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Non-perturbative corrections to the one-loop free energy induced by a massive scalar field on a stationary slowly varying in space gravitational background

Journal of High Energy Physics volume 2014, Article number: 111 (2014) Cite this article

A preprint version of the article is available at arXiv.

Abstract

The explicit expressions for the one-loop non-perturbative corrections to the gravitational effective action induced by a scalar field on a stationary gravitational background are obtained both at zero and finite temperatures. The perturbative and non-perturbative contributions to the one-loop effective action are explicitly separated. It is proved that, after a suitable renormalization, the perturbative part of the effective action at zero temperature can be expressed in a covariant form solely in terms of the metric and its derivatives. This part coincides with the known large mass expansion of the one-loop effective action. The non-perturbative part of the renormalized one-loop effective action at zero temperature is proved to depend explicitly on the Killing vector defining the vacuum state of quantum fields. This part cannot be expressed in a covariant way through the metric and its derivatives alone. The implications of this result for the structure and symmetries of the effective action for gravity are discussed.

References

  1. C.J. Isham, Canonical quantum gravity and the problem of time, gr-qc/9210011 [INSPIRE].

  2. B.S. DeWitt, The quantization of geometry, Gravitation: An Introduction to Current Research, in L. Witten ed., Wiley, New York U.S.A. (1962).

  3. B.S. DeWitt, Quantum theory of gravity. I. The canonical theory, Phys. Rev. 160 (1967) 1113 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  4. A.A. Grib, S.G. Mamayev and V.M. Mostepanenko, Vacuum Quantum Effects in Strong Fields, Friedmann Lab. Publ., St. Petersburg Russia (1994).

    Google Scholar 

  5. S.G. Mamaev, V.M. Mostepanenko and A.A. Starobinskii, Particle creation from the vacuum near a homogeneous isotropic singularity, Zh. Eksp. Teor. Fiz. 70 (1976) 1577 [Sov. Phys. JETP 43 (1976) 823].

  6. C.J. Isham and K.V. Kuchar, Representations of Space-time Diffeomorphisms. 1. Canonical Parametrized Field Theories, Annals Phys. 164 (1985) 288 [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  7. C.J. Isham and K.V. Kuchar, Representations of Space-time Diffeomorphisms. 2. Canonical Geometrodynamics, Annals Phys. 164 (1985) 316 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  8. K.V. Kuchar and C.G. Torre, Gaussian reference fluid and interpretation of quantum geometrodynamics, Phys. Rev. D 43 (1991) 419 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  9. J.D. Brown and K.V. Kuchar, Dust as a standard of space and time in canonical quantum gravity, Phys. Rev. D 51 (1995) 5600 [gr-qc/9409001] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  10. B.S. DeWitt, The Global Approach to Quantum Field Theory, Volumes I and II, Clarendon Press, Oxford U.K. (2003).

  11. I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, Effective action in quantum gravity, IOP, Bristol U.K. (1992).

    Google Scholar 

  12. E. Elizalde and S.D. Odintsov, Renormalization group improved effective potential for gauge theories in curved space-time, Phys. Lett. B 303 (1993) 240 [hep-th/9302074] [INSPIRE].

    ADS  Article  Google Scholar 

  13. E. Elizalde and S.D. Odintsov, Renormalization group improved effective Lagrangian for interacting theories in curved space-time, Phys. Lett. B 321 (1994) 199 [hep-th/9311087] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  14. A.I. Zel’nikov, The vacuum polarization of massive fields in algebraically special spaces (in Russian), Abstracts of VI-th Soviet Gravitational Conference, Moscow Russia, 1984, V.N. Ponomareva ed., Moscow State Pedagogical Insitute, Moscow Russia (1984), pg. 197.

  15. S.P. Gavrilov and D.M. Gitman, Vacuum instability in external fields, Phys. Rev. D 53 (1996) 7162 [hep-th/9603152] [INSPIRE].

    ADS  Google Scholar 

  16. D.N. Page, Thermal stress tensors in static Einstein spaces, Phys. Rev. D 25 (1982) 1499 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  17. M.R. Brown, A.C. Ottewill and D.N. Page, Conformally invariant quantum field theory in static Einstein space-times, Phys. Rev. D 33 (1986) 2840 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  18. V.P. Frolov and A.I. Zelnikov, Killing approximation for vacuum and thermal stress-energy tensor in static space-times, Phys. Rev. D 35 (1987) 3031 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  19. V.P. Frolov and I.D. Novikov, Black Hole Physics: Basic Concepts and New Developments, Springer-Verlag, New York U.S.A. (1998).

