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{{Short description|Extension of quantum field theory to curved spacetime}}
{{Use American English|date = February 2019}}
{{quantum field theory}}

In [[theoretical physics]], '''quantum field theory in curved spacetime (QFTCS)'''<ref>{{cite book
 | first = B. S. |last = Kay 
 | year = 2025
 | title = Encyclopedia of Mathematical Physics
 |chapter = Quantum Field Theory in Curved Spacetime 
 |pages = 357–381 
 |doi = 10.1016/B978-0-323-95703-8.00085-9 
 | arxiv = 2308.14517 
 |isbn = 978-0-323-95706-9 
 }}</ref> is an extension of [[quantum field theory]] from [[Minkowski space|Minkowski spacetime]] to a general [[curved spacetime]]. This theory uses a semi-classical approach; it 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 field]]s (multi[[graviton]] [[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 hole]]s.

==Overview==
Ordinary [[quantum field theory|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 constant]]s, on curved spacetimes quantum fields lose their interpretation as asymptotic [[Elementary particle|particle]]s.<ref name="wald">{{cite book |first=R. M. |last=Wald |author-link=Robert Wald |title=Quantum field theory in curved space-time and black hole thermodynamics |publisher=Chicago U. |year=1995 |isbn=0-226-87025-1 }}</ref> Only in certain situations, such as in asymptotically flat spacetimes (zero [[Physical cosmology|cosmological]] [[Curvature#Curvature of space|curvature]]), can the notion of incoming and outgoing particle be recovered, thus enabling one to define an [[S-matrix|''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 [[diffeomorphism]]s. This is because a mode decomposition of a field into positive and negative frequency modes is not invariant under diffeomorphisms. If ''{{prime|t}}''(''t'') is a diffeomorphism, in general, the [[Fourier transform]] of exp[''{{prime|ikt}}''(''t'')] will contain negative frequencies even if ''k'' > 0. [[Creation and annihilation operators|Creation operators]] correspond to positive frequencies, while [[Creation and annihilation operators|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.<ref>{{cite web
 | first = C. J. |last = Fewster 
 | year = 2008
 | title = Lectures on quantum field theory in curved spacetime (Lecture Note 39/2008 Max Planck Institute for Mathematics in the Natural Sciences (2008))
 | url = https://www.mis.mpg.de/preprints/ln/lecturenote-3908.pdf
}}</ref><ref>{{Citation|last1=Khavkine|first1=Igor|title=Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction|date=2015|url=https://link.springer.com/10.1007/978-3-319-21353-8_5|work=Advances in Algebraic Quantum Field Theory|pages=191–251|editor-last=Brunetti|editor-first=Romeo|place=Cham|publisher=Springer International Publishing|arxiv=1412.5945|bibcode=2014arXiv1412.5945K|doi=10.1007/978-3-319-21353-8_5|isbn=978-3-319-21352-1|access-date=2022-01-14|last2=Moretti|first2=Valter|series=Mathematical Physics Studies |s2cid=119179440 |editor2-last=Dappiaggi|editor2-first=Claudio|editor3-last=Fredenhagen|editor3-first=Klaus|editor4-last=Yngvason|editor4-first=Jakob}}</ref>

