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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{AQFT} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory} [[!include AQFT and operator algebra contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{axioms}{Axioms}\dotfill \pageref*{axioms} \linebreak \noindent\hyperlink{theorems}{Theorems}\dotfill \pageref*{theorems} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{FreeScalarField}{Free scalar field / Klein Gordon field}\dotfill \pageref*{FreeScalarField} \linebreak \noindent\hyperlink{free_fermion__dirac_field}{Free fermion / Dirac field}\dotfill \pageref*{free_fermion__dirac_field} \linebreak \noindent\hyperlink{electromagnetic_field}{Electromagnetic field}\dotfill \pageref*{electromagnetic_field} \linebreak \noindent\hyperlink{proca_field}{Proca field}\dotfill \pageref*{proca_field} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{axioms_2}{Axioms}\dotfill \pageref*{axioms_2} \linebreak \noindent\hyperlink{lecture_notes_and_textbooks}{Lecture notes and Textbooks}\dotfill \pageref*{lecture_notes_and_textbooks} \linebreak \noindent\hyperlink{reviews}{Reviews}\dotfill \pageref*{reviews} \linebreak \noindent\hyperlink{ExamplesReferences}{Examples}\dotfill \pageref*{ExamplesReferences} \linebreak \noindent\hyperlink{LocalGaugeTheory}{Local gauge theory}\dotfill \pageref*{LocalGaugeTheory} \linebreak \noindent\hyperlink{ReferencesPerturbationTheory}{Perturbation theory and renormalization}\dotfill \pageref*{ReferencesPerturbationTheory} \linebreak \noindent\hyperlink{further_developments}{Further developments}\dotfill \pageref*{further_developments} \linebreak \noindent\hyperlink{relation_to_functorial_qft}{Relation to functorial QFT}\dotfill \pageref*{relation_to_functorial_qft} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \textbf{Algebraic Quantum Field Theory} or \textbf{Axiomatic Quantum Field Theory} or \textbf{AQFT} for short is a formalization of [[quantum field theory]] (and specifically full, hence [[non-perturbative quantum field theory]]) that axiomatizes the assignment of \emph{[[algebras of observables]]} to patches of parameter space ([[spacetime]], [[worldvolume]]) that one expects a quantum field theory to provide. As such, the approach of AQFT is roughly dual to that of [[FQFT]], where instead \emph{spaces of states} are assigned to boundaries of [[cobordism]]s and propagation maps between state spaces to cobordisms themselves. One may roughly think of AQFT as being a formalization of what in basic [[quantum mechanics]] textbooks is called the \textbf{[[Heisenberg picture]]} of quantum mechanics. On the other hand [[FQFT]] axiomatizes the \emph{[[Schrödinger picture]]} . The axioms of traditional AQFT encode the properties of a [[local net]] of observables and are called the [[Haag-Kastler axioms]]. They are one of the oldest systems of axioms that seriously attempt to put [[quantum field theory]] on a solid conceptual footing. From the [[nPOV]] we may think of a [[local net]] as a co-flabby [[presheaf|copresheaf]] of [[algebra|algebras]] on spacetime which satisfies a certain \emph{locality} axiom with respect to the [[smooth Lorentzian manifold|Lorentzian structure]] of [[spacetime]]: \begin{itemize}% \item \textbf{locality:} algebras assigned to spacelike separated regions commute with each other when embedded into any joint superalgebra. \end{itemize} This is traditionally formulated (implicitly) as a structure in ordinary [[category theory]]. More recently, with the proof of the [[cobordism hypothesis]] and the corresponding [[(∞,n)-category]]-formulation of [[FQFT]] also [[higher category theory|higher categorical]] versions of systems of local algebras of observables are being put forward and studied. Three structures are curently being studied, that are all conceptually very similar and similar to the Haag-Kastler axioms: \begin{itemize}% \item [[factorization algebra]]s \item [[topological chiral homology]] \item [[blob homology]]. \end{itemize} Initially, all three of these encoded what in physics are called \emph{Euclidean} quantum field theories, whereas only the notion of [[local net]] incorporated the fact that the underlying spacetime of a quantum field theory is a [[smooth Lorentzian space]]. Recent developments in the formalism of [[factorization algebra]]s have extended their theory to globally hyperbolic [[Lorentzian manifolds]]. In the context of the Haag-Kastler axioms there is a precise theorem, the [[Osterwalder-Schrader theorem]], relating the Euclidean to the Lorentzian formulation: this is the operation known as [[Wick rotation]]. Sheaves are used explicitly in: \begin{itemize}% \item Roberts, John E.: \href{http://books.google.com/books?id=IFjzuLjE43kC&lpg=PA297&ots=5Ld1B3I45m&dq=Operator%20algebras%20and%20applications%2C%20Part%202&pg=PA523#v=onepage&q=Operator%20algebras%20and%20applications,%20Part%202&f=false}{New light on the mathematical structure of algebraic field theory.} Operator algebras and applications, Part 2 (Kingston, Ont., 1980), pp. 523--550, Proc. Sympos. Pure Math., 38, Amer. Math. Soc., Providence, R.I., 1982. \item Roberts, John E.: \href{http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103921341}{Localization in algebraic field theory}. Comm. Math. Phys. 85 (1982), no. 1, 87--98. \end{itemize} --- much information to be filled in --- \hypertarget{axioms}{}\subsection*{{Axioms}}\label{axioms} \begin{itemize}% \item [[Wightman axioms]] \item [[Haag-Kastler axioms]] \end{itemize} \hypertarget{theorems}{}\subsection*{{Theorems}}\label{theorems} \begin{itemize}% \item [[Reeh-Schlieder theorem]] \item [[Osterwalder-Schrader theorem]] \item [[PCT theorem]] \item [[Bisognano-Wichmann theorem]] \item [[spin-statistics theorem]] \end{itemize} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} Generically the algebra of a relativistic AQFT turns out to be a ([[generalized the|the]]) hyperfinite type $III_1$ [[von Neumann algebra factor]]. See (\hyperlink{Yngvason}{Yngvason}) \hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples} Examples of AQFT [[local nets of observables]] that encode interacting quantum field theories are not easy to construct. The construction of \emph{free} field theories is well understood, see the references \hyperlink{ExamplesReferences}{below}. In [[perturbation theory]] also interacting theories can be constructed, see the references \hyperlink{ReferencesPerturbationTheory}{here}. \hypertarget{FreeScalarField}{}\subsubsection*{{Free scalar field / Klein Gordon field}}\label{FreeScalarField} A survey of the AQFT description of the [[free field theor|free]] [[scalar field]] on [[Minkowski spacetime]] is in (\hyperlink{Montoya}{Motoya, slides 11-17}). Discussion in more general context of [[AQFT on curved spacetimes]] in (\hyperlink{BrunettiFredenhagen}{Brunetti-Fredenhagen, section 5.2}) \hypertarget{free_fermion__dirac_field}{}\subsubsection*{{Free fermion / Dirac field}}\label{free_fermion__dirac_field} The [[free field theory|free]] [[Dirac field]] and its deformations is discussed for instance in (\hyperlink{DLM}{DLM, section 3.2}), (\hyperlink{Dimock82}{Dimock 83}). \hypertarget{electromagnetic_field}{}\subsubsection*{{Electromagnetic field}}\label{electromagnetic_field} The quantized [[electromagnetic field]] is discussed for instance in (\hyperlink{Dimock92}{Dimock 92}). \hypertarget{proca_field}{}\subsubsection*{{Proca field}}\label{proca_field} (\hyperlink{Furliani}{Furliani}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[quantum mechanics]], [[quantum field theory]], [[perturbative quantum field theory]] \begin{itemize}% \item \textbf{AQFT} \begin{itemize}% \item [[Haag-Kastler axioms]], [[Wightman axioms]] \item [[local net of observables]], [[field net]] \item [[quantum lattice systems]], [[string-localized quantum field]] \item [[locally covariant AQFT]] \item [[perturbative AQFT]] \item [[homotopical algebraic quantum field theory]] \item [[Haag-Ruelle scattering theory]] \end{itemize} \item [[FQFT]] \end{itemize} \item [[constructive quantum field theory]] \end{itemize} [[!include Isbell duality - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{axioms_2}{}\subsubsection*{{Axioms}}\label{axioms_2} The original article that introduced