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\section*{AQFT on curved spacetimes}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{aqft}{}\paragraph*{{AQFT}}\label{aqft}

[[!include AQFT and operator algebra contents]]

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{vacuum_energy_and_cosmological_constant}{Vacuum energy and Cosmological constant}\dotfill \pageref*{vacuum_energy_and_cosmological_constant} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Where the [[Haag-Kastler axioms]] formulate [[quantum field theory]] on [[Minkowski spacetime]], known as \emph{[[algebraic quantum field theory]]} (AQFT) there is a generalization of these axioms to [[curved spacetimes]] (\hyperlink{BrunettiFredenhagen01}{Brunetti-Fredenhagen 01}), also known as \emph{locally covariant algebraic quantum field theory}. For the case of [[perturbative quantum field theory]] this is \emph{[[locally covariant perturbative quantum field theory]]}, see there for more.

(This falls short of being a theory of [[quantum gravity]], instead it describes [[quantum field theory]] on classical [[background field]] configurations of [[gravity]].)

This is the mathematically rigorous framework for studying subjects such as the [[cosmological constant]] (see \href{cosmological+constant#InPerturbativeQuantumGravity}{there}), [[Hawking raditation]] or the [[cosmic microwave background]] (\hyperlink{FredenhagenHack13}{Fredenhagen-Hack 13}).

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{vacuum_energy_and_cosmological_constant}{}\subsubsection*{{Vacuum energy and Cosmological constant}}\label{vacuum_energy_and_cosmological_constant}

The [[renormalization]] freedom in [[perturbative QFT|perturbative]] [[quantization]] of [[gravity]] ([[perturbative quantum gravity]]) induces freedom in the choice of [[vacuum expectation value]] of the [[stress-energy tensor]] and hence in the [[cosmological constant]].

Review includes (\hyperlink{Hack15}{Hack 15, section 3.2.1}).

For more see at \emph{[[cosmological constant]]} \href{cosmological+constant#InPerturbativeQuantumGravity}{here}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[general covariance]]


\item [[causal structure]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

To some extent the problem of AQFT on curved spacetime was formulated in

\begin{itemize}%
\item [[Freeman Dyson]] \emph{Missed opportunities}, Bulletin of the AMS, Volume 78, Number 5, September 1972 (\href{https://www.math.uh.edu/~tomforde/Articles/Missed-Opportunities-Dyson.pdf}{pdf})

\begin{quote}%
$[$ the [[Haag-Kastler axioms]] $]$ taken together with the axioms defining a C\emph{-algebra are a distillation into abstract mathematical language of all the general truths that we have learned about the physics of microscopic systems during the last 50 years. They describe a mathematical structure of great elegance whose properties correspond in many respects to the facts of experimental physics. In some sense, the axioms represent the most serious attempt that has yet been made to define precisely what physicists mean by the words ``observability, causality, locality, relativistic invariance,'' which they are constantly using or abusing in their everyday speech. $[$\ldots{}$]$ I therefore propose as an outstanding opportunity still open to the pure mathematicians, to create a mathematical structure preserving the main features of the Haag-Kastler axioms but possessing E-invariance instead of P-invariance.}


\end{quote}


\end{itemize}
$P$ here denotes the [[Poincaré group]], while $E$ denotes what Dyson calls the `Einstein group', which is now called the [[diffeomorphism group]].

General accounts of (perturbative, algebraic) quantum field theory on curved spacetimes include

\begin{itemize}%
\item N. Birrell, P. Davies, \emph{Quantum Fields in Curved Space}, Cambridge: Cambridge University Press, 1982


\item [[Robert Wald]], \emph{Quantum field theory in curved spacetime and black hole thermodynamics}. Univ. of Chicago Press 1994 (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0842.53052&format=complete}{ZMATH entry}).


