.: On the construction of Hartle–Hawking–Israel states across a static bifurcate Killing horizon. Lett. Math. Phys. 105(4), 575–640 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    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
  60. 60.
    Strominger, A.: The dS/CFT correspondence. JHEP 0110, 341–346 (2001)Google Scholar
  61. 61.
    Vasy, A.: Microlocal Analysis of Asymptotically Hyperbolic Spaces and High Energy Resolvent Estimates. MSRI Publications, vol. 60. Cambridge University Press, Cambridge (2012)Google Scholar
  62. 62.
    Vasy, A.: Analytic continuation and high energy estimates for the resolvent of the Laplacian on forms on asymptotically hyperbolic spaces. Adv. Math. 306, 1019–1045 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Vasy, A.: On the positivity of propagator differences. Ann. Henri Poincaré 18(3), 983–1007 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Vasy, A.: Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces, (with an appendix by S. Dyatlov). Invent. Math. 194, 381–513 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Vasy, A.: Propagation of singularities for the wave equation on manifolds with corners. Ann. Math. 168(3), 749–812 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Vasy, A.: Propagation Phenomena. Lecture Notes. Stanford University, Stanford (2014)Google Scholar
  67. 67.
    Vasy, A.: Resolvents, Poisson operators and scattering matrices on asymptotically hyperbolic and de Sitter spaces. J. Spect. Theory 4(4), 643–673 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Vasy, A.: The wave equation on asymptotically de Sitter-like spaces. Adv. Math. 223(1), 49–97 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Vasy, A.: A minicourse on microlocal analysis for wave propagation. In: Asymptotic Analysis in General Relativity, London Mathematical Society Lecture Note Series 443, Cambridge University Press, (2018)Google Scholar
  70. 70.
    Wald, R.M.: Dynamics in nonglobally hyperbolic, static spacetimes. J. Math. Phys. 21, 2802–2805 (1980)ADSMathSciNetCrossRefGoogle Scholar
  71. 71.
    Zahn, J.: Generalized Wentzell boundary conditions and quantum field theory. Ann. Henri Poincaré 19, 163 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Zworski, M.: Resonances for asymptotically hyperbolic manifolds: Vasy’s method revisited. J. Spect. Theor. 6, 1087–1114 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Stanford UniversityStanfordUSA
  2. 2.Institut FourierUniversité Grenoble AlpesGrenoble Cedex 09France

Personalised recommendations