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Stochastic Gravity

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Abstract

We give a summary of the status of currentresearch in stochastic semiclassical gravity and suggestdirections for further investigations. This theorygeneralizes the semiclassical Einstein equation to an Einstein-Langevin equation with a stochasticsource term arising from the fluctuations of theenergy-momentum tensor of quantum fields. We mentionrecent efforts in applying this theory to the study of black hole fluctuation and backreactionproblems, linear response of hot flat space, andstructure formation in inflationary cosmology. Toexplore the physical meaning and implications of thisstochastic regime in relation to both classical andquantum gravity, we find it useful to take the view thatsemiclassical gravity is mesoscopic physics and thatgeneral relativity is the hydrodynamic limit of certain spacetime quantum substructures. We view theclassical spacetime depicted by general relativity as acollective state and the metric or connection functionsas collective variables. Three basic issues —stochasticity, collectivity, correlations — andthree processes — dissipation, fluctuations,decoherence — underscore the transformation fromquantum microstructure and interaction to the emergenceof classical macrostructure and dynamics. We discuss ways toprobe into the high-energy activity from below and maketwo suggestions: via effective field theory and thecorrelation hierarchy. We discuss how stochastic behavior at low energy in an effective theoryand how correlation noise associated with coarse-grainedhigher correlation functions in an interacting quantumfield could carry nontrivial information about the high-energy sector. Finally, we describeprocesses deemed important at the Planck scale,including tunneling and pair creation, wave scatteringin random geometry, growth of fluctuations and forms, Planck-scale resonance states, and spacetimefoams.

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REFERENCES

  1. N. D. Birrell, and P. C. Davis, Quantum Fields in Curved Space (Cambridge University Press, New York, 1982).

    Google Scholar 

  2. Ya. Zel'dovich and A. Starobinsky, Zh. Eksp. Teor. Fiz. 61, 2161 (1971) [Sov. Phys. JETP 34, 1159 (1971) ]; L. Grishchuk, Ann. N. Y. Acad. Sci. 302, 439 (1976); B. L. Hu and L. Parker, Phys. Lett. 63A, 217 (1977); B. L. Hu and L. Parker, Phys. Rev. D 17, 933 (1978); F. V. Fischetti, J. B. Hartle, and B. L. Hu, Phys. Rev. D 20, 1757 (1979); J. B. Hartle and B. L. Hu, Phys. Rev. D 20, 1772 (1979); 21, 2756 (1980); J. B. Hartle, Phys. Rev. D 23, 2121 (1981); P.A. Anderson, Phys. Rev. D 28, 271 (1983); 29, 615 (1984).

    Google Scholar 

  3. E. Calzetta and B. L. Hu, Phys. Rev. D 35, 495 (1987).

    Google Scholar 

  4. J. Schwinger, J. Math. Phys. 2, 407 (1961), P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4, 1, 12 (1963); L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JEPT 20, 1018 (1965) ]; G. Zhou, Z. Su, B. Hao, and L. Yu, Phys. Rep. 118, 1 (1985); Z. Su, L. Y. Chen, X. Yu, and K. Chou, Phys. Rev. B 37, 9810 (1988); B. S. DeWitt, In Quantum Concepts in Space and Time, R. Penrose and C. J. Isham, eds. (Clarendon Press, Oxford, 1986); R. D. Jordan, Phys. Rev. D 33, 44 (1986); E. Calzetta and B. L. Hu, Phys. Rev. D 35, 495 (1987); A. Campos and E. Verdaguer, Phys. Rev. D 49, 1861 (1994).

    Google Scholar 

  5. B. L. Hu, Physica A 158, 399 (1989).

    Google Scholar 

  6. C. Kiefer, Class. Quant. Grav. 4, 1369 (1987); J. J. Halliwell, Phys. Rev. D 39, 2912 (1989); T. Padmanabhan, Phys. Rev. D 39, 2924 (1989); B. L. Hu, Quantum and statistical effects in superspace cosmology, in Quantum Mechanics in Curved Space time, J. Audretsch and V. de Sabbata, eds. (Plenum, New York, 1990); E. Calzetta, Class. Quant. Grav. 6, L227 (1989); Phys. Rev. D 43, 2498 (1991); J. P. Paz and S. Sinha, Phys. Rev. D 44, 1038 (1991); 45, 2823 (1992); B. L. Hu, J. P. Paz, and S. Sinha, Minisuperspace as a quantum open system, in Directions in General Relativity, Vol. 1, B. L. Hu, M. P. Ryan, and C. V. Vishveswara, eds. (Cambridge University Press, Cambridge, 1993).

