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Letters in Mathematical Physics

, Volume 104, Issue 2, pp 217–232 | Cite as

“Tunnelling” Black-Hole Radiation with ϕ 3 Self-Interaction: One-Loop Computation for Rindler Killing Horizons

  • Giovanni Collini
  • Valter Moretti
  • Nicola Pinamonti
Article

Abstract

Tunnelling processes through black hole horizons have recently been investigated in the framework of WKB theory discovering interesting interplay with the Hawking radiation. A more precise and general account of that phenomenon has been subsequently given within the framework of QFT in curved spacetime by two of the authors of the present paper. In particular, it has been shown that, in the limit of sharp localization on opposite sides of a Killing horizon, the quantum correlation functions of a scalar field appear to have thermal nature, and the tunnelling probability is proportional to exp{−β Hawking E}. This local result is valid in every spacetime including a local Killing horizon, no field equation is necessary, while a suitable choice for the quantum state is relevant. Indeed, the two-point function has to verify a short-distance condition weaker than the Hadamard one. In this paper we consider a massive scalar quantum field with a ϕ 3 self-interaction and we investigate the issue whether or not the black hole radiation can be handled at perturbative level, including the renormalisation contributions. We prove that, for the simplest model of the Killing horizon generated by the boost in Minkowski spacetime, and referring to Minkowski vacuum, the tunnelling probability in the limit of sharp localization on opposite sides of the horizon preserves the thermal form proportional to exp{−β H E} even taking the one-loop renormalisation corrections into account. A similar result is expected to hold for the Unruh state in the Kruskal manifold, since that state is Hadamard and looks like Minkowski vacuum close to the horizon.

Mathematics Subject Classification (2010)

81T15 83C57 81T20 

Keywords

algebraic quantum field theory black hole radiation renormalisation Rindler space 

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References

  1. 1.
    Akhmedov E.T., Pilling T., Singleton D.: Subtleties in the quasi-classical calculation of Hawking radiation. Int. J. Mod. Phys. D17, 2453–2458 (2008)ADSCrossRefMathSciNetGoogle Scholar
  2. 2.
    Akhmedov E.T., Pilling T., de Gill A., Singleton D.: Temporal contribution to gravitational WKB-like calculations. Phys. Lett. B666, 269–271 (2008)ADSCrossRefGoogle Scholar
  3. 3.
    Angheben M., Nadalini M., Vanzo L., Zerbini S.: Hawking radiation as tunneling for extremal and rotating black holes. JHEP 0505, 014 (2005)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buchholz D.: Quarks, gluons, colour: facts or fiction?. Nucl. Phys. B469, 333–356 (1996)ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Brunetti R., Dütsch M., Fredenhagen K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13, 1–56 (2009)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Calogeracos A., Volovik G.E.: Rotational quantum friction in superfluids: radiation from object rotating in superfluid vacuum. JETP Lett. 69, 281–287 (1999)ADSCrossRefGoogle Scholar
  8. 8.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Academic Press, New York (2007)Google Scholar
  9. 9.
    Dappiaggi C., Moretti V., Pinamonti N.: Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime. Adv. Theor. Math. Phys. 15, 355–448 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Di Criscienzo, R., Nadalini, M., Vanzo, L., Zerbini, S., Zoccatelli, G.: On the Hawking radiation as tunneling for a class of dynamical black holes. Phys. Lett. B 657, 107–111 (2007)Google Scholar
  11. 11.
    Epstein H., Glaser V.: The role of locality in perturbation theory. Ann. Inst. Henri Poincar. 19(3), 211–295 (1973)MathSciNetGoogle Scholar
  12. 12.
    Fredenhagen K., Haag R.: Generally covariant quantum field theory and scaling limits. Commun. Math. Phys. 108, 91 (1987)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Fredenhagen K., Haag R.: On the derivation of Hawking radiation associated with the formation of a black hole. Commun. Math. Phys. 127, 273 (1990)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hollands S., Wald R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289–326 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hollands S., Wald R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Hayward, S.A., Di Criscienzo, R., Vanzo, L., Nadalini, M., Zerbini, S.: Local Hawking temperature for dynamical black holes. Class. Quant. Grav. 26, 062001 (2009)Google Scholar
  17. 17.
    Moretti V., Pinamonti N.: State independence for tunneling processes through black hole horizons. Commun. Math. Phys. 309, 295–311 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Moretti V.: Comments on the stress–energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189–222 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Vanzo, L., Acquaviva, G., Di Criscienzo, R.: Tunnelling methods and Hawking’s radiation: achievements and prospects. Class. Quant. Grav. 28, 183001 (2011)Google Scholar
  20. 20.
    Volovik, G.E.: Simulation of Panleve–Gullstrand black hole in thin 3He-A film. JETP Lett. 69, 705–713 (1999). arXiv:gr-qc/9901077Google Scholar
  21. 21.
    Hawking S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199 (1975)ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kerner R., Mann R.B.: Fermions tunnelling from black holes. Class. Quant. Grav. 25, 095014 (2008)ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Medved A.J.M., Vagenas E.C.: On Hawking radiation as tunneling with back-reaction. Mod. Phys. Lett. A 20, 2449–2454 (2005)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Parikh, M.K., Wilczek, F.: Hawking radiation as tunneling. Phys. Rev. Lett. 85, 5042. hep-th/9907001 (2000)Google Scholar
  25. 25.
    Radzikowski M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Strohmaier, A.: Microlocal analysis. In: Bär, C.K. (eds.) Quantum Field Theory on Curved Spacetime Lecture Notes in Physics, vol. 786. Fredenhagen Springer, Berlin (2009)Google Scholar
  27. 27.
    Wald R.M.: General Relativity. Chicago University Press, Chicago (1984)CrossRefzbMATHGoogle Scholar
  28. 28.
    Wald R.M.: Quantum field theory in curved space–time and black hole thermodynamics. The University of Chicago Press, Chicago (1994)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Giovanni Collini
    • 1
  • Valter Moretti
    • 2
  • Nicola Pinamonti
    • 3
  1. 1.Institut für Theoretische PhysikLeipzigGermany
  2. 2.Dipartimento di MatematicaUniversità di Trento and Istituto Nazionale di Fisica Nucleare-Gruppo Collegato di TrentoPovoItaly
  3. 3.Dipartimento di MatematicaUniversità di GenovaGenovaItaly

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