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\section*{causal perturbation 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{MainTheoremOfRenormalization}{Main theorem of renormalization}\dotfill \pageref*{MainTheoremOfRenormalization} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{on_minkowski_spacetime}{On Minkowski spacetime}\dotfill \pageref*{on_minkowski_spacetime} \linebreak
\noindent\hyperlink{ReferencesOnCurvedSpacetimes}{On curved spacetimes}\dotfill \pageref*{ReferencesOnCurvedSpacetimes} \linebreak
\noindent\hyperlink{local_covariance}{Local covariance}\dotfill \pageref*{local_covariance} \linebreak


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

What is called \emph{causal perturbation theory} is a mathematically rigorous construction of \emph{[[perturbative quantum field theory|perturbative]]} ([[gauge theory|gauge]]-) [[quantum field theory]], such as [[quantum electrodynamics]], based on a mathematical formulation of [[renormalization]] by St\"u{}ckelberg-Bogoliubov-Epstein-Glaser (\hyperlink{EpsteinGlaser73}{Epstein-Glaser 73}).

Causal perturbation theory may be regarded as providing a well-defined consistent generalization from [[quantum mechanics]] to [[quantum field theory]] on [[Lorentzian manifold|Lorentzian]] [[spacetimes]] of the construction of the [[S-matrix]] via the [[Dyson formula]] (``[[time-ordered products]]'') in the [[interaction picture]] .

A key idea of causal perturbation theory is that the [[interaction]] term $V$ is considered multiplied with some [[smooth function]] $g$ which has [[compact support]] on [[spacetime]]. This hence serves as a spacetime-dependent ``[[coupling constant]]'' which ``switches off'' the interaction outside a compact region, but not discontinuously as in many other schemes, but smoothly, hence ``adiabatically'' in terminology borrowed from [[thermodynamics]]. Therefore this is often called the ``[[adiabatic switching]]'' function, or similar.

The corresponding [[S-matrix]] would naively be given by the [[Dyson formula]]

\begin{displaymath}
S_g \coloneqq T \exp\left( \int_X g(x) V(x) dvol(x) \right)
\end{displaymath}
for $V$ the [[interaction]] term, and ``$T$'' indicating the [[time-ordered product]]. Causal perturbation theory proceeds by [[axiom|axiomatizing]] the key structural properties of this ``[[adiabatic switching|adiabatically switched]]'' [[S-matrix]], in particular its \emph{[[causal additivity]]}, making sense of the ``[[time-ordered product]]'' by appropriate [[causal locality|causal ordering]] (whence the name of the approach) and then proving by [[induction]] that solutions to these axioms exist.

It turns out that at each step in the [[induction]] (corresponding to each loop order) a [[distribution]] has to be \href{extension%20of%20distributions#SolutionSpaceOfPointExtensions}{extended to a point}, namely to the point at which the [[interaction]] takes place. A key theorem (\hyperlink{EpsteinGlaser73}{Epstein-Glaser 73, section 5}, \href{extension+of+distributions#SpaceOfPointExtensions}{this prop.}) states that this extension is parameterized by a finite-dimensional space of [[point-supported distributions]] at that point. One may organize the distributions that need to be extended into [[Feynman diagrams]] (\hyperlink{DuetschFredenhagenKellerