Communications in Mathematical Physics

, Volume 319, Issue 1, pp 1–68 | Cite as

Correlators, Feynman Diagrams, and Quantum No-hair in deSitter Spacetime

  • Stefan HollandsEmail author


We provide a parametric representation for position-space Feynman integrals of a massive, self-interacting scalar field in deSitter spacetime, for an arbitrary graph. The expression is given as a multiple contour integral over a kernel whose structure is determined by the set of all trees (or forests) within the graph, and it belongs to a class of generalized hypergeometric functions. We argue from this representation that connected deSitter n-point vacuum correlation functions have exponential decay for large proper time-separation, and also decay for large spatial separation, to arbitrary orders in perturbation theory. Our results may be viewed as an analog of the so-called cosmic-no-hair theorem in the context of a quantized test scalar field. This work has significant overlap with a paper by Marolf and Morrison, which is being released simultaneously.


Feynman Diagram Operator Product Expansion Minkowski Spacetime Feynman Graph Feynman Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Allen B.: Vacuum states in deSitter space. Phys. Rev. D 32, 3136 (1985)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Axler S., Bourdon P., Ramey W.: Harmonic Function Theory. Springer, New York (2001)zbMATHGoogle Scholar
  3. 3.
    Anderson M.T.: Existence and stability of even dimensional asymptotically de Sitter spaces. Ann. Henri Poincare 6, 801 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Binoth T., Guillet J.P., Heinrich G.: Reduction formalism for dimensionally regulated one-loop N-point integrals. Nucl. Phys. B 572, 361 (2000)ADSCrossRefGoogle Scholar
  5. 5.
    Birke L., Fröhlich J.: KMS, etc. Rev. Math. Phys. 14, 829 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bogner C., Weinzierl S.: Feynman graph polynomials. Int. J. Mod. Phys. A 25, 2585 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Bogner C., Weinzierl S.: Resolution of singularities for multi-loop integrals. Comput. Phys. Commun. 178, 596 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Bros J., Epstein H., Moschella U.: Particle decays and stability on the de Sitter universe. Ann. Henri Poincare 11, 611 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Bros J., Epstein H., Moschella U.: Lifetime of a massive particle in a de Sitter universe. JCAP 0802, 003 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    Bros J., Epstein H., Moschella U.: Analyticity properties and thermal effects for general quantum field theory on de Sitter space-time. Commun. Math. Phys. 196, 535 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Bros J., Moschella U.: Two-point Functions and Quantum Fields in de Sitter Universe. Rev. Math. Phys. 8, 327 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bros, J., Viano, G.A.: Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid: an analytic continuation viewpoint I, II, III. Forum Math. 8, 621–658 (1996); 8, 659–722 (1996); 9, 165–192 (1997)Google Scholar
  13. 13.
    Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Brunetti R., Fredenhagen K., Hollands S.: A remark on alpha vacua for quantum field theories on de Sitter space. JHEP 0505, 063 (2005)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Brunetti R., Fredenhagen K., Kohler M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Brunetti R., Duetsch M., Fredenhagen K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13(5), 1541–1599 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Bunch T.S.: BPHZ Renormalization of \({\lambda \varphi^{\ast\ast}4}\) field theory in curved space-time. Ann. Phys. 131, 118 (1981)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Buchholz D., Schlemmer J.: Local temperature in curved spacetime. Class. Quant. Grav. 24, F25 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Camporesi R., Higuchi A.: On the eigen functions of the dirac operator on spheres and real hyperbolic spaces. J. Geom. Phys. 20, 1 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Cardoso V., Natario J., Schiappa R.: Asymptotic quasinormal frequencies for black holes in non-asymptotically flat spacetimes. J. Math. Phys. 45, 4698 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Epstein H., Glaser V.: The role of locality in quantum field theory. Ann. Poincare Theor. Phys. A 19, 211–295 (1973)MathSciNetGoogle Scholar
  22. 22.
    Friedlander F.G.: The wave equation on a curved space-time. Cambridge University Press, Cambridge (1975)zbMATHGoogle Scholar
  23. 23.
    Friedrich H.: Existence and structure of n-geodesically complete of future complete solutions of Einstein’s equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Friedrich H.: Existence and structure of past asymptotically simple solutions of Einstein’s field equations with positive cosmological constant. J. Geom. Phys. 3, 101 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. 25.
    Gracia-Bondia, J.M., Lazzarini, S.: Connes-Kreimer-Epstein-Glaser renormalization., 2000
  26. 26.
    Gibbons G.W., Hawking S.W.: Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 15, 2738 (1977)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Goldstein K.: A note on α-vacua and interacting field theory on deSitter spacetime. Nucl. Phys. B 699, 325 (2003)ADSCrossRefGoogle Scholar
  28. 28.
    Heinrich G., Binoth T.: A general reduction method for one-loop N-point integrals. Nucl. Phys. Proc. Suppl. 89, 246 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    Hepp K.: Proof of the Bogolyubov-Parasiuk theorem on renormalization. Commun. Math. Phys. 2, 301 (1966)ADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Higuchi A., Kouris S.S.: The covariant graviton propagator in deSitter spacetime. Class. Quant. Grav. 18, 4317 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    Higuchi A., Kouris S.S.: On the scalar sector of the covariant graviton two-point function in deSitter spacetime. Class. Quant. Grav. 18, 2933 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. 32.
    Higuchi A., Weeks R.H.: The physical graviton two-point function in deSitter spacetime with S 3 spatial sections. Class. Quant. Grav. 20, 3005 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. 33.
    Higuchi, A.: Tree level vacuum instability in an interacting field theory in deSitter spacetime, [gr-gc], 2008
  34. 34.
    Higuchi A.: Decay of the free-theory vacuum of scalar field theory in deSitter spacetime in the interaction picture. Class. Quant. Grav. 26, 072001 (2009)MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    Higuchi A., Lee Y.C., Nicholas J.R.: More on the covariant retarded Green’s function for the electromagnetic field in de Sitter spacetime. Phys. Rev. D 80, 107502 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    Faizal M., Higuchi A.: On the FP-ghost propagators for Yang-Mills theories and perturbative quantum gravity in the covariant gauge in de Sitter spacetime. Phys. Rev. D 78, 067502 (2008)MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Hollands S., Wald R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  38. 38.
    Hollands S., Wald R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  39. 39.
    Hollands S., Wald R.M.: Axiomatic quantum field theory in curved spacetime. Commun. Math. Phys. 293, 85 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  40. 40.
    Hollands S.: The operator product expansion for perturbative quantum field theory in curved spacetime. Commun. Math. Phys. 273, 1 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  41. 41.
    Hollands S.: Renormalized quantum Yang-Mills fields in curved spacetime. Rev. Math. Phys. 20, 1033 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Hollands, S.: Renormalization via flow equations on the sphere and deSitter spacetime, In preparationGoogle Scholar
  43. 43.
    Hormander, L.: The analysis of linear partial differential operators I. 2nd edition, Berlin heidelberg-New York: Springer Verlag, 1990Google Scholar
  44. 44.
    Itzykson C., Zuber J.B.: Quantum Field Theory. Dover Publ., New York (2005)Google Scholar
  45. 45.
    Jaekel, C.: In preparationGoogle Scholar
  46. 46.
    Jones D.S.: Asymptotics of the hypergeometric function. Math. Meth. Appl. Sci. 24, 369 (2001)zbMATHCrossRefGoogle Scholar
  47. 47.
    Kay B.S., Wald R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on space-times with a bifurcate killing horizon. Phys. Rept. 207, 49 (1991)MathSciNetADSCrossRefGoogle Scholar
  48. 48.
    Kopper Ch., Meunier F.: Large momentum bounds from flow equations. Ann. Henri Poincaré 3, 435–450 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Kopper C., Muller V.F.: Renormalization proof for massive \({\phi^{4}_{4}}\) theory on Riemannian manifolds. Commun. Math. Phys. 275, 331–372 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  50. 50.
    Kopper, Ch.: Renormierungstheorie mit Flußgleichungen. Aachen: Shaker Verlag, 1998Google Scholar
  51. 51.
    Keller K.J.: Euclidean Epstein-Glaser renormalization. J. Math. Phys. 50, 103503 (2009)MathSciNetADSCrossRefGoogle Scholar
  52. 52.
    Keller, K.J.: Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization. [math-ph], 2010
  53. 53.
    Kodama H., Ishibashi A.: A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions. Prog. Theor. Phys. 110, 701 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  54. 54.
    Krajewski T., Rivasseau V., Tanasa A., Wang Z.: Topological graph polynomials and quantum field theory, part I: heat kernel theories. J. Noncommut. Geom. 4, 29–82 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Magnen J., Rivasseau V.: Constructive ϕ4 field theory without tears. Ann. Henri Poincare 9, 403–424 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  56. 56.
    Marolf D., Morrison I.A.: The IR stability of de Sitter: Loop corrections to scalar propagators. Phys. Rev. D 82, 105032 (2010)ADSCrossRefGoogle Scholar
  57. 57.
    Marolf D., Morrison I.A.: The IR-stability of deSitter QFT: results at all orders. Phys. Rev. D 84, 044040 (2011)ADSCrossRefGoogle Scholar
  58. 58.
    Moretti V.: Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt’s coefficients in C(infinity) Lorentzian manifolds by a ‘local Wick rotation’. Commun. Math. Phys. 212, 165 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  59. 59.
