Quantum field theory in curved spacetime

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In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory treats spacetime as a fixed, classical background, while giving a quantum-mechanical description of the matter and energy propagating through that spacetime. A general prediction of this theory is that particles can be created by time-dependent gravitational fields (multigraviton pair production), or by time-independent gravitational fields that contain horizons. The most famous example of the latter is the phenomenon of Hawking radiation emitted by black holes.


Ordinary quantum field theories, which form the basis of standard model, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.

For non-zero cosmological constants, on curved spacetimes quantum fields lose their interpretation as asymptotic particles.[1] Only in certain situations, such as in asymptotically flat spacetimes (zero cosmological curvature), can the notion of incoming and outgoing particle be recovered, thus enabling one to define an S-matrix. Even then, as in flat spacetime, the asymptotic particle interpretation depends on the observer (i.e., different observers may measure different numbers of asymptotic particles on a given spacetime).

Another observation is that unless the background metric tensor has a global timelike Killing vector, there is no way to define a vacuum or ground state canonically. The concept of a vacuum is not invariant under diffeomorphisms. This is because a mode decomposition of a field into positive and negative frequency modes is not invariant under diffeomorphisms. If t′(t) is a diffeomorphism, in general, the Fourier transform of exp[ikt′(t)] will contain negative frequencies even if k > 0. Creation operators correspond to positive frequencies, while annihilation operators correspond to negative frequencies. This is why a state which looks like a vacuum to one observer cannot look like a vacuum state to another observer; it could even appear as a heat bath under suitable hypotheses.

Since the end of the 1980s, the local quantum field theory approach due to Rudolf Haag and Daniel Kastler has been implemented in order to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is suitable to generalize the renormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in the presence of a black hole have been obtained. In particular the algebraic approach allows one to deal with the problems mentioned above arising from the absence of a preferred reference vacuum state, the absence of a natural notion of particle and the appearance of unitarily inequivalent representations of the algebra of observables.[2][3]


Using perturbation theory in quantum field theory in curved spacetime geometry is known as the semiclassical approach to quantum gravity. This approach studies the interaction of quantum fields in a fixed classical spacetime and among other thing predicts the creation of particles by time-varying spacetimes[4] and Hawking radiation.[5] The latter can be understood as a manifestation of the Unruh effect where an accelerating observer observes black body radiation.[6] Other prediction of quantum fields in curved spaces include,[7] for example, the radiation emitted by a particle moving along a geodesic[8][9][10][11] and the interaction of Hawking radiation with particles outside black holes.[12][13][14][15]

This formalism is also used to predict the primordial density perturbation spectrum arising in different models of cosmic inflation. These predictions are calculated using the Bunch–Davies vacuum or modifications thereto.[16]

Approximation to quantum gravity[edit]

The theory of quantum field theory in curved spacetime may be considered as an intermediate step towards quantum gravity.[17] QFT in curved spacetime is expected to be a viable approximation to the theory of quantum gravity when spacetime curvature is not significant on the Planck scale.[18][19][20] However, the fact that the true theory of quantum gravity remains unknown means that the precise criteria for when QFT on curved spacetime is a good approximation are also unknown.[1]: 1 

Gravity is not renormalizable in QFT, so merely formulating QFT in curved spacetime is not a true theory of quantum gravity.

See also[edit]