    Google Scholar 

  7. E. B. Davies, The Quantum Theory of Open Systems (Academic Press, London, 1976); K. Lindenberg and B. J. West, The Nonequilibrium Statistical Mechanics of Open and Closed Systems (VCH Press, New York, 1990); U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1993).

    Google Scholar 

  8. R. Feynman and F. Vernon, Ann. Phys. (N.Y.) 24, 118 (1963); R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965); A. O. Caldeira and A. J. Leggett, Physica 121A, 587 (1983); H. Grabert, P. Schramm, and G. L. Ingold, Phys. Rep. 168, 115 (1988); B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1902); 47, 1576 (1993).

    Google Scholar 

  9. A. O. Caldeira and A. J. Leggett, Physica 121A, 587 (1983); Ann. Phys. (N.Y.) 149, 374 (1983); H. Grabert, P. Schramm, and G. L. Ingold, Phys. Rep. 168, 115 (1988); B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992); 47, 1576 (1993).

    Google Scholar 

  10. B. L. Hu, Statistical mechanics and quantum cosmology, in Proceedings Second International Workshop on Thermal Fields and Their Applications, H. Ezawa et al., eds. (North-Holland, Amsterdam, 1991).

    Google Scholar 

  11. W. H. Zurek, Phys. Rev. D 24, 1516 (1981); 26, 1862 (1982); in Frontiers of Nonequilibrium Statistical Physics, G. T. Moore and M. O. Scully, eds. (Plenum, New York, 1986); Physics Today 44, 36 (1991); E. Joos and H. D. Zeh, Z. Phys. B 59, 223 (1985); A. O. Caldeira and A. J. Leggett, Phys. Rev. A 31, 1059 (1985); W. G. Unruh and W. H. Zurek, Phys. Rev. D 40, 1071 (1989); B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992); W. H. Zurek, Prog. Theor. Phys. 89, 281 (1993); D. Giulini et al., Decoherence and the Appearance of a Classical World in Quantum Theory (Springer-Verlag, Berlin, 1996).

    Google Scholar 

  12. R. B. Griffiths, J. Stat. Phys. 36, 219 (1984); R. Omnés, J. Stat. Phys. 53, 893, 933, 957 (1988); Ann. Phys. (N.Y.) 201, 354 (1990); Rev. Mod. Phys. 64, 339 (1992); The Interpretation of Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1994); M. Gell-Mann and J. B. Hartle, In Complexity, Entropy and the Physics of Information, W. H. Zurek, ed. (Addison-Wesley, Reading, Massachusetts, 1990); Phys. Rev. D 47, 3345 (1993); J. B. Hartle, Quantum mechanics of closed systems, in Directions in General Relativity, Vol. 1, B. L. Hu, M. P. Ryan, and C. V. Vishveswara, eds. (Cambridge University Press, Cambridge, 1993); H. F. Dowker and J. J. Halliwell, Phys. Rev. D 46, 1580 (1992); T. Brun, Phys. Rev. D 47, 3383 (1993); J. P. Paz and W. H. Zurek, Phys. Rev. D 48, 2728 (1993); J. Twamley, Phys. Rev. D 48, 5730 (1993); F. Dowker and A. Kent, Phys. Rev. Lett. 75, 3038 (1995); J. Stat. Phys. 82, 1575 (1996); A. Kent, Phys. Rev. A 54, 4670 (1996); Phys. Rev. Lett. 78, 2874 (1997); Phys. Rev. Lett. 81, 1982 (1998).