    Mottola E.: Particle creation in De Sitter space. Phys. Rev. D 31, 754 (1985)MathSciNetADSCrossRefGoogle Scholar
  60. 60.
    Mazur P., Mottola E.: Spontaneous breaking of De Sitter symmetry by radiative effects. Nucl. Phys. B 278, 694 (1986)MathSciNetADSCrossRefGoogle Scholar
  61. 61.
    Antoniadis I., Mottola E.: Graviton fluctuations in De Sitter space. J. Math. Phys. 32, 1037 (1991)MathSciNetADSzbMATHCrossRefGoogle Scholar
  62. 62.
    Müller V.F.: Perturbative renormalization by flow equations. Rev. Math. Phys. 15, 491–557 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Osterwalder K., Schrader R.: Axioms for euclidean green’s functions. 1. Commun. Math. Phys. 31, 83 (1973)MathSciNetADSzbMATHCrossRefGoogle Scholar
  64. 64.
    Osterwalder K., Schrader R.: Axioms for euclidean green’s functions. 2. Commun. Math. Phys. 42, 281 (1975)MathSciNetADSzbMATHCrossRefGoogle Scholar
  65. 65.
    Srivastava H.M., Panda R.: Some bilateral generating functions for a class of generalized hypergeometric polynomials. J. Reine Angew. Math. 283, 265 (1976)MathSciNetGoogle Scholar
  66. 66.
    Srivastava H.M., Panda R.: Expansion theorems for the H-function of several complex variables. J. Reine Angew. Math. 288, 129 (1976)MathSciNetGoogle Scholar
  67. 67.
    Sarivastava, H.M., Sarivastava, A.: A new class of orthogonal expansions for the H-function of several variables, Comment. Math. Univ. St Paul, 27, 59-69Google Scholar
  68. 68.
    Mathur B.L.: Some results concerning a special function of several complex variables. Indian J. Pure Appl. Math. 12(8), 1001 (1980)MathSciNetGoogle Scholar
  69. 69.
    Pinter G.: The Hopf algebra structure of Connes and Kreimer in Epstein-Glaser renormalization. Lett. Math. Phys. 54, 227 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Polyakov A.M.: DeSitter space and eternity. Nucl. Phys. B 797, 199 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  71. 71.
    Polyakov A.M.: Decay of vacuum energy. Nucl. Phys. B 834, 316–329 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  72. 72.
    Radzikowski M.J.: Micro-local approach to the hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  73. 73.
    Sanders K.: Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime. Commun. Math. Phys. 295, 485 (2010)ADSzbMATHCrossRefGoogle Scholar
  74. 74.
    Strohmaier A., Verch R., Wollenberg M.: Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems. J. Math. Phys. 43, 5514 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  75. 75.
    Tsamis N.C., Woodard R.P.: Quantum gravity slows inflation. Nucl. Phys. B 474, 235 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  76. 76.
    Tsamis N.C., Woodard R.P.: The quantum gravitational back-reaction on inflation. Ann. Phys. 253, 1 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  77. 77.
    Tsamis N.C., Woodard R.P.: Strong infrared effects in quantum gravity. Ann. Phys. 238, 1 (1995)MathSciNetADSCrossRefGoogle Scholar
  78. 78.
    Tsamis N.C., Woodard R.P.: The Structure of perturbative quantum gravity on a De Sitter background. Commun. Math. Phys. 162, 217 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  79. 79.
    Tsamis N.C., Woodard R.P.: Physical Green’s functions in quantum gravity. Ann. Phys. 215, 96 (1992)MathSciNetADSCrossRefGoogle Scholar
  80. 80.
    Tutte W.T.: Graph Theory. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  81. 81.
    Urakawa Y., Tanaka T.: Natural selection of inflationary vacuum required by infra-red regularity and gauge-invariance. Prog. Theor. Phys. 125, 1067–1089 (2011)ADSzbMATHCrossRefGoogle Scholar
  82. 82.
    Urakawa Y., Tanaka T.: IR divergence does not affect the gauge-invariant curvature perturbation. Phys. Rev. D 82, 121301 (2010)ADSCrossRefGoogle Scholar
  83. 83.
    Urakawa Y., Tanaka T.: Influence on observation from IR divergence during inflation – Multi field inflation –. Prog. Theor. Phys. 122, 1207 (2010)ADSCrossRefGoogle Scholar
  84. 84.
    Urakawa Y., Tanaka T.: Influence on observation from IR divergence during inflation I. Prog. Theor. Phys. 122, 779 (2009)ADSzbMATHCrossRefGoogle Scholar
  85. 85.
    Vilenken, N.Y., Klimyk, A.U.: Representations of Lie-Groups and Special Functions. Vol. 1–3, Dordrecht: Kluwer Acad. Publ., 1991Google Scholar
  86. 86.
    Polchinski J.: Renormalization and effective Lagrangians. Nucl. Phys. B 231, 269–295 (1984)ADSCrossRefGoogle Scholar
  87. 87.
    Wegner F., Houghton A.: Renormalization group equations for critical phenomena. Phys. Rev. A 8, 401–412 (1973)ADSCrossRefGoogle Scholar
  88. 88.
    Wilson K.: Renormalization group and critical phenomena I. Renormalization group and the Kadanoff scaling picture. Phys. Rev. B4, 3174–3183 (1971)ADSGoogle Scholar
  89. 89.
    Wilson K.: Renormalization group and critical phenomena II. Phase cell analysis of critical behaviour. Phys. Rev. B4, 3184–3205 (1971)ADSGoogle Scholar
  90. 90.
    Zimmermann, W.: Convergence of Bogolyubov’s method of renormalization in momentum space. Commun. Math. Phys. 15, 208 (1969) [Lect. Notes Phys. 558, 217 (2000)]Google Scholar

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Authors and Affiliations

  1. 1.School of MathematicsCardiff UniversityCardiffUK

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