  1. ^ a b Wald, R. M. (1995). Quantum field theory in curved space-time and black hole thermodynamics. Chicago U. ISBN 0-226-87025-1.
  2. ^ Fewster, C. J. (2008). "Lectures on quantum field theory in curved spacetime (Lecture Note 39/2008 Max Planck Institute for Mathematics in the Natural Sciences (2008))" (PDF).
  3. ^ Khavkine, Igor; Moretti, Valter (2015), Brunetti, Romeo; Dappiaggi, Claudio; Fredenhagen, Klaus; Yngvason, Jakob (eds.), "Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction", Advances in Algebraic Quantum Field Theory, Cham: Springer International Publishing, pp. 191–251, arXiv:1412.5945, Bibcode:2014arXiv1412.5945K, doi:10.1007/978-3-319-21353-8_5, ISBN 978-3-319-21352-1, S2CID 119179440, retrieved 2022-01-14
  4. ^ Parker, L. (1968-08-19). "Particle Creation in Expanding Universes". Physical Review Letters. 21 (8): 562–564. Bibcode:1968PhRvL..21..562P. doi:10.1103/PhysRevLett.21.562.
  5. ^ Hawking, S. W. (1993-05-01), "Particle Creation by Black Holes", Euclidean Quantum Gravity, World Scientific, pp. 167–188, doi:10.1142/9789814539395_0011, ISBN 978-981-02-0515-7, retrieved 2021-08-15
  6. ^ Crispino, Luís C. B.; Higuchi, Atsushi; Matsas, George E. A. (2008-07-01). "The Unruh effect and its applications". Reviews of Modern Physics. 80 (3): 787–838. arXiv:0710.5373. Bibcode:2008RvMP...80..787C. doi:10.1103/RevModPhys.80.787. hdl:11449/24446. S2CID 119223632.
  7. ^ Birrell, N. D. (1982). Quantum fields in curved space. P. C. W. Davies. Cambridge [Cambridgeshire]: Cambridge University Press. ISBN 0-521-23385-2. OCLC 7462032.
  8. ^ Crispino, L. C. B.; Higuchi, A.; Matsas, G. E. A. (November 1999). "Scalar radiation emitted from a source rotating around a black hole". Classical and Quantum Gravity. 17 (1): 19–32. arXiv:gr-qc/9901006. doi:10.1088/0264-9381/17/1/303. ISSN 0264-9381. S2CID 14018854.
  9. ^ Crispino, L. C. B.; Higuchi, A.; Matsas, G. E. A. (September 2016). "Corrigendum: Scalar radiation emitted from a source rotating around a black hole (2000 Class. Quantum Grav. 17 19)". Classical and Quantum Gravity. 33 (20): 209502. doi:10.1088/0264-9381/33/20/209502. hdl:11449/162073. ISSN 0264-9381. S2CID 126192949.
  10. ^ Oliveira, Leandro A.; Crispino, Luís C. B.; Higuchi, Atsushi (2018-02-16). "Scalar radiation from a radially infalling source into a Schwarzschild black hole in the framework of quantum field theory". The European Physical Journal C. 78 (2): 133. Bibcode:2018EPJC...78..133O. doi:10.1140/epjc/s10052-018-5604-8. ISSN 1434-6052. S2CID 55070002.
  11. ^ Brito, João P. B.; Bernar, Rafael P.; Crispino, Luís C. B. (2020-06-11). "Synchrotron geodesic radiation in Schwarzschild--de Sitter spacetime". Physical Review D. 101 (12): 124019. arXiv:2006.08887. Bibcode:2020PhRvD.101l4019B. doi:10.1103/PhysRevD.101.124019. S2CID 219708236.
  12. ^ Higuchi, Atsushi; Matsas, George E. A.; Sudarsky, Daniel (1998-10-22). "Interaction of Hawking radiation with static sources outside a Schwarzschild black hole". Physical Review D. 58 (10): 104021. arXiv:gr-qc/9806093. Bibcode:1998PhRvD..58j4021H. doi:10.1103/PhysRevD.58.104021. hdl:11449/65552. S2CID 14575175.
  13. ^ Crispino, Luís C. B.; Higuchi, Atsushi; Matsas, George E. A. (1998-09-22). "Interaction of Hawking radiation and a static electric charge". Physical Review D. 58 (8): 084027. arXiv:gr-qc/9804066. Bibcode:1998PhRvD..58h4027C. doi:10.1103/PhysRevD.58.084027. hdl:11449/65534. S2CID 15522105.
  14. ^ Castiñeiras, J.; Costa e Silva, I. P.; Matsas, G. E. A. (2003-03-27). "Do static sources respond to massive scalar particles from the Hawking radiation as uniformly accelerated ones do in the inertial vacuum?". Physical Review D. 67 (6): 067502. arXiv:gr-qc/0211053. Bibcode:2003PhRvD..67f7502C. doi:10.1103/PhysRevD.67.067502. hdl:11449/23239. S2CID 33007353.
  15. ^ Castiñeiras, J.; Costa e Silva, I. P.; Matsas, G. E. A. (2003-10-31). "Interaction of Hawking radiation with static sources in de Sitter and Schwarzschild--de Sitter spacetimes". Physical Review D. 68 (8): 084022. arXiv:gr-qc/0308015. Bibcode:2003PhRvD..68h4022C. doi:10.1103/PhysRevD.68.084022. hdl:11449/23527. S2CID 41250020.
  16. ^ Greene, Brian R.; Parikh, Maulik K.; van der Schaar, Jan Pieter (28 April 2006). "Universal correction to the inflationary vacuum". Journal of High Energy Physics. 2006 (4): 057. arXiv:hep-th/0512243. Bibcode:2006JHEP...04..057G. doi:10.1088/1126-6708/2006/04/057. S2CID 16290999.
  17. ^ Brunetti, Romeo; Fredenhagen, Klaus; Rejzner, Katarzyna (2016). "Quantum Gravity from the Point of View of Locally Covariant Quantum Field Theory". Communications in Mathematical Physics. 345 (3): 741–779. Bibcode:2016CMaPh.345..741B. doi:10.1007/s00220-016-2676-x. S2CID 55608399. Quantum field theory on curved spacetime, which might be considered as an intermediate step towards quantum gravity, already has no distinguished particle interpretation.
  18. ^ Bär, Christian; Fredenhagen, Klaus (2009). "Preface". Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations. Springer. ISBN 9783642027802. In particular, due to the weakness of gravitational forces, the back reaction of the spacetime metric to the energy momentum tensor of the quantum fields may be neglected, in a first approximation, and one is left with the problem of quantum field theory on Lorentzian manifolds. Surprisingly, this seemingly modest approach leads to far-reaching conceptual and mathematical problems and to spectacular predictions, the most famous one being the Hawking radiation of black holes.
  19. ^ Kay, Bernard S. (2006). "Quantum field theory in curved spacetime". Encyclopedia of Mathematical Physics. Academic Press (Elsevier). pp. 202–214. arXiv:gr-qc/0601008. One expects it to be a good approximation to full quantum gravity provided the typical frequencies of the gravitational background are very much less than the Planck frequency [...] and provided, with a suitable measure for energy, the energy of created particles is very much less than the energy of the background gravitational field or of its matter sources.
  20. ^ Yang, Run-Qiu; Liu, Hui; Zhu, Shining; Luo, Le; Cai, Rong-Gen (2020). "Simulating quantum field theory in curved spacetime with quantum many-body systems". Physical Review Research. 2 (2): 023107. arXiv:1906.01927. Bibcode:2020PhRvR...2b3107Y. doi:10.1103/PhysRevResearch.2.023107. S2CID 218502756. Quantum field theory in curved spacetime is a semiclassical approximation to quantum gravity theory, where the curved background spacetime is treated classically, while the matter fields in the curved spacetime are quantized.

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