    Google Scholar 

  13. W. Bernard and H. B. Callen, Rev. Mod. Phys. 31, 1017 (1959); R. Kubo, Rep. Prog. Phys. 29, 255 (1966); L. Landau, E. Lifshitz, and L. Pitaevsky, Statistical Physics, Vol. 1 (Pergamon, London, 1980); R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II (Springer-Verlag, Berlin, 1985).

    Google Scholar 

  14. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics, Vol. 2 (Springer-Verlag, Berlin, 1991), Chapter 4.

    Google Scholar 

  15. M. Gell-Mann and J. B. Hartle, Phys. Rev. D 47, 3345 (1993).

    Google Scholar 

  16. Y. Zhang, Ph.D. Thesis, University of Maryland (1991); B. L. Hu, J. P. Paz, and Y. Zhang, Quantum origin of noise and fluctuation in cosmology, in The Origin of Structure in the Universe, E. Gunzig and P. Nardone, eds. (Plenum Press, New York, 1993), p. 227.

  17. B. L. Hu and A. Matacz, Phys. Rev. D 49, 6612 (1994).

    Google Scholar 

  18. E. Calzetta and B. L. Hu, Phys. Rev D 49, 6636 (1994).

    Google Scholar 

  19. Z. Su, L. Y. Chen, X. Yu, and K. Chou, Phys. Rev. B 37, 9810 (1988).

    Google Scholar 

  20. H.F. Dowker and J. J. Halliwell, Phys. Rev. D 46, 1580 (1992); J. P. Paz and W. H. Zurek, Phys. Rev. D 48, 2728 (1993).

    Google Scholar 

  21. E. Calzetta and B. L. Hu, Decoherence of correlation histories, in Directions in General Relativity, Vol. II: Brill Festschrift, B. L. Hu and T. A. Jacobson, eds. (Cambridge University Press, Cambridge, 1993).

    Google Scholar 

  22. A. Campos and E. Verdaguer, Phys. Rev. D 49, 1861 (1994).

    Google Scholar 

  23. R. Martin and E. Verdaguer, Int. J. Theor. Phys., this issue; gr-qc/9812063, 9811070.

  24. B. L. Hu, In Proceedings of the Third International Workshop on Thermal Fields and Its Applications, R. Kobes and G. Kunstatter, eds. (World Scientific, Singapore, 1994).

    Google Scholar 

  25. B. L. Hu and S. Sinha, Phys. Rev. D 51, 1587 (1995).

    Google Scholar 

  26. B. L. Hu and A. Matacz, Phys. Rev. D 51, 1577 (1995).

    Google Scholar 

  27. A. Campos and E. Verdaguer, Phys. Rev. D 53, 1927 (1996).

    Google Scholar 

  28. S. A. Fulling and L. Parker, Ann. Phys. (N.Y.) 87, 176 (1974).

    Google Scholar 

  29. E. Calzetta, A. Campos, and E. Verdaguer, Phys. Rev. D 56, 2163 (1997).

    Google Scholar 

  30. G. Horowitz, Phys. Rev. D 21, 1445 (1980).

    Google Scholar 

  31. F. C. Lombardo and F. D. Mazzitelli, Phys. Rev. D. 55, 3889 (1997).

    Google Scholar 

  32. F. C. Lombardo and F. D. Mazzitelli, Phys. Rev. D. 58, 024009 (1998); F. C. Lombardo, F. D. Mazzitelli, and J. Russo, Phys. Rev. D. 59,064007 (1999).

    Google Scholar 

  33. P. Johnson and B. L. Hu, Quantum stochastic theory of relativistic particle-field interaction: Effect of strong fields on the trajectory of a particle detector, in preparation.

  34. E. Calzetta and B. L. Hu, Correlations, decoherence, disspation and noise in quantum field theory, in Heat Kernel Techniques and Quantum Gravity, S. Fulling, ed. (Texas A&M Press, College Station, Texas, 1995); hep-th/9501040.

    Google Scholar 

  35. E. Calzetta and B. L. Hu, Nonequilibrium quantum fields: Master effective action, correlation noise and stochastic Boltzmann equation, in preparation.