the [[Haag-Kastler axioms]] is \begin{itemize}% \item [[Rudolf Haag]], [[Daniel Kastler]], \emph{An algebraic approach to quantum field theory} , Journal of Mathematical Physics, Bd.5, 1964, S.848-861 \end{itemize} following \begin{itemize}% \item [[Rudolf Haag]], \emph{Discussion des ``axiomes'' et des propri\'e{}t\'e{}s asymptotiques d'une th\'e{}orie des champs locales avec particules compos\'e{}es, Les Probl\`e{}mes Math\'e{}matiques de la Th\'e{}orie Quantique des Champs}, Colloque Internationaux du CNRS LXXV (Lille 1957), CNRS Paris (1959), 151. \end{itemize} The generalization of the [[spacetime]] [[site]] from open in [[Minkowski space]] to more general and [[curvature|curved]] spacetimes (see [[AQFT on curved spacetimes]]) is due to \begin{itemize}% \item [[Romeo Brunetti]], [[Klaus Fredenhagen]], \emph{Quantum field theory on curved spacetimes} \href{http://arxiv.org/abs/0901.2063}{arXiv:0901.2063} \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 \end{itemize} See also \emph{[[AQFT on curved spacetimes]]} . \hypertarget{lecture_notes_and_textbooks}{}\subsubsection*{{Lecture notes and Textbooks}}\label{lecture_notes_and_textbooks} Introductory lecture notes include \begin{itemize}% \item [[Klaus Fredenhagen]], \emph{Algebraische Quantenfeldtheorie}, lecture notes, 2003 ([[FredenhagenAQFT2003.pdf:file]]) \item [[Christopher Fewster]], [[Kasia Rejzner]], \emph{Algebraic Quantum Field Theory - an introduction} (\href{https://arxiv.org/abs/1904.04051}{arXiv:1904.04051}) \end{itemize} and for just [[quantum mechanics]] in the algebraic perspective: \begin{itemize}% \item Jonathan Gleason, \emph{The $C*$-algebraic formalism of quantum mechanics}, 2009 (\href{http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gleason.pdf}{pdf}, [[GleasonAlgebraic.pdf:file]]) \end{itemize} Classical textbooks are \begin{itemize}% \item [[Huzihiro Araki]], \emph{[[Mathematical Theory of Quantum Fields]]} \item [[Rudolf Haag]], \emph{[[Local Quantum Physics -- Fields, Particles, Algebras]]} \end{itemize} A good account of the mathematical axiomatics of Haag-Kastler AQFT is \begin{itemize}% \item [[Hans Halvorson]], [[Michael Müger]], \emph{Algebraic Quantum Field Theory} (\href{http://arxiv.org/abs/math-ph/0602036}{arXiv}) \end{itemize} This is, among other things, the ideal starting point for pure mathematicians who have always been left puzzled or otherwise unsatisfied by accounts of quantum field theory, even those tagged as being ``for mathematicians''. AQFT is truly axiomatic and rigorously formal. An account written by mathematicians for mathematicians is this: \begin{itemize}% \item [[Hellmut Baumgärtel]], Manfred Wollenberg, \emph{Causal nets of operator algebras.} Berlin: Akademie Verlag 1992 (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0749.46038&format=complete}{ZMATH entry}) \end{itemize} and this: \begin{itemize}% \item [[Hellmut Baumgärtel]], \emph{Operator algebraic Methods in Quantum Field Theory. A series of lectures.} Akademie Verlag 1995 (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0839.46063&format=complete}{ZMATH entry}) \end{itemize} A classic of the trade is this one: \begin{itemize}% \item [[Nikolay Bogolyubov]], Logunov, Oksak, Todorov: \emph{General principles of quantum field theory} (Mathematical Physics and Applied Mathematics, 10. Dordrecht etc.: Kluwer Academic Publishers, 1990) \end{itemize} \hypertarget{reviews}{}\subsubsection*{{Reviews}}\label{reviews} Recent account of the principle of locality in AQFT from the point of view of traditional school \begin{itemize}% \item [[Franco Strocchi]], \emph{Relativistic Quantum Mechanics and Field Theory}, Found.Phys. 34 (2004) 501-527 (\href{http://arxiv.org/abs/hep-th/0401143}{arXiv:hep-th/0401143}) \item [[Sergio Doplicher]], \emph{The principle of locality: Effectiveness, fate, and challenges}, J. Math. Phys. \textbf{51}, 015218 (2010), \href{http://dx.doi.org/10.1063/1.3276100}{doi} \item [[Franco Strocchi]], \emph{An Introduction to Non-Perturbative Foundations of Quantum Field Theory}, Oxford University Press, 2013 \end{itemize} Talk slides include \begin{itemize}% \item Edison