    Book  MATH  Google Scholar 

  20. P.R. Anderson, W.A. Hiscock and D.A. Samuel, Stress - energy tensor of quantized scalar fields in static spherically symmetric space-times, Phys. Rev. D 51 (1995) 4337 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  21. P.O. Kazinski, One-loop effective potential of the Higgs field on the Schwarzschild background, Phys. Rev. D 80 (2009) 124020 [arXiv:0909.3048] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  22. P.O. Kazinski, Gravitational mass-shift effect in the standard model, Phys. Rev. D 85 (2012) 044008 [arXiv:1107.4714] [INSPIRE].

    ADS  Google Scholar 

  23. I.S. Kalinichenko and P.O. Kazinski, High-temperature expansion of the one-loop free energy of a scalar field on a curved background, Phys. Rev. D 87 (2013) 084036 [arXiv:1301.5103] [INSPIRE].

    ADS  Google Scholar 

  24. G.G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York U.S.A. (1972).

    MATH  Google Scholar 

  25. B.S. DeWitt, Quantum field theory in curved spacetime, Phys. Rept. 19 (1975) 295 [INSPIRE].

    ADS  Article  Google Scholar 

  26. P.O. Kazinski and M.A. Shipulya, One-loop omega-potential of quantum fields with ellipsoid constant-energy surface dispersion law, Annals Phys. 326 (2011) 2658 [arXiv:1103.5146] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  27. D.V. Fursaev, Kaluza-Klein method in theory of rotating quantum fields, Nucl. Phys. B 596 (2001) 365 [Erratum ibid. B 664 (2003) 403] [hep-th/0006217] [INSPIRE].

  28. D.V. Fursaev, Statistical mechanics, gravity and Euclidean theory, Nucl. Phys. Proc. Suppl. 104 (2002) 33 [hep-th/0107089] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  29. D.V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  30. N.D. Birrel and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge U.K. (1982).

    Book  Google Scholar 

  31. J.S. Dowker and G. Kennedy, Finite temperature and boundary effects in static space-times, J. Phys. A 11 (1978) 895 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  32. N. Nakazawa and T. Fukuyama, On the energy-momentum tensor at finite temperature in curved space-time, Nucl. Phys. B 252 (1985) 621 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  33. J.S. Dowker and J.P. Schofield, High temperature expansion of the free energy of a massive scalar field in a curved space, Phys. Rev. D 38 (1988) 3327 [INSPIRE].

    ADS  Google Scholar 

  34. J.S. Dowker and J.P. Schofield, Chemical potentials in curved space, Nucl. Phys. B 327 (1989) 267 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  35. P.O. Kazinski, Propagator of a scalar field on a stationary slowly varying gravitational background, arXiv:1211.3448 [INSPIRE].

  36. B.L. Hu and D.J. O’Connor, Effective Lagrangian for λϕ 4 theory in curved space-time with varying background fields: quasilocal approximation, Phys. Rev. D 30 (1984) 743 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  37. I.G. Avramidi, Non-perturbative effective action in gauge theories and quantum gravity, Adv. Theor. Math. Phys. 14 (2010) 309 [arXiv:0903.1295] [INSPIRE].

    Article  MATH  MathSciNet  Google Scholar 

  38. I.M. Gel’fand and G.E. Shilov, Generalized Functions, Vol. I: Properties and Operations, Academic Press, New York U.S.A., London U.K., (1964).

  39. R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain. I. Three-dimensional problem with smooth boundary surface, Annals Phys. 60 (1970) 401 [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  40. R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain. II. Electromagnetic field. Riemannian spaces, Annals Phys. 64 (1971) 271.

    ADS  Article  MATH  MathSciNet  Google Scholar 

  41. R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillations, Annals Phys. 69 (1972) 76 [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  42. R. Balian and C. Bloch, Solution of the Schrödinger Equation in Terms of Classical Paths, Annals Phys. 85 (1974) 514 [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  43. L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Butterworth-Heinemann, San Francisco U.S.A. (1994).