\item [[Stefan Hollands]], [[Robert Wald]], \emph{Quantum fields in curved spacetime}, Physics Reports Volume 574, 16 April 2015, Pages 1-35 (\href{https://arxiv.org/abs/1401.2026}{arXiv:1401.2026})


\item [[Christopher Fewster]], [[Rainer Verch]], \emph{Algebraic quantum field theory in curved spacetimes} (\href{https://arxiv.org/abs/1504.00586}{arXiv:1504.00586})



\end{itemize}
Foundations for [[perturbative quantum field theory]] on curved spacetimes in terms of [[causal perturbation theory]] were laid in

\begin{itemize}%
\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], \emph{Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds}, Commun. Math. Phys. 208 : 623-661, 2000 (\href{https://arxiv.org/abs/math-ph/9903028}{math-ph/9903028})


\item [[Stefan Hollands]], [[Robert Wald]], \emph{Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime}, Commun. Math. Phys. 223:289-326,2001 (\href{https://arxiv.org/abs/gr-qc/0103074}{arXiv:gr-qc/0103074})


\item [[Stefan Hollands]], [[Robert Wald]], \emph{On the Renormalization Group in Curved Spacetime}, Commun. Math. Phys. 237 (2003) 123-160 (\href{https://arxiv.org/abs/gr-qc/0209029}{arXiv:gr-qc/0209029})


\item [[Stefan Hollands]], [[Robert Wald]], \emph{Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes}, Rev.Math.Phys. 17 (2005) 227-312 (\href{https://arxiv.org/abs/gr-qc/0404074}{arXiv:gr-qc/0404074})



\end{itemize}
The [[AQFT]]-style axiomatization via [[local nets]] on a category of [[Lorentzian manifolds]] ([[locally covariant perturbative quantum field theory]]) is due to

\begin{itemize}%
\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Rainer Verch]], \emph{The generally covariant locality principle -- A new paradigm for local quantum physics}, Commun. Math. Phys. 237:31-68 (2003) (\href{http://arxiv.org/abs/math-ph/0112041}{arXiv:math-ph/0112041})


\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], \emph{Quantum Field Theory on Curved Backgrounds} , Proceedings of the Kompaktkurs ``Quantenfeldtheorie auf gekruemmten Raumzeiten'' held at Universitaet Potsdam, Germany, in 8.-12.10.2007, organized by C. Baer and K. Fredenhagen (\href{http://arxiv.org/abs/0901.2063}{arXiv:0901.2063})



\end{itemize}
Reviews with emphasis on this AQFT-point of view include

\begin{itemize}%
\item [[Robert Wald]], \emph{The Formulation of Quantum Field Theory in Curved Spacetime} (\href{https://arxiv.org/abs/0907.0416}{arXiv:0907.0416})


\item [[Robert Wald]], \emph{The History and Present Status of Quantum Field Theory in Curved Spacetime} (\href{https://arxiv.org/abs/gr-qc/0608018}{arXiv:gr-qc/0608018})


\item [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], \emph{QFT on curved spacetimes: axiomatic framework and examples} (\href{http://arxiv.org/abs/1412.5125}{arXiv:1412.5125})



\end{itemize}
Papers about the application of [[microlocal analysis]] include

\begin{itemize}%
\item Alexander Strohmaier, [[Rainer Verch]], Manfred Wollenberg: \emph{Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems} (\href{http://xxx.uni-augsburg.de/abs/math-ph/0202003}{arXiv}).

\end{itemize}
Discussion of [[renormalization]] in AQFT on curved spacetimes includes

\begin{itemize}%
\item [[Igor Khavkine]], [[Valter Moretti]], \emph{Analytic Dependence is an Unnecessary Requirement in Renormalization of Locally Covariant QFT}, Communications in Mathematical Physics, March 2016 (\href{http://arxiv.org/abs/1411.1302}{arXiv:1411.1302}, \href{http://link.springer.com/article/10.1007%2Fs00220-016-2618-7}{publisher})

\end{itemize}
Discussion of the [[cosmology]] in the context of AQFT on curved spacetimes includes

\begin{itemize}%
\item [[Klaus Fredenhagen]], [[Thomas-Paul Hack]], \emph{Quantum field theory on curved spacetime and the standard cosmological model} (\href{http://arxiv.org/abs/1308.6773}{arXiv:1308.6773})


\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Thomas-Paul Hack]], [[Nicola Pinamonti]], [[Katarzyna Rejzner]], \emph{Cosmological perturbation theory and quantum gravity} (\href{https://arxiv.org/abs/1605.02573}{arXiv:1605.02573})


\item [[Thomas-Paul Hack]], \emph{Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes}, Springer 2016 (\href{https://arxiv.org/abs/1506.01869}{arXiv:1506.01869}, \href{https://doi.org/10.1007/978-3-319-21894-6}{doi:10.1007/978-3-319-21894-6})



\end{itemize}
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[[!redirects locally covariant algebraic quantum field theory]] [[!redirects locally covariant AQFT]]

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[[!redirects QFT on curved spacetime]] [[!redirects QFT on curved spacetimes]]

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