Montoya, \emph{Algebraic quantum field theory} (2009) (\href{http://www.matmor.unam.mx/~robert/sem/20091021_Montoya.pdf}{pdf}) \end{itemize} \hypertarget{ExamplesReferences}{}\subsubsection*{{Examples}}\label{ExamplesReferences} Construction of examples is considered for instance in \begin{itemize}% \item [[Jonathan Dimock]], \emph{Dirac quantum fields on a manifold}, Trans. Amer. Math. Soc. 269 (1982), 133-147. (\href{http://www.ams.org/journals/tran/1982-269-01/S0002-9947-1982-0637032-8/home.html}{web}) \end{itemize} \begin{itemize}% \item [[Jonathan Dimock]], \emph{Quantized electromagnetic field on a manifdold}, Reviews in mathematical physics, Volume 4, Issue 2 (1992) (\href{http://www.worldscinet.com/rmp/04/0402/S0129055X92000078.html}{web}) \end{itemize} \begin{itemize}% \item Edward Furliani, \emph{Quantization of massive vector fields in curved space--time}, J. Math. Phys. 40, 2611 (1999) (\href{http://jmp.aip.org/resource/1/jmapaq/v40/i6/p2611_s1?isAuthorized=no}{web}) \end{itemize} General discussion of AQFT quantization of [[free quantum fields]] is in \begin{itemize}% \item [[Christian Bär]], N. Ginoux, [[Frank Pfäffle]], \emph{Wave Equations on Lorentzian Manifolds and Quantization}, (EMS, 2007) (\href{http://arxiv.org/abs/0806.1036}{arXiv:0806.1036}) \item [[Christian Bär]], N. Ginoux, \emph{Classical and quantum fields on lorentzian manifolds} (2011) (\href{http://arxiv.org/abs/1104.1158}{arXiv:1104.1158}) \end{itemize} Examples of [[non-perturbative quantum field theory|non-perturbative]] [[interacting quantum field theory|interacting]] [[scalar field theory]] in \emph{any} [[spacetime]] [[dimension]] (in particular in $d \geq 4$) are claimed in \begin{itemize}% \item [[Detlev Buchholz]], [[Klaus Fredenhagen]], \emph{A $C^\ast$-algebraic approach to interacting quantum field theories} (\href{https://arxiv.org/abs/1902.06062}{arXiv:1902.06062}) \end{itemize} \hypertarget{LocalGaugeTheory}{}\subsubsection*{{Local gauge theory}}\label{LocalGaugeTheory} Discussion of aspects of [[gauge theory]] includes \begin{itemize}% \item Fabio Ciolli, [[Giuseppe Ruzzi]], Ezio Vasselli, \emph{Causal posets, loops and the construction of nets of local algebras for QFT} (\href{http://arxiv.org/abs/1109.4824}{arXiv:1109.4824}) \item Fabio Ciolli, [[Giuseppe Ruzzi]], Ezio Vasselli, \emph{QED Representation for the Net of Causal Loops} (\href{http://arxiv.org/abs/1305.7059}{arXiv:1305.7059}) \item [[Giuseppe Ruzzi]], \emph{Nets of local algebras and gauge theories}, 2014 (\href{http://www.aqft14.eu/wp-content/uploads/2014/05/Ruzzi.pdf}{pdf slides}) \end{itemize} Construction and axiomatization of gauge field AQFT via [[homotopy theory]] and [[homotopical algebra]] (see also at \emph{[[field bundle]]}) is being developed in \begin{itemize}% \item [[Marco Benini]], [[Claudio Dappiaggi]], [[Alexander Schenkel]], \emph{Quantized Abelian principal connections on Lorentzian manifolds}, Communications in Mathematical Physics 2013 (\href{http://arxiv.org/abs/1303.2515}{arXiv:1303.2515}) \item [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], \emph{Homotopy colimits and global observables in Abelian gauge theory} (\href{http://arxiv.org/abs/1503.08839}{arXiv:1503.08839}) \item [[Marco Benini]], [[Alexander Schenkel]], \emph{Quantum field theories on categories fibered in groupoids} (\href{https://arxiv.org/abs/1610.06071}{arXiv:1610.06071}) \end{itemize} The issue of the tension between local gauge invariance and locality and the need to pass to [[stacks]]/[[higher geometry]] is made explicit in \begin{itemize}% \item [[Alexander Schenkel]], \emph{On the problem of gauge theories in locally covariant QFT}, talk at \emph{\href{http://www.science.unitn.it/~moretti/convegno/convegno.html}{Operator and Geometric Analysis on Quantum Theory}} Trento, 2014 ([[SchenkelTrento2014.pdf:file]]) (with further emphasis on this point in the companion talk \href{field+bundle#Schreiber14}{Schreiber 14}) \end{itemize} Further development of this [[homotopical algebraic quantum field theory]] includes \begin{itemize}% \item [[Marco Benini]], [[Alexander Schenkel]], \emph{Quantum field theories on