Quantum Fields in Curved Spacetime

doi:10.1017/CBO9781139583961.015

10 - Quantum Fields in Curved Spacetime

from Part Four - Beyond Einstein

Published online by Cambridge University Press: 05 June 2015

Stefan Hollands and

Edited by

James Isenberg and

Stefan Hollands: Affiliation:
Universität Leipzig
Robert M. Wald: Affiliation:
University of Chicago
Abhay Ashtekar: Affiliation:
Pennsylvania State University
Beverly K. Berger: Affiliation:
Formerly Program Director for Gravitational Physics, National Science Foundation
James Isenberg: Affiliation:
University of Oregon
Malcolm MacCallum: Affiliation:
University of Bristol

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Summary

The nature of quantum field theory in curved spacetime

Quantum field theory in curved spacetime (QFTCS) is the theory of quantum fields propagating in a background, classical, curved spacetime (M, g). On account of its classical treatment of the metric, QFTCS cannot be a fundamental theory of nature. However, QFTCS is expected to provide an accurate description of quantum phenomena in a regime where the effects of curved spacetime may be significant, but effects of quantum gravity itself may be neglected. In particular, it is expected that QFTCS should be applicable to the description of quantum phenomena occurring in the early universe and near (and inside of) black holes – provided that one does not attempt to describe phenomena occurring so near to singularities that curvatures reach Planckian scales and the quantum nature of the spacetime metric would have to be taken into account.

It should be possible to derive QFTCS by taking a suitable limit of a more fundamental theory wherein the spacetime metric is treated in accord with the principles of quantum theory. However, this has not been done – except in formal and/or heuristic ways – simply because no present quantum theory of gravity has been developed to the point where such a well defined limit can be taken in general situations. Rather, the framework of QFTCS that we shall describe in this review has been obtained by suitably merging basic principles of classical general relativity with the basic principles of quantum field theory in Minkowski spacetime. As we shall explain further below, the basic principles of classical general relativity are relatively easy to identify and adhere to, but it is far less clear what to identify as the “basic principles” of quantum field theory in Minkowski spacetime. Indeed, many of the concepts normally viewed as fundamental to quantum field theory in Minkowski spacetime, such as Poincaré invariance, do not even make sense in the context of curved spacetime, and therefore cannot be considered as “fundamental” from the viewpoint of QFTCS.

Type: Chapter
Information: General Relativity and Gravitation
A Centennial Perspective
, pp. 513 - 552

DOI: https://doi.org/10.1017/CBO9781139583961.015 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2015

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References

[1] R. F., Streater and A. S., Wightman, PCT, spin and statistics, and all that, Princeton, NJ, Princeton University Press (2000).Google Scholar

[2] R., Haag, Local quantum physics: fields, particles, algebras, Berlin, Springer (1992).Google Scholar

[3] J., Glimm and A., Jaffe: Quantum physics: afunctional integral point of view, Berlin, Springer (1987).Google Scholar

[4] S., Hollands and R.M., Wald, Commun. Math. Phys. 293, 85 (2010).

[5] C., Kopper and V. F., Muller, Commun. Math. Phys. 275, 331 (2007).

[6] S., Hollands and R. M., Wald, arXiv:1401. 2026 [gr-qc].

[7] S., Hollands, Rev. Math. Phys. 20, 1033 (2008).

[8] Antonio N., Bernal and Miguel, Sànchez, Commun. Math. Phys. 243, 461 (2003).

[9] C., BärN., Ginoux and F., Pfäffle, Wave equations on Lorentzian manifolds and quantization, ZÜrich, European, Mathematical Society (2007).Google Scholar

[10] J., Dimock, Commun. Math. Phys. 77, 219 (1980).

[11] R. M., Wald, Quantum field theory in curved spacetime and black hole thermodynamics, Chicago, IL, University of Chicago Press (1994).Google Scholar

[12] M., Reed and B., Simon, Fourier analysis and self-adjointness, New York, Academic Press (1975).Google Scholar

[13] N. I., Akhizer, The classical moment problem and some related questions in analysis, Oliver and Boyd (1965).Google Scholar

[14] M. J., Radzikowski, Commun. Math. Phys. 179, 529 (1996).

[15] M. J., Radzikowski, Commun. Math. Phys. 180, 1 (1996).

[16] R., BrunettiK., Fredenhagen and M., Kohler, Commun. Math. Phys. 180, 633 (1996).

[17] K., Sanders, Commun. Math. Phys. 295, 485 (2010).

[18] B. S., Kay and R. M., Wald, Phys. Rept. 207, 49 (1991).

[19] B. S., DeWitt and R. W., Brehme, Annals of Physics 9, 220 (1960).

