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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*{perturbative quantum field theory} \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{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{ReferencesNonConvergenceOfThePerturbationSeries}{Non-convergence of the perturbation series}\dotfill \pageref*{ReferencesNonConvergenceOfThePerturbationSeries} \linebreak \noindent\hyperlink{linfinity_algebra_structure}{L-infinity algebra structure}\dotfill \pageref*{linfinity_algebra_structure} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} What is called \emph{perturbative quantum field theory} (pQFT) is [[quantum field theory]] where the [[interaction]] (between [[field (physics)|fields]]/[[particles]]) is treated as a tiny [[perturbation]] of the ``[[free field theory]]'' where no [[interaction]] is assumed to takes place (``[[perturbation theory]]''). This is meant to be an approximation to the actual \emph{[[non-perturbative quantum field theory]]}. However, the latter remains elusive except for toy examples of low spacetime dimension, vanishing [[interaction]] and/or [[topological field theory|topological invariance]] and most of the ``quantum field theory'' in the literature is tacitly understood to be perturbative. Hence pQFT studies the \emph{[[infinitesimal neighbourhood]]} (also called the \emph{[[formal neighbourhood]]}) of [[free quantum field theories]] in the space of all quantum field theories. Mathematically this means that the resulting [[quantum observables]] are [[formal power series]] in the [[coupling constant]] $g$ which measures the strength of the [[interaction]] (as well as in \emph{[[Planck's constant]]}, which measures the general strength of [[quantum physics|quantum]]). This distinguishes perturbative quantum field theory from [[non-perturbative quantum field theory]], where the algebras of [[quantum observables]] are supposed to be not formal power series algebras, but [[C\emph{-algebras]].} The key object of perturbative QFT is the perturbative \emph{[[scattering matrix]]} which expresses, as a [[formal power series]] in the ratio of the [[coupling constant]] over [[Planck's constant]], the [[probability amplitude]] of [[scattering]] processes, namely of processes where [[free field theory|free fields]] in a certain [[quantum state|state]] come in from the far past, interact and hence scatter off each other, and then go off in some other [[quantum state]] into the far future. The [[scattering cross sections]] thus defined are the quantities which may be directly measured in scattering [[experiments]], such as the [[LHC]] accelerator. The perturbative [[S-matrix]] turns out to have an expression as a sum over separate [[scattering amplitudes]] for elementary processes labeled by \emph{[[Feynman diagrams]]}, each of which depicts one specific way for fields ([[particles]]) to interact with each other. That the full S-matrix is the sum over all amplitudes for all these possible scattering processes, the \emph{[[Feynman perturbation series]]}, is an incarnation of the informal heuristic of the [[path integral]] and the [[superposition principle]] in [[quantum physics]], which says that the [[probability amplitude]] for a specific outcome is the sum over the probability amplitudes of all the possible processes that can contribute to this outcome. For all interesting [[interacting field theories]], such as [[quantum electrodynamics]] and [[quantum chromodynamics]], this [[scattering matrix]] [[formal power series]] necessarily has \emph{vanishing} [[radius of convergence]] (\hyperlink{Dyson52}{Dyson 52}). If it is assumed that the [[formal power series|formal]] [[Feynman perturbation series]] is the [[Taylor series]] of an actual [[smooth function]] given by the actual [[non-perturbative quantum field theory]] that is being approximated, then this means that it is at least an [[asymptotic series]] (by \href{asymptotic+series#TaylorSeriesOfSmoothFunctionIsAsymptoticSeries}{this example}) whose first couple of terms could sum to a good approximation of the actual value to be computed. Indeed, the sum of the first few [[loop orders]] in the [[S-matrix]] for [[QED]] and [[QCD]] in the [[standard model of particle physics]] turns out to be in agreement with [[experiment]] to good precision. (There are however known [[non-perturbative effects]] which are not captured in perturbation theory, such as [[confinement]] in [[QCD]], supposed related to [[instantons in QCD]]. In [[resurgence theory]] one tries to identify these from the [[asymptotic series|asymptotic]] nature of the [[Feynman perturbation series]].) A key step in the construction of perturbative quantum field theory is the \emph{[[renormalization]]} of the point interactions. This comes about because given \begin{enumerate}% \item a [[local Lagrangian density]] defining the nature of the [[field (physics)|fields]] and their [[interactions]], \item a [[vacuum state]] (generally: [[Hadamard state]]) that defines the [[free field theory|free]] [[quantum field theory]] to be perturbed about \end{enumerate} it turns out that the construction of the perturbative [[S-matrix]] (the [[Feynman perturbation series]]) still involves at each order a finite-dimensional space of choices to be made. Physically, these are the specification of further high energy interactions not seen in the original [[local Lagrangian density]]; mathematically, this is the choice of extending the [[time-ordered product]] of the interaction, which is an [[operator-valued distribution]], to the locus of coinciding interaction points, in the sense of [[extensions of distributions]]. Historically, perturbative quantum field theory as originally conceived informally by [[Schwinger-Tomonaga-Feynman-Dyson]] in the 1940s, had been notorious for the mysterious conceptual nature of its mathematical principles (``divergences''). The mathematically rigorous formulation of [[renormalization]] (``removal of [[UV-divergences]]'') in perturbative quantum field theory on [[Minkowski spacetime]] was established by \hyperlink{EpsteinGlaser73}{Epstein-Glaser 73}, based on \hyperlink{BogoliubovShirkov59}{Bogoliubov-Shirkov 59} and \hyperlink{Stueckelberg51}{St\"u{}ckelberg 51}), now known as \textbf{[[causal perturbation theory]]}; laid out in the seminal Erice summer school proceeding (\hyperlink{VeloWightman76}{Velo-Wightman 76}). The correct definition of the [[adiabatic switching|adiabatic limit]] (``removal of IR divergencies'') was understood in \hyperlink{IlinSlavnov78}{Il'in-Slavnov 78} and eventually developed by \hyperlink{DuetschFredenhagen01}{D\"u{}tsch-Fredenhagen 01}, \hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-D\"u{}tschFredenhagen 09}, this is now called \textbf{[[perturbative algebraic quantum field theory]]}. The rigorous derivation of the previously informal [[Feynman rules]] and their [[dimensional regularization]] for computation of [[scattering amplitudes]] was achieved in \hyperlink{Keller10}{Keller 10 (IV.12)}, \hyperlink{DuetschFredenhagenKellerRejzner14}{D\"u{}tsch-Fredenhagen-Keller-Rejzner 14}. Quantization of [[gauge theories]] ([[Yang-Mills theory]]) in [[causal perturbation theory]]/[[perturbative AQFT]] was then discussed (for trivial [[principal bundles]] and restricted to [[gauge invariant observables]]) in the spirit of [[BRST-complex]]/[[BV-formalism]] in (\hyperlink{FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b}). The generalization of all these constructions from [[Minkowski spacetime]] to perturbative quantum fields on more general [[spacetimes]] (i.e. for more general [[gravity|gravitational]] [[background fields]] such as appearing in [[cosmology]] or [[black hole]] physics) was made possible due to the identification of the proper generalization of [[vacuum states]] and their [[Feynman propagators]] to [[Hadamard states]] on [[globally hyperbolic spacetimes]] in \hyperlink{Radzikowski96}{Radzikowski 96}. The resulting rigorous perturbative [[QFT on curved spacetimes]] was developed in a long series of articles by [[Stefan Hollands|Hollands]], [[Robert Wald|Wald]], [[Romeo Brunetti|Brunetti]], [[Klaus Fredenhagen|Fredenhagen]] and others, now called \emph{[[locally covariant perturbative AQFT]]}. While this establishes a rigorous construction of