  36. B. L. Hu, Alpan Raval, and S. Sinha, Notes on black hole fluctuations and backreaction, in Black Holes, Gravitational Radiation and the Universe, B. R. Iyer and B. Bhawal, eds. (Kluwer, Dordtrecht, 1999).

    Google Scholar 

  37. N. G. Phillips, Ph.D. Thesis, University of Maryland (1999).

  38. B. L. Hu, N. Phillips, and A. Raval, Fluctuations of the energy momentum tensor of a quantum field in a black hole spacetime, in preparation.

  39. B. L. Hu, Alpan Raval, and S. Sinha, Backreaction of a radiating quantum black hole and fluctuation-dissipation relation, in preparation.

  40. E. Calzetta and E. Verdaguer, Phys. Rev. D 59, 083513 (1999).

    Google Scholar 

  41. A. Campos and B. L. Hu, Phys. Rev. D 58, 125021 (1998).

    Google Scholar 

  42. J. A. Wheeler, Ann. Phys. (N.Y.) 2, 604 (1957); Geometrodynamics (Academic Press, London, 1962); in Relativity, Groups and Topology, B. and C. DeWitt, eds. (Gordon and Breach, New York, 1964).

    Google Scholar 

  43. L. H. Ford, Phys. Rev. D 51, 1692 (1995).

    Google Scholar 

  44. C.-I. Kuo and L. H. Ford, Phys. Rev. D 47, 4510 (1993).

    Google Scholar 

  45. N. Phillips and B. L. Hu, Phys. Rev. D 55, 6123 (1997).

    Google Scholar 

  46. E. Calzetta and B.-L. Hu, Phys. Rev. D 52, 6770 (1995); E. Calzetta and S. Gonorazky, Phys. Rev. D 55, 1812 (1997); A. Matacz, Phys. Rev. D 55, 1860 (1997); 56, R1836 (1997).

    Google Scholar 

  47. B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, 1965); Phys. Rep. C 19, 295 (1975); S. Christensen, Phys. Rev. D 14, 2490 (1976).

  48. P. R. Anderson, W. A. Hiscock, and D. A. Samuel, Phys. Rev. Lett. 70, 1739 (1993); Phys. Rev. D 51, 4337 (1995).

    Google Scholar 

  49. D. N. Page, Phys. Rev. D 25, 1499 (1982).

    Google Scholar 

  50. L. H. Ford, Cosmological and black hole horizon fluctuations, gr-qc/9704050.

  51. A. Casher, F. Englert, N. Itzhaki, and R. Parentani, Black hole horizon fluctuations, hepth/ 9606106.

  52. R. D. Sorkin, How wrinkled is the surface of a black hole?, gr-qc/9701056.

  53. C. Barrabès, V. Frolov, and R. Parentani, Metric fluctuation correction to Hawking radiation, gr-qc/9812076.

  54. P. Anderson et al., In Heat Kernei Techniques and Quantum Gravity, S. A. Fulling, ed. (Texas A&M University Press, College Station, Texas, 1995).

    Google Scholar 

  55. J. W. York, Jr., Phys. Rev. D 28, 2929 (1983); 31, 775 (1985); 33, 2092 (1986).

    Google Scholar 

  56. J. M. Bardeen, Phys. Rev. Lett. 46, 382 (1981); P. Hajicek and W. Israel, Phys. Lett. 80A, 9 (1980).

    Google Scholar 

  57. S. Massar, Phys. Rev. D 52, 5857 (1995).

    Google Scholar 

  58. P. Candelas and D. W. Sciama, Phys. Rev. Lett. 38, 1372 (1977)

    Google Scholar 

  59. E. Mottola, Phys. Rev. D 33, 2136 (1986)

    Google Scholar 

  60. A. Campos and B. L. Hu, Int. J. Theor. Phys. 38, 1253 (1999).

    Google Scholar 

  61. D. J. Gross, M. J. Perry, and L. G. Yaffe, Phys. Rev. D 25, 330 (1982).

    Google Scholar 

  62. A. Rebhan, Nucl. Phys. B 351, 706 (1991).

    Google Scholar 

  63. A. P. de Almeida, F. T. Brandt, and J. Frenkel, Phys. Rev. D 49, 4196 (1994); F. T. Brandt and J. Frenkel, Phys. Rev. D 58,085012 (1998), and references therein.