    Google Scholar 

  44. A.L. Zelmanov, Chronometric invariants and comoving coordinates in General Relativity, Dokl. Akad. Nauk SSSR 107 (1956) 815.

    MathSciNet  Google Scholar 

  45. Yu.S. Vladimirov, Reference Frames in Theory of Gravity (in Russian), Energoizdat, Moscow Russia (1982).

    Google Scholar 

  46. N.V. Mitskevich, A.P. Efremov and A.I. Nesterov, Dynamics of Fields in General Relativity (in Russian), Energoatomizdat, Moscow Russia (1985).

    Google Scholar 

  47. R.M. Wald, General Relativity, University of Chicago Press, Chicago U.S.A. (1984).

    Book  MATH  Google Scholar 

  48. P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or. Perish, Wilmington, Delaware U.S.A. (1984).

  49. B.S. DeWitt, Transition from discrete to continuous spectra, Phys. Rev. 103 (1956) 1565 [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  50. A.B. Migdal, Vacuum polarization in strong fields and pion condensation, Sov. Phys. Usp. 20 (1977) 879.

    ADS  Article  Google Scholar 

  51. L.D. Landau and E.M. Lifshitz, Statistical Physics. Part I, Pergamon, Oxford U.K. (1978).

    Google Scholar 

  52. E.M. Lifshits and L.P. Pitaevskii, Statistical Physics. Part II, Pergamon, New York U.S.A. (1980).

    Google Scholar 

  53. I.M. Lifshits, M.Ya. Azbel and M.I. Kaganov, Electron theory of metals, Consultants Bureau, New York U.S.A. (1973).

    Google Scholar 

  54. D. Shoenberg, Magnetic Oscillations in Metals, (Cambridge University Press, Cambridge U.S.A. (1984).

    Book  Google Scholar 

  55. Y. Gusev and A. Zelnikov, Finite temperature nonlocal effective action for quantum fields in curved space, Phys. Rev. D 59 (1999) 024002 [hep-th/9807038] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  56. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2, Wiley, New York U.S.A. (1962).

    MATH  Google Scholar 

  57. P.B. Gilkey, Asymptotic Formulae in Spectral Geometry, CRC Press LLC, Boca Raton U.S.A. (2004).

    MATH  Google Scholar 

  58. A.Z. Petrov, Einstein Spaces, Pergamon, Oxford U.K. (1969).

    MATH  Google Scholar 

  59. J.D. Bekenstein and L. Parker, Path Integral Evaluation of Feynman Propagator in Curved Space-time, Phys. Rev. D 23 (1981) 2850 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  60. S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].

    ADS  Google Scholar 

  61. J.F. Donoghue, E. Golowich and B.R. Holstein, Dynamics of the Standard Model, Cambridge University Press, Cambridge U.K. (1994).

    Google Scholar 

  62. P.O. Kazinski, Comment on “Quantum versus classical instability of scalar fields in curved backgrounds”, arXiv:1310.6252 [INSPIRE].

  63. G.E.A. Matsas, R.F.P. Mendes and D.A.T. Vanzella, Reply to “Comment on ‘Quantum versus classical instability of scalar fields in curved backgrounds”, arXiv:1310.7849 [INSPIRE].

  64. L.V. Prokhorov, Hamiltonian path integrals (in Russian), Fizika Elementarnykh Chastits i Atomnogo Yadra 13 (1982) 1094.

    MathSciNet  Google Scholar 

  65. L. Parker and D.J. Toms, New Form for the Coincidence Limit of the Feynman Propagator, or Heat Kernel, in Curved Space-time, Phys. Rev. D 31 (1985) 953 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  66. I. Jack and L. Parker, Proof of Summed Form of Proper Time Expansion for Propagator in Curved Space-time, Phys. Rev. D 31 (1985) 2439 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  67. L. Parker and D.J. Toms, Quantum Field Theory in Curved Spacetime, Cambridge University Press, Cambridge U.K. (2009).

    Book  MATH  Google Scholar 

  68. K.W. Howard, Vacuum \( \left\langle {T}_{\mu}^{\nu}\right\rangle \) in Schwarzschild spacetime, Phys. Rev. D 30 (1984) 2532 [INSPIRE].