perturbative quantum field theory on general gravitational backgrounds, the construction principles had remained somewhat ad-hoc: The [[axioms]] for the perturbative [[S-matrix]] (equivalently for the [[time-ordered products]] or [[retarded products]] of field operators) were well motivated by comparison with the [[Dyson series]] in [[quantum mechanics]], by the heuristics of the [[path integral]] and not the least by their excellent confirmation by [[experiment]], but had not been derived from first principles of [[quantization]]. Then in \hyperlink{DuetschFredenhagen01}{D\"u{}tsch Fredenhagen 01} it was observed that the [[Wick algebras]] of [[quantum observables]] in [[free quantum field theory]] are equivalently the [[Moyal deformation quantization]] of the canonical [[Poisson bracket]] (the \emph{[[Peierls bracket]]} or \emph{[[causal propagator]]}) on the [[covariant phase space]] of the free field theory (or rather of a choice of [[Hadamard state]] for it) and \hyperlink{Collini16}{Collini 16} showed that under suitable conditions the perturbative [[interacting observable algebra]] is the [[Fedosov deformation quantization]] of [[covariant phase space]] of the interacting theory. A general argument to this extent was given in \hyperlink{HawkinsRejzner16}{Hawkins-Rejzner 16}. This suggests that the construction of the full [[non-perturbative quantum field theory]] ought to be given by a [[strict deformation quantization]] of the [[covariant phase space]]. But presently no example of such for non-trivial interaction in [[spacetime]] [[dimension]] $\geq 4$ is known. In particular the [[phenomenology|phenomenologically]] interesting case of a complete construction of interacting field theories on 4-dimensional spacetimes is presently unknown. For the case of [[Yang-Mills theory]] this open problem to go beyond perturbative quantum field theory is one of the ``Millenium Problems'' (see at \emph{[[quantization of Yang-Mills theory]]}). For the case of [[quantum gravity]] this is possibly the $10^4$-year problem that the field is facing. But observe that as a perturbative ([[effective quantum field theory|effective]]``) quantum field theory, [[quantum gravity]] does fit into the framework of perturbative QFT, is mathematically well-defined and makes predictions, see the references \href{quantum%20gravity#ReferencesAsAnEffectiveFieldTheory}{there}. \hypertarget{details}{}\subsection*{{Details}}\label{details} A comprehensive introduction is at \emph{[[geometry of physics -- perturbative quantum field theory]]}. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \begin{itemize}% \item [[main theorem of perturbative renormalization theory]] \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[renormalization]], [[radiative correction]] \item [[causal perturbation theory]], [[perturbative AQFT]] \end{itemize} [[!include products in pQFT -- table]] \begin{itemize}% \item [[non-perturbative field theory]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The original informal conception of perturbative QFT is due to [[Schwinger-Tomonaga-Feynman-Dyson]]: \begin{itemize}% \item [[Freeman Dyson]], \emph{The raditation theories of Tomonaga, Schwinger and Feynman}, Phys. Rev. 75, 486, 1949 (\href{http://web.ihep.su/dbserv/compas/src/dyson49b/eng.pdf}{pdf}) \end{itemize} The rigorous formulation of renormalized perturbative quantum field theory in terms of [[causal perturbation theory]] was first accomplished in \begin{itemize}% \item [[Henri Epstein]], [[Vladimir Glaser]], \emph{[[The Role of locality in perturbation theory]]}, Annales Poincar\'e{} Phys. Theor. A 19 (1973) 211 (\href{http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0}{Numdam}) \end{itemize} with precursors in \begin{itemize}% \item [[Ernst Stückelberg]], D. Rivier, Helv. Phys. Acta, 22 (1949) 215. \item [[Ernst Stückelberg]], J. Green, Helv. Phys. Acta, 24 (1951) 153. \item [[Ernst Stückelberg]], A. Peterman, , \emph{La normalisation des constants dans la theorie des quanta}, Helv. Phys. Acta 26, 499 (1953); \item [[Nikolay Bogoliubov]], [[Dmitry Shirkov]], \emph{Introduction to the Theory of Quantized Fields}, New York (1959) \end{itemize} A