    Google Scholar 

  64. B. L. Hu, In Quantum Classical Correspondence, D. S. Feng and B. L. Hu, eds. (International Press, Boston, 1997).

    Google Scholar 

  65. A. D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Doklady Akad. Nauk SSR 177, 70-71 (1987) [Sov. Phys. Doklady 12, 1040-1041 (1968) ]; see also S. L. Adler, Rev. Mod. Phys. 54, 729 (1982).

    Google Scholar 

  66. B. L. Hu, General relativity as geometro-hydrodynamics, gr-qc/9607070.

  67. E. Calzetta and B. L. Hu, Phys. Rev. D 55, 1795 (1997).

    Google Scholar 

  68. B. L. Altshuler, P. A. Lee, and R. A. Webb, eds., Mesoscopic Phenomena in Solids (North-Holland, Amsterdam, 1991); B. K.Kramer, ed., Quantum Coherence in Mesoscopic Systems (Plenum Press, New York, 1991); W. P. Kirk and M. A. Reed, eds., Nanostructures and Mesocopic Systems (Academic Press, San Diego, 1992); Y. Imry, Introduction to Mesoscopic Physics (Wiley, New York, 1997).

    Google Scholar 

  69. A. O Caldeira and A. J. Leggett, Ann. Phys. (N.Y) 149, 374 (1993).

    Google Scholar 

  70. S. Weinberg, Quantum Field Theory (Wiley, New York, 1996).

    Google Scholar 

  71. B. L. Hu and Y. Zhang, Coarse-Graining, scaling, and inflation, University of Maryland Preprint 90-186 (1990); B. L. Hu, In Relativity and Gravitation: Classical and Quantum, J. C. D' Olivo et al., eds. (World Scientific, Singapore, 1991).

  72. T. Jacobson, Introduction to black hole microscopy, hep-th/9510026.

  73. W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981); Phys. Rev. D 51, 2827 (1995); T. Jacobsen, Phys. Rev. D 44, 1731 (1991); 53, 7082 (1994).

    Google Scholar 

  74. B. L. Hu, Fluctuation, dissipation and irreversibility in cosmology, in The Physical Origin of Time-Asymmetry, J. J. Halliwell, J. Perez-Mercader, and W. H. Zurek, eds. (Cambridge University Press, Cambridge, 1994).

    Google Scholar 

  75. J. B. Hartle, R. Laflamme, and D. Marolf, Phys. Rev. D 51, 7007 (1995);T. Brun and J. J. Halliwell, Phys. Rev. D 54, 2899 (1996); J. J. Halliwell, Phys. Rev. D 58, 105015 (1998); C. Anastopoulos, gr-qc/9805074; T. Brun and J. B. Hartle, quantphys/9808024.

    Google Scholar 

  76. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1971), Chapter 15.

    Google Scholar 

  77. M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory (Cambridge University Press, Cambridge, 1990); E. Witten, Physics Today 49, 24 (1996); J. Polchinsky, Superstring Theory (Cambridge University Press, Cambridge, 1998).

    Google Scholar 

  78. A. Ashtekar, Lectures on Non-Perturbativ e Canonical Gravity (World Scientific, Singapore, 1991); in Knot Theory and Quantum Gravity, J. Baez, ed. (Oxford University, Press, Oxford, 1995); R. Gambini and J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity (Cambridge University Press, Cambridge, 1996); A. Ashtekar, K. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998).

    Google Scholar 

  79. T. Regge, Nuovo Cimento 19, 558 (1961); J. B. Hartle, J. Math. Phys. 26, 804 (1985); 27, 287 (1986); 30, 452 (1989); H. W. Hamber, In Critical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1986); H. W. Hamber, Nucl. Phys. B (Proc. Suppl.) 20, 728 (1991); 25A, 150 (1992); Phys. Rev. D 45, 507 (1992); Nucl. Phys. B 400, 347 (1993); R. M. Williams and P. A. Tucker, Class. Quant. Grav. 9, 1409 (1992); H. W. Hamber and R. M. Williams, Phys. Rev. D 47, 510 (1993); Nucl. Phys. 415, 463 (1994).