==Applications==
Using [[Perturbation theory (quantum mechanics)|perturbation theory]] in quantum field theory in curved spacetime geometry is known as the [[Semiclassical physics|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<ref>{{Cite journal|last=Parker|first=L.|date=1968-08-19|title=Particle Creation in Expanding Universes|url=https://link.aps.org/doi/10.1103/PhysRevLett.21.562|journal=Physical Review Letters|volume=21|issue=8|pages=562–564|doi=10.1103/PhysRevLett.21.562|bibcode=1968PhRvL..21..562P |url-access=subscription}}</ref> and [[Hawking radiation]].<ref>{{Citation|last=Hawking|first=S. W.|title=Particle Creation by Black Holes|date=1993-05-01|url=https://www.worldscientific.com/doi/abs/10.1142/9789814539395_0011|work=Euclidean Quantum Gravity|pages=167–188|publisher=World Scientific|doi=10.1142/9789814539395_0011|isbn=978-981-02-0515-7|access-date=2021-08-15|author-link=Stephen Hawking|url-access=subscription}}</ref> The latter can be understood as a manifestation of the [[Unruh effect]] where an accelerating observer observes black body radiation.<ref>{{Cite journal|last1=Crispino|first1=Luís C. B.|last2=Higuchi|first2=Atsushi|last3=Matsas|first3=George E. A.|date=2008-07-01|title=The Unruh effect and its applications|url=https://link.aps.org/doi/10.1103/RevModPhys.80.787|journal=Reviews of Modern Physics|volume=80|issue=3|pages=787–838|doi=10.1103/RevModPhys.80.787|arxiv=0710.5373 |bibcode=2008RvMP...80..787C |hdl=11449/24446|s2cid=119223632 |hdl-access=free}}</ref> Other prediction of quantum fields in curved spaces include,<ref>{{Cite book|last=Birrell|first=N. D.|title=Quantum fields in curved space|date=1982|publisher=Cambridge University Press|others=P. C. W. Davies|isbn=0-521-23385-2|location=Cambridge [Cambridgeshire]|oclc=7462032}}</ref> for example, the radiation emitted by a particle moving along a geodesic<ref>{{Cite journal|last1=Crispino|first1=L. C. B.|last2=Higuchi|first2=A.|last3=Matsas|first3=G. E. A.|date=November 1999|title=Scalar radiation emitted from a source rotating around a black hole|journal=Classical and Quantum Gravity|language=en|volume=17|issue=1|pages=19–32|doi=10.1088/0264-9381/17/1/303|issn=0264-9381|arxiv=gr-qc/9901006|s2cid=14018854 }}</ref><ref>{{Cite journal|last1=Crispino|first1=L. C. B.|last2=Higuchi|first2=A.|last3=Matsas|first3=G. E. A.|date=September 2016|title=Corrigendum: Scalar radiation emitted from a source rotating around a black hole (2000 Class. Quantum Grav. 17 19)|journal=Classical and Quantum Gravity|language=en|volume=33|issue=20|article-number=209502|doi=10.1088/0264-9381/33/20/209502|issn=0264-9381|hdl=11449/162073|s2cid=126192949 |hdl-access=free}}</ref><ref>{{Cite journal|last1=Oliveira|first1=Leandro A.|last2=Crispino|first2=Luís C. B.|last3=Higuchi|first3=Atsushi|date=2018-02-16|title=Scalar radiation from a radially infalling source into a Schwarzschild black hole in the framework of quantum field theory|journal=The European Physical Journal C|language=en|volume=78|issue=2|page=133|doi=10.1140/epjc/s10052-018-5604-8|bibcode=2018EPJC...78..133O |s2cid=55070002 |issn=1434-6052|doi-access=free}}</ref><ref>{{Cite journal|last1=Brito|first1=João P. B.|last2=Bernar|first2=Rafael P.|last3=Crispino|first3=Luís C. B.|date=2020-06-11|title=Synchrotron geodesic radiation in Schwarzschild--de Sitter spacetime|url=https://link.aps.org/doi/10.1103/PhysRevD.101.124019|journal=Physical Review D|volume=101|issue=12|article-number=124019|doi=10.1103/PhysRevD.101.124019|arxiv=2006.08887|bibcode=2020PhRvD.101l4019B |s2cid=219708236 }}</ref> and the interaction of [[Hawking radiation]] with particles outside black holes.<ref>{{Cite journal|last1=Higuchi|first1=Atsushi|last2=Matsas|first2=George E. A.|last3=Sudarsky|first3=Daniel|date=1998-10-22|title=Interaction of Hawking radiation with static sources outside a Schwarzschild black hole|url=https://link.aps.org/doi/10.1103/PhysRevD.58.104021|journal=Physical Review D|volume=58|issue=10|article-number=104021|doi=10.1103/PhysRevD.58.104021|arxiv=gr-qc/9806093|bibcode=1998PhRvD..58j4021H |hdl=11449/65552|s2cid=14575175 |hdl-access=free}}</ref><ref>{{Cite journal|last1=Crispino|first1=Luís C. B.|last2=Higuchi|first2=Atsushi|last3=Matsas|first3=George E. A.|date=1998-09-22|title=Interaction of Hawking radiation and a static electric charge|url=https://link.aps.org/doi/10.1103/PhysRevD.58.084027|journal=Physical Review D|volume=58|issue=8|article-number=084027|doi=10.1103/PhysRevD.58.084027|arxiv=gr-qc/9804066 |bibcode=1998PhRvD..58h4027C |hdl=11449/65534|s2cid=15522105 |hdl-access=free}}</ref><ref>{{Cite journal|last1=Castiñeiras|first1=J.|last2=Costa e Silva|first2=I. P.|last3=Matsas|first3=G. E. A.|date=2003-03-27|title=Do static sources respond to massive scalar particles from the Hawking radiation as uniformly accelerated ones do in the inertial vacuum?|url=https://link.aps.org/doi/10.1103/PhysRevD.67.067502|journal=Physical Review D|volume=67|issue=6|article-number=067502|doi=10.1103/PhysRevD.67.067502 | arxiv=gr-qc/0211053 |bibcode=2003PhRvD..67f7502C |hdl=11449/23239|s2cid=33007353 |hdl-access=free}}</ref><ref>{{Cite journal|last1=Castiñeiras|first1=J.|last2=Costa e Silva|first2=I. P.|last3=Matsas|first3=G. E. A.|date=2003-10-31|title=Interaction of Hawking radiation with static sources in de Sitter and Schwarzschild--de Sitter spacetimes|url=https://link.aps.org/doi/10.1103/PhysRevD.68.084022|journal=Physical Review D|volume=68|issue=8|article-number=084022|doi=10.1103/PhysRevD.68.084022 | arxiv=gr-qc/0308015 |bibcode=2003PhRvD..68h4022C |hdl=11449/23527|s2cid=41250020 |hdl-access=free}}</ref>