categories fibered in groupoids} (\href{https://arxiv.org/abs/1610.06071}{arXiv:1610.06071}) \end{itemize} \hypertarget{ReferencesPerturbationTheory}{}\subsubsection*{{Perturbation theory and renormalization}}\label{ReferencesPerturbationTheory} [[perturbation theory|Perturbation theory]] and [[renormalization]] in the context of AQFT and is discussed in the following articles. The observation that in [[perturbation theory]] the [[renormalization|Stückelberg-Bogoliubov-Epstein-Glaser]] local [[S-matrix|S-matrices]] yield a [[local net of observables]] was first made in \begin{itemize}% \item V. Il'in, D. Slavnov, \emph{Observable algebras in the S-matrix approach} Theor. Math. Phys. \textbf{36} , 32 (1978) \end{itemize} which was however mostly ignored and forgotten. It is taken up again 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{http://arxiv.org/abs/math-ph/9903028}{arXiv}) \end{itemize} (a quick survey is in section 8, details are in section 2). Further developments along these lines are in \begin{itemize}% \item [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Perturbative algebraic quantum field theory and deformation quantization}, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) (\href{http://xxx.uni-augsburg.de/abs/hep-th/0101079}{arXiv:hep-th/0101079}) \end{itemize} (relation to [[deformation quantization]]) \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{http://arxiv.org/abs/math-ph/9903028}{arXiv}) \item [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Perturbative Algebraic Quantum Field Theory and the Renormalization Groups} Adv. Theor. Math. Physics 13 (2009), 1541-1599 (\href{http://arxiv.org/abs/0901.2038}{arXiv:0901.2038}) \end{itemize} (relation to [[renormalization]]) \begin{itemize}% \item [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{A local (perturbative) construction of observables in gauge theores: the example of qed} , Commun. Math. Phys. 203 (1999), no.1, 71-105, (\href{http://xxx.uni-augsburg.de/abs/hep-th/9807078}{arXiv:hep-th/9807078}). \end{itemize} (relation to [[gauge theory]] and [[QED]]) Lecture notes are in \begin{itemize}% \item [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], \emph{Perturbative algebraic quantum field theory}, In \emph{Mathematical Aspects of Quantum Field Theories}, Springer 2016 (\href{https://arxiv.org/abs/1208.1428}{arXiv:1208.1428}) \item [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], \emph{Perturbative Construction of Models of Algebraic Quantum Field Theory} (\href{https://arxiv.org/abs/1503.07814}{arXiv:1503.07814}) \end{itemize} and a textbook acount is in \begin{itemize}% \item [[Katarzyna Rejzner]], \emph{Perturbative Algebraic Quantum Field Theory}, Mathematical Physics Studies, Springer 2016 (\href{https://link.springer.com/book/10.1007%2F978-3-319-25901-7}{pdf}) \end{itemize} \hypertarget{further_developments}{}\subsubsection*{{Further developments}}\label{further_developments} \begin{itemize}% \item [[Claudio Dappiaggi]], [[Gandalf Lechner]], E. Morfa-Morales, \emph{Deformations of quantum field theories on spacetimes with Killing vector fields}, Commun.Math.Phys.305:99-130, (2011), (\href{http://arxiv.org/abs/1006.3548}{arXiv:1006.3548}) \item [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], \emph{Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds}, (\href{https://arxiv.org/abs/1903.03396}{arXiv:1903.03396v2}) \end{itemize} \hypertarget{relation_to_functorial_qft}{}\subsubsection*{{Relation to functorial QFT}}\label{relation_to_functorial_qft} A relation to [[FQFT]] is discussed in \begin{itemize}% \item [[Urs Schreiber]], \emph{AQFT from $n$-functorial QFT} , Comm. Math. Phys., Volume 291, Issue 2, pp.357-401 (\href{http://ncatlab.org/schreiber/files/AQFTfromFQFT.pdf}{pdf}) \end{itemize} The role of [[von Neumann algebra factor]]s is discussed in \begin{itemize}% \item J. Yngvason, \emph{The role of type III factors in quantum field theory} (\href{http://arxiv.org/abs/math-ph/0411058}{arXiv:math-ph/0411058}) \end{itemize} [[!redirects algebraic quantum field theory]] [[!redirects algebraic quantum field theories]] \end{document}