    Google Scholar 

  80. J. D. Bekenstein, Phys. Rev. D 7, 1333 (1973); S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).

    Google Scholar 

  81. R. M. Wald, Phys. Rev. D 48, R3427 (1993);T. Jacobson, G. Kang, and R. Myers, Phys. Rev. D 49, 6587 (1994); T. Jacobson, Black hole entropy and induced gravity, gr-qc/9404039; M. Banados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 72, 957 (1994); J. D. Bekenstein, Do we understand black hole entropy? in Proceedings Seventh Marcel Grossmann Meeting (Stanford University, 1994), gr-qc/9409015; D. M. Page, Black hole information, in Proceedings 5th Canadian Conference on General Relativity and Relativistic Astrophysics, R. B. Mann and R. G. McLenaghan, eds. (World Scientific, Singapore, 1994), hep-th/9305040; V. P. Frolov, D. V. Fursaev, and A. I. Zelnikov, Black hole entropy: Off-shell vs on-shell, hep-th/9512184; Black hole entropy and/induced gravity, hep-th/9607104.

    Google Scholar 

  82. T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).

    Google Scholar 

  83. L. Susskind and J. Uglam, Phys. Rev. D 50, 2700 (1994);D. Kabat, S. H. Shenker, and M. J. Strassler, Phys. Rev. D 52, 7027 (1995); J. D. Bekenstein and V. F. Mukhanov, Phys. Lett. B 360, 7 (1995).

    Google Scholar 

  84. A. Strominger and C. Vafa, Phys. Lett. B 379, 99 (1996); G. T. Horowitz, The origin of black hole entropy in string theory, in Proceedings of the Pacific Conference on Gravitation and Cosmology (Seoul, Korea, 1996), gr-qc/9604051; G. Horowitz and J. Polchinski, Phys. Rev. D 55, 6189 (1997); J. M. Maldacena, Black holes and D-branes, Nucl. Phys. Proc. Suppl. 61A, 111 (1998).

    Google Scholar 

  85. A. Ashtekhar, In String Gravity and Physics at the Planck Energy Scale, N. Sanchez and A. Zichichi, eds. (Kluwer, Dordrecht, 1996).

    Google Scholar 

  86. T. Regge, Nuovo Cimento 19, 558 (1961);J. B. Hartle, J. Math Phys. 26, 804 (1985); 27, 287 (1986); 30, 452 (1989); H. W. Hamber, In Critical Phenomena, Random Systems, Gauge Theories K Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1986).

    Google Scholar 

  87. G. Ponzano and T. Regge, Semiclassical limit of Racah coefficients, in Spectroscopic and Group Theoretical Methods in Physics, F. Bloch, ed. (North-Holland, Amsterdam, 1968); J. Iwasaki, A reformulation of the Ponzano-Regge quantum gravity models in terms of surfaces (1994); J. W. Barrett and T. J. Foxon, Class. Quant. Grav. 11, 543 (1994); J. W. Barrett, Quantum gravity as topological quantum field theory (1995).

    Google Scholar 

  88. D. Nelson et al., eds., Statistical Mechanics of Membranes and Surfaces (World Scientific, Singapore, 1989); D. J. Gross, T. Piran, and S.Weinberg, eds., Two-Dimensional Quantum Gravity and Random Surfaces (World Scientific, Singapore, 1992); F. David, P. Ginsparg, and J. Jinn-Justin, eds. Fluctuating Geometries in Statistical Mechanics and Field Theory (North-Holland, Amsterdam, 1996); J. Ambjorn, M. Carfora, and A. Marzuoli, The Geometry of Dynamical Triangulations (Springer, Berlin, 1997).

    Google Scholar 

  89. B. L. Hu, Cosmology as “condensed matter” physics, in Proceedings Third Asia Pacific Physics Conference, Y. W. Chan et al., eds. (World Scientific, Singapore, 1988), Vol. 1, p. 301; gr-qc/9511076.