This formalism is also used to predict the primordial density [[perturbation theory|perturbation]] spectrum arising in different models of [[cosmic inflation]]. These predictions are calculated using the [[Bunch&ndash;Davies vacuum]] or modifications thereto.<ref name="green(2006)">{{cite journal|last1=Greene|first1=Brian R.|author-link=Brian Greene|last2=Parikh|first2=Maulik K.|last3=van der Schaar|first3=Jan Pieter|date=28 April 2006|title=Universal correction to the inflationary vacuum|journal=Journal of High Energy Physics|volume=2006|issue=4|page=057|arxiv=hep-th/0512243|bibcode=2006JHEP...04..057G|doi=10.1088/1126-6708/2006/04/057|s2cid=16290999}}</ref>

==Approximation to quantum gravity==

The theory of quantum field theory in curved spacetime may be considered as an intermediate step towards [[quantum gravity]].<ref>{{cite journal|last1=Brunetti|first1=Romeo|last2=Fredenhagen|first2=Klaus|author-link2=Klaus Fredenhagen|last3=Rejzner|first3=Katarzyna|year=2016|title=Quantum Gravity from the Point of View of Locally Covariant Quantum Field Theory|journal=Communications in Mathematical Physics|volume=345|issue=3 |pages=741–779|doi=10.1007/s00220-016-2676-x|bibcode=2016CMaPh.345..741B |s2cid=55608399 |quote=Quantum field theory on curved spacetime, which might be considered as an intermediate step towards quantum gravity, already has no distinguished particle interpretation.|doi-access=free|arxiv=1306.1058}}</ref> 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.<ref>{{cite book|last1=Bär|first1=Christian|title=Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations|last2=Fredenhagen|first2=Klaus|publisher=Springer|year=2009|isbn=978-3-642-02780-2|chapter=Preface|quote=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.|author-link2=Klaus Fredenhagen}}</ref><ref>{{cite book|first=Bernard S. |last=Kay |chapter=Quantum field theory in curved spacetime |title=Encyclopedia of Mathematical Physics |publisher=Academic Press (Elsevier) |year=2006 |pages=202–214 |arxiv=gr-qc/0601008 |quote=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.}}</ref><ref>{{cite journal|doi=10.1103/PhysRevResearch.2.023107 |year=2020 |title=Simulating quantum field theory in curved spacetime with quantum many-body systems |first1=Run-Qiu |last1=Yang |first2=Hui |last2=Liu |first3=Shining |last3=Zhu |first4=Le |last4=Luo |first5=Rong-Gen |last5=Cai |journal=Physical Review Research |volume=2 |issue=2 |article-number=023107 |arxiv=1906.01927 |bibcode=2020PhRvR...2b3107Y |s2cid=218502756 |quote=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.|doi-access=free }}</ref> 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.<ref name="wald"/>{{rp|1}}

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

==See also==
{{Div col|colwidth=20em}}
* [[General relativity]]
* [[History of quantum field theory]]
* [[Local quantum field theory]]
* [[Statistical field theory]]
* [[Topological quantum field theory]]
* [[Quantum geometry]]
* [[Quantum spacetime]]
{{Div col end}}

==References==
{{reflist}}

== Further reading ==
*{{cite book |first1=N. D. |last1=Birrell |first2=P. C. W. |last2=Davies |title=Quantum fields in curved space |publisher=CUP |year=1982 |isbn=0-521-23385-2 }}
*{{cite book |first=S. A. |last=Fulling |title=Aspects of quantum field theory in curved space-time |publisher=CUP |year=1989 |isbn=0-521-34400-X }}
*{{cite book |first1=V. |last1=Mukhanov |first2=S. |last2=Winitzki |title=Introduction to Quantum Effects in Gravity |publisher=CUP |year=2007 |isbn=978-0-521-86834-1 }}
*{{cite book |first1=L. |last1=Parker |author-link=Leonard Parker |first2=D. |last2=Toms |title=Quantum Field Theory in Curved Spacetime |year=2009 |publisher=Cambridge University Press |isbn=978-0-521-87787-9 }}

==External links==
* [https://www.quantumfieldtheory.info/Wh_Ch_Scalar_Fields_in_GR.pdf  Summary Chart of Intro Steps to Quantum Fields in Curved Spacetime]  A two-page chart outline of the basic principles governing the behavior of quantum fields in general relativity.

{{quantum gravity}}
{{quantum field theories|state=expanded}}

[[Category:Quantum field theory]]
[[Category:Quantum gravity]]