seminal compilation of the resulting rigorous understanding of [[renormalization]] is \begin{itemize}% \item G. Velo and [[Arthur Wightman]] (eds.) \emph{Renormalization Theory} Proceedings of the 1975 Erice summer school, NATO ASI Series C 23, D. Reidel, Dordrecht, 1976 \end{itemize} Concrete computations in rigorous [[causal perturbation theory]] have been spelled out for [[quantum electrodynamics]] in \begin{itemize}% \item [[Günter Scharf]], \emph{[[Finite Quantum Electrodynamics -- The Causal Approach]]}, Berlin: Springer-Verlag, 1995, 2nd edition \end{itemize} and for [[Yang-Mills theory]], [[quantum chromodynamics]] and [[perturbative quantum gravity]] in \begin{itemize}% \item [[Günter Scharf]], \emph{[[Quantum Gauge Theories -- A True Ghost Story]]}, Wiley 2001 \end{itemize} The treatment of the IR-divergencies by organizing the perturbative [[quantum observables]] into a [[local net of observables]] was first suggested in \begin{itemize}% \item V. A. Il'in and D. S. Slavnov, \emph{Observable algebras in the S-matrix approach}, Theor. Math. Phys. 36 (1978) 32 (\href{http://inspirehep.net/record/135575}{spire}, \href{http://dx.doi.org/10.1007/BF01035870}{doi}) \end{itemize} and then developed to \emph{[[perturbative algebraic quantum field theory]]} in \begin{itemize}% \item [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion}, Commun. Math. Phys. 219 (2001) 5-30 (\href{https://arxiv.org/abs/hep-th/0001129}{arXiv:hep-th/0001129}) \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} Quantization of [[gauge theories]] ([[Yang-Mills theory]]) in [[causal perturbation theory]]/[[perturbative AQFT]] is discussed (for trivial [[principal bundles]] and restricted to [[gauge invariant observables]]) in the spirit of [[BRST-complex]]/[[BV-formalism]] in \begin{itemize}% \item [[Klaus Fredenhagen]], [[Kasia Rejzner]], \emph{Batalin-Vilkovisky formalism in the functional approach to classical field theory}, Commun. Math. Phys. 314(1), 93--127 (2012) (\href{https://arxiv.org/abs/1101.5112}{arXiv:1101.5112}) \item [[Klaus Fredenhagen]], [[Kasia Rejzner]], \emph{Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory}, Commun. Math. Phys. 317(3), 697--725 (2012) (\href{https://arxiv.org/abs/1110.5232}{arXiv:1110.5232}) \end{itemize} and surveyed in \begin{itemize}% \item [[Kasia Rejzner]], section 7 of \emph{[[Perturbative Algebraic Quantum Field Theory]]}, Springer 2016 \item [[Michael Dütsch]], \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018 \end{itemize} The generalization of all these constructions to quantum fields on general [[globally hyperbolic spacetimes]] (perurbative [[AQFT on curved spacetimes]]) was made possible by th results on [[Hadamard states]] and [[Feynman propagators]] in \begin{itemize}% \item [[Marek Radzikowski]], \emph{Micro-local approach to the Hadamard condition in quantum field theory on curved space-time}, Commun. Math. Phys. 179 (1996), 529--553 (\href{http://projecteuclid.org/euclid.cmp/1104287114}{Euclid}) \end{itemize} and then developed in a long series of articles by [[Stefan Hollands]], [[Robert Wald]], [[Romeo Brunetti]], [[Klaus Fredenhagen]] and others. For this see the references at \emph{[[AQFT on curved spacetimes]]}. The observation that perturbative quantum field theory is equivalently the [[formal deformation quantization]] of the defining [[local Lagrangian density]] is for [[free field theory]] due to \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}) \item A. C. Hirshfeld, P. Henselder, \emph{Star Products and Perturbative Quantum Field Theory}, Annals Phys. 298 (2002) 382-393 (\href{https://arxiv.org/abs/hep-th/0208194}{arXiv:hep-th/0208194}) \end{itemize} and for interacting field theories ([[causal perturbation theory]]/[[perturbative AQFT]]) due \begin{itemize}% \item [[Giovanni Collini]], \emph{Fedosov Quantization and Perturbative Quantum Field Theory} (\href{https://arxiv.org/abs/1603.09626}{arXiv:1603.09626}) \item [[Eli Hawkins]], [[Kasia Rejzner]], \emph{The Star Product in Interacting Quantum Field Theory} (\href{https://arxiv.org/abs/1612.09157}{arXiv:1612.09157}) \end{itemize} For more see the references at \emph{[[perturbative algebraic quantum field theory]]}. The relation of the construction via [[causal perturbation theory]] to the [[Feynman perturbation series]] in terms of [[Feynman diagrams]] was understood in \begin{itemize}% \item [[Jose Gracia-Bondia]], S. Lazzarini, \emph{Connes-Kreimer-Epstein-Glaser Renormalization} (\href{https://arxiv.org/abs/hep-th/0006106}{arXiv:hep-th/0006106}) \item [[Kai Keller]], chapter IV of \emph{Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization}, PhD thesis (\href{https://arxiv.org/abs/1006.2148}{arXxiv:1006.2148}) \item [[Michael Dütsch]], [[Klaus Fredenhagen]], [[Kai Keller]], [[Katarzyna Rejzner]], \emph{Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization}, J. Math. Phy. 55(12), 122303 (2014) (\href{https://arxiv.org/abs/1311.5424}{arXiv:1311.5424}) \end{itemize} \hypertarget{ReferencesNonConvergenceOfThePerturbationSeries}{}\subsubsection*{{Non-convergence of the perturbation series}}\label{ReferencesNonConvergenceOfThePerturbationSeries} The argument that the perturbation series of realistic pQFTs necessarily [[divergent series|diverges]] (is at best an [[asymptotic series]]) goes back to \begin{itemize}% \item [[Freeman Dyson]], \emph{Divergence of perturbation theory in quantum electrodynamics}, Phys. Rev. 85, 631, 1952 (\href{http://inspirehep.net/record/29799?ln=en}{spire}) \end{itemize} and is made more precise in \begin{itemize}% \item [[Lev Lipatov]], \emph{Divergence of the Perturbation Theory Series and the Quasiclassical Theory}, Sov.Phys.JETP 45 (1977) 216--223 (\href{http://jetp.ac.ru/cgi-bin/dn/e_045_02_0216.pdf}{pdf}) \end{itemize} recalled for instance in \begin{itemize}% \item [[Igor Suslov]], section 1 of \emph{Divergent perturbation series}, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 (\href{https://arxiv.org/abs/hep-ph/0510142}{arXiv:hep-ph/0510142}) \item Justin Bond, last section of \emph{Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle} (\href{https://mcgreevy.physics.ucsd.edu/s13/final-papers/2013S-215C-Bond-Justin.pdf}{pdf}) \item [[Stefan Hollands]], [[Robert Wald]], section 4.1 of \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 Marco Serone, from 2:46 on in \emph{A look at $\phi^4_2$ using perturbation theory} (\href{https://www.youtube.com/watch?v=J4nxvY1rOhI}{recording}) \end{itemize} For the example of [[phi{\tt \symbol{94}}4 theory]] this non-convergence of the perturbation series is discussed in \begin{itemize}% \item Robert Helling, p. 4 of \emph{Solving classical field equations} (\href{http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf}{pdf}, [[HellingClassicalQFT.pdf:file]]) \item Alexander P. Bakulev, [[Dmitry Shirkov]], section 1.1 of \emph{Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory}, Proceedings of the 6th Mathematical Physics Meeting, Sept. 14--23, 2010, Belgrade, Serbia (ISBN 978-86-82441-30-4), pp. 27--54 (\href{https://arxiv.org/abs/1102.2380}{arXiv:1102.2380}) \end{itemize} See also \begin{itemize}% \item Carl M. Bender, Carlo Heissenberg, \emph{Convergent and Divergent Series in Physics} (\href{https://arxiv.org/abs/1703.05164}{arXiv:1703.05164}) \end{itemize} And see at \emph{[[perturbation theory]] -- \href{perturbation+theory#ReferencesDivergenceConvergence}{On divergence/convergence}} \hypertarget{linfinity_algebra_structure}{}\subsubsection*{{L-infinity algebra structure}}\label{linfinity_algebra_structure} Further identification of [[L-infinity algebra]]-[[structure]] in the [[Feynman amplitudes]]/[[S-matrix]] of [[Lagrangian field theory|Lagrangian]] [[perturbative quantum field theory]]: \begin{itemize}% \item [[Markus Fröb]], \emph{Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories} (\href{https://arxiv.org/abs/1803.10235}{arXiv:1803.10235}) \item [[Alex Arvanitakis]], \emph{The $L_\infty$-algebra of the S-matrix} (\href{https://arxiv.org/abs/1903.05643}{arXiv:1903.05643}) \end{itemize} [[!redirects perturbative quantum field theories]] [[!redirects perturbative field theory]] [[!redirects perturbative field theorys]] [[!redirects perturbative QFT]] [[!redirects perturbative QFTs]] [[!redirects pQFT]] [[!redirects pQFTs]] \end{document}