    Google Scholar 

  90. L. Susskind, J. Math. Phys. 36, 6377 (1995).

    Google Scholar 

  91. G. 'tHooft, Quantization of point particles in 21 1 dimensional gravity and spacetime discreteness, gr-qc/9601014.

  92. T. Appelquist and J. Carazzone, Phys. Rev. D 11, 2856 (1975);S. Weinberg, Phys. Lett. 83B, 339 (1979); B. Ovrut and H. J. Schnitzer, Phys. Rev. D 21, 3369 (1980); 22, 2518 (1980); L. Hall, Nucl. Phys. B 178, 75 (1981); P. Lapage, In From Actions to Answers, T. DeGrand and D. Toussaint, eds. (World Scientific, Singapore, 1990), p. 483; S. Weinberg, The Quantum Theory of Fields, Vol. 1 (Cambridge University Press, Cambridge, 1995).

    Google Scholar 

  93. J. B. Hartle and B. L. Hu, Phys. Rev. D 20, 1772 (1979); 21, 2756 (1980).

    Google Scholar 

  94. J. Schwinger, Phys. Rev. 82, 664 (1951).

    Google Scholar 

  95. E. Mottola, Phys. Rev. D 33, 2136 (1986).

    Google Scholar 

  96. E. Calzetta and B.-L. Hu, Phys. Rev. D 37, 2878 (1988).

    Google Scholar 

  97. P. Danielewicz, Ann. Phys. (N.Y.) 152, 239 (1984);1984); S-P. Li and L. McLerran, Nucl. Phys. B 214, 417 (1983).

    Google Scholar 

  98. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962);D. F. Du Bois, In Lectures in Theoretical Physics, Vol 9c, W. Brittin, ed. (Gordon and Breach, New York, 1967); B. Bezzerides and D. F. Du Bois, Ann. Phys. (N.Y.) 70, 10 (1972).

    Google Scholar 

  99. St. Mrowczynski and P. Danielewicz, Nucl. Phys. B 342, 345 (1990);0); St. Mrowczynski and U. Heinz, Ann. Phys. 229, 1 (1994); P. Henning, Phys. Rep. 253, 235 (1995); Nucl. Phys. A 582, 633 (1995); D. Boyanovsky, I. Lawrie, and D.-S. Lee, Phys. Rev. D 54, 4013 (1996); E. Wang and U. Heinz, Phys. Rev. D 53, 899 (1996).

    Google Scholar 

  100. M. Kac and J. Logan, Fluctuations, in Fluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (Elsevier, New York, 1979), p. 1.

    Google Scholar 

  101. R. Jackiw, Phys. Rev. D 9, 1686 (1974); J. Iliopoulos, C. Itzykson, and A. Martin, Rev. Mod. Phys. 47, 165 (1975).

    Google Scholar 

  102. H. D. Dahmen and G. Jona-Lasinio, Nuovo Cimento 52A, 807 (1962);C. de Dominicis and P. Martin, J. Math. Phys. 5, 14 (1964); J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D 10, 2428 (1974); R. E. Norton and J. M. Cornwall, Ann. Phys. (N.Y.) 91, 106 (1975).

    Google Scholar 

  103. J. B. Hartle and G. Horowitz, Phys. Rev. D 24, 257 (1981).

    Google Scholar 

  104. Y. Kluger, E. Mottola, and J. M. Eisenberg, Phys. Rev. D 58, 125015 (1998).

    Google Scholar 

  105. F. A. Buot, Phys. Rep. 234, 73-174 (1993);S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995); P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic Press, New York, 1995).

    Google Scholar 

  106. S. Chandrasekhar and B. C. Xanthopoulos, Proc. Roy. Soc. A 408, 175 (1986);410, 311 (1987); A. G. Mirzbekian and G. Vilkovisky, gr-qc/9803006.