    ADS  Google Scholar 

  69. L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, Oxford U.K. (1965).

    MATH  Google Scholar 

  70. D.R. Brill and J.A. Wheeler, Interaction of neutrinos and gravitational fields, Rev. Mod. Phys. 29 (1957) 465 [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  71. S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].

  72. V.S. Buldyrev and V.E. Nomofilov, Asymptotic solutions of an elliptic equation system on a Riemannian manifold concentrated in the vicinity of a phase trajectory, J. Phys. A 14 (1981) 1577 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  73. V.G. Bagrov, V.V. Belov, A.Y. Trifonov and A.A. Evseevich, Quasiclassical trajectory coherent approximation in quantum mechanics of charged particle in curved space-time, Class. Quant. Grav. 8 (1991) 515 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  74. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Acad. Press, Boston U.S.A. (1994).

    MATH  Google Scholar 

  75. L. Landau, Diamagnetismus der Metalle, Z. Phys. 64 (1930) 629.

    ADS  Article  MATH  Google Scholar 

  76. A.S. Vshivtsev, K.G. Klimenko and B.V. Magnitsky, Landau oscillations in (2+1)-dimensional quantum electrodynamics, J. Exp. Theor. Phys. 80 (1995) 162 [INSPIRE].

    ADS  Google Scholar 

  77. A.S. Vshivtsev and K.G. Klimenko, An exact expression for magnetic oscillations in quantum electrodynamics, Zh. Eksp. Theor. Fiz. 109 (1996) 954 [J. Exp. Theor. Phys. 154 (1996) 82].

  78. W. Heisenberg and H. Euler, Consequences of Dirac’s theory of positrons, Z. Phys. 98 (1936) 714 [physics/0605038] [INSPIRE].

    ADS  Article  Google Scholar 

  79. J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  80. N.N. Bogolyubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields, Wiley, New York U.S.A. (1980).

    Google Scholar 

  81. J.C. Collins, Renormalization, Cambridge University Press, Cambridge U.K. (1984).

    Book  MATH  Google Scholar 

  82. V.A. Fock, The Theory of Space, Time and Gravitation, Pergamon Press, London U.K. (1959).

    MATH  Google Scholar 

  83. A.H. Taub, General Relativistic Variational Principle for Perfect Fluids, Phys. Rev. 94 (1954) 1468 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  84. B.F. Schutz, Perfect Fluids in General Relativity: Velocity Potentials and a Variational Principle, Phys. Rev. D 2 (1970) 2762 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  85. J.D. Brown, Action functionals for relativistic perfect fluids, Class. Quant. Grav. 10 (1993) 1579 [gr-qc/9304026] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  86. P. Hajicek and J. Kijowski, Lagrangian and Hamiltonian formalism for discontinuous fluid and gravitational field, Phys. Rev. D 57 (1998) 914 [Erratum ibid. D 61 (2000) 129901] [gr-qc/9707020] [INSPIRE].

  87. R. Jackiw, V.P. Nair, S.Y. Pi and A.P. Polychronakos, Perfect fluid theory and its extensions, J. Phys. A 37 (2004) R327 [hep-ph/0407101] [INSPIRE].

    ADS  Article  Google Scholar 

  88. P.O. Kazinski, Radiation reaction for multipole moments, J. Exp. Theor. Phys. 105 (2007) 327 [hep-th/0604168] [INSPIRE].

    ADS  Article  Google Scholar 

  89. R.G. Newton, Scattering Theory of Waves and Particles, Springer-Verlag, New York U.S.A. (1982)

    Book  MATH  Google Scholar 

  90. A.N. Vasil’ev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics, Chapman & Hall/CRC, Boca Raton U.S.A. (2004).

    Book  MATH  Google Scholar 

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  1. Physics Faculty, Tomsk State University, Tomsk, 634050, Russia

    Igor Kalinichenko & Peter Kazinski

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  1. Igor Kalinichenko
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Kalinichenko, I., Kazinski, P. Non-perturbative corrections to the one-loop free energy induced by a massive scalar field on a stationary slowly varying in space gravitational background. J. High Energ. Phys. 2014, 111 (2014). https://doi.org/10.1007/JHEP08(2014)111

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Keywords

  • Nonperturbative Effects
  • Models of Quantum Gravity
  • Anomalies in Field and String Theories
  • Renormalization Regularization and Renormalons