[20] S. A., FullingF. J., Narcowich and R. M., Wald, Annals of Physics 136, 243 (1981).

[21] W., Junker and E., Schrohe, Ann. Inst. Henri Poincare Phys. Theor. 3, 1113 (2002).

[22] C., Gerard and M., Wrochna, arXiv:1209.2604 [math-ph].

[23] H., Araki and S., Yamagami, Publ. RIMS, Kyoto U. 18 (1982).

[24] R., Verch, Commun. Math. Phys. 160, 507 (1994).

[25] O., Bratteli and D. W., Robinson, Operator algebras and quantum statistical mechanics, Berlin, Springer (2002).Google Scholar

[26] H., Sahlmann and R., Verch, Commun. Math. Phys. 214, 705 (2000).

[27] W. G., Unruh and R. M., Wald, Phys. Rev. D 29, 1047 (1984).

[28] S., De Bievre and M., Merkli, Class. Quant. Grav. 23, 6525 (2006).

[29] B., Allen, Phys. Rev. D 32, 3136 (1985).

[30] J., Bros and U., Moschella, Rev. Math. Phys. 8, 327 (1996).

[31] J., Bros, U., Moschella and J.P., Gazeau, Phys. Rev. Lett. 73, 1746 (1994).

[32] S., Hollands, Ann. Inst. Henri Poincaré 13, 1039 (2012).

[33] S., Hollands, Commun. Math. Phys. 319, 1 (2013).

[34] D., Marolf and I. A., Morrison, Phys. Rev. D 84, 044040 (2011).

[35] A., HiguchiD., Marolf and I. A., Morrison, Phys.Rev. D 83, 084029 (2011).

[36] K., Sanders, arXiv: 1310. 5537 [gr-qc].

[37] D. G., Boulware, Phys. Rev. D 11, 1404 (1975).

[38] W. G., Unruh, Phys.Rev. D 14, 870 (1976).

[39] C., DappiaggiV., Moretti and N., Pinamonti, Adv. Theor. Math. Phys. 15, 355 (2011).

[40] K., Fredenhagen and R., Haag, Commun. Math. Phys. 127, 273 (1990).

[41] K., Fredenhagen and R., Haag, Commun. Math. Phys. 108, 91 (1987).

[42] M., Dafermos and I., Rodnianski, Commun. Pure Appl. Math. 62, 859 (2009).

[43] S. W., Hawking, Commun. Math. Phys. 43, 199 (1975) [Erratum ibid. 46, 206 (1976)].

[44] R. M., Wald, Commun. Math. Phys. 45, 9 (1975).

[45] J. M., BardeenB., Carter and S. W., Hawking, Commun. Math. Phys. 31, 161 (1973).

[46] R. M., WaldLiving Rev. Rel. 4, 6 (2001).

[47] C. J., Fewster, Class. Quant. Grav. 17, 1897 (2000).

[48] L. H., Ford and T. A., Roman, Phys.Rev. D 55, 2082 (1997).

[49] S., Hollands and R. M., Wald, Commun. Math. Phys. 237, 123 (2003).

[50] M., Duetsch and K., Fredenhagen, Commun. Math. Phys. 219, 5 (2001).

[51] H., Kodama and M., Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984).

[52] V. F., MukhanovH. A., Feldman and R. H., Brandenberger, Phys. Rept. 215, 203 (1992).

[53] C., Schomblond and P., Spindel, Ann. Inst. Poincare Phys. Theor. 25, 67 (1976).

[54] S., Hollands and R. M., Wald, Commun. Math. Phys. 223, 289 (2001).

[55] R., BrunettiK., Fredenhagen and R., Verch, Commun. Math. Phys. 237, 31 (2003).

[56] H., Epstein and V., Glaser, Ann. Inst. Poincare Phys. Theor. A 19, 211 (1973).

[57] N. N., Bogoliubov and D. V., Shirkov, Introduction to the theory of quantized fields, New York, Wiley (1980).Google Scholar

[58] S., Hollands and R. M., Wald, Commun. Math. Phys. 231, 309 (2002).

[59] S., Hollands and R. M., Wald, Rev. Math. Phys. 17, 227 (2005).

[60] R., Brunetti and K., Fredenhagen, Commun. Math. Phys. 208, 623 (2000).

[61] M., Duetsch and K., Fredenhagen, Rev. Math. Phys. 16, 1291 (2004).

[62] L., Hormander, The analysis of linear partial differential operators I: distribution theory and Fourier analysis, Berlin, Springer, 2nd edition (1990).Google Scholar

[63] S., Hollands, Commun. Math. Phys. 273, 1 (2007).