    Google Scholar 

  107. G. Stephens, E. Calzetta, B. L. Hu, and S. A. Ramsey, Phys. Rev. D 59, 045009 (1999).

    Google Scholar 

  108. V. A. Rubakov, Nucl. Phys. B 245, 481 (1991).

    Google Scholar 

  109. T. Vachaspati and A. Vilenkin, Phys. Rev. D 37, 898 (1988).

    Google Scholar 

  110. S.A. Ramsey and B. L. Hu, Phys. Rev. D 56, 678 (1997).

    Google Scholar 

  111. A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983).

    Google Scholar 

  112. A. Vilenkin, Phys. Rev. D 27, 2848 (1983);30, 509 (1984); Phys. Lett. 117B, 25 (1985); Nucl. Phys. B 226, 527 (1983).

    Google Scholar 

  113. S. W. Hawking, Comm. Math. Phys. 87, 395 (1982).

    Google Scholar 

  114. D. N. Page, Phys. Rev. D 34, 2267 (1986).

    Google Scholar 

  115. P. Ginzparg and M. J. Perry, Nucl. Phys. B 222, 245 (1983).

    Google Scholar 

  116. B. F. Whiting and J. W. York, Jr., Phys. Rev. Lett. 61, 1336 (1988).

    Google Scholar 

  117. R. K. P. Zia, Driven diffusive system, in Phase Transitions and Critical Phenomena, Vol. 20, C. Domb and J. Lebowitz, eds. (Academic Press, New York, 1995).

    Google Scholar 

  118. M. Kadar, G. Parisi, and Y. C. Zhang, Phys. Rev. Lett. 56, 342 (1986).

    Google Scholar 

  119. B. L. Hu and Y. Zhang, Mod. Phys. Lett. A 8, 3575 (1993); Int. J. Mod. Phys. 10, 4537 (1995).

    Google Scholar 

  120. A. Anderson and J. J. Halliwell, Phys. Rev D 48, 2753 (1993); A. Anastopoulos and J. J. Halliwell, Phys. Rev. D. 51, 6870 (1995).

    Google Scholar 

  121. B. L. Hu and K. Shiokawa, Phys. Rev. D 57, 3474 (1998).

    Google Scholar 

  122. S. W. Hawking, Nucl. Phys. B 144, 349 (1978); S. W. Hawking, D. N. Page, and C. N. Pope, Nucl. Phys. B 170[FS1], 283 (1980); N. P. Warner, Comm. Math. Phys. 86, 419 (1982).

    Google Scholar 

  123. S. W. Hawking, Phys. Rev. D 37, 904 (1988); S. Coleman, Nucl. Phys. B 307, 867 (1988); S. Coleman, J. B. Hartle, T. Piran, and S. Weinberg, eds., Quantum Cosmology and Baby Universes (World Scientific, Singapore, 1991)

    Google Scholar 

  124. S. W. Hawking, Phys. Rev. D 53, 3099 (1996).

    Google Scholar 

  125. D. Garfunkle, S. B. Giddings, and A. Stominger, Phys. Rev. D 49, 958 (1994);F. Dowker et al., Phys. Rev. D 50, 2662 (1994); S. W. Hawking, G. T. Horowitz, and S. F. Ross, Phys. Rev. D 51, 4302 (1995); R. Busso and S. W. Hawking, Phys. Rev. 52, 5659 (1995); 54, 6312 (1996).

    Google Scholar 

  126. S. Carlip, Phys. Rev. Lett. 79, 4071 (1997); Class. Quant. Grav. 15, 2629 (1998); L. J. Garay, Phys. Rev. Lett. 80, 2508 (1998); Phys. Rev. D 58, 124015 (1998).

    Google Scholar 

  127. J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Quantum decoherence in a D-foam background, Mod. Phys. Lett A. (1997).

  128. K. Huang, Statistical Mechanics, 2nd ed. (Wiley, New York, 1987).

    Google Scholar 

  129. A. I. Akhiezer and S. V. Peletminsky, Methods of Statistical Physics (Pergamon, London, 1981).

    Google Scholar 

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Hu, B.L. Stochastic Gravity. International Journal of Theoretical Physics 38, 2987–3037 (1999). https://doi.org/10.1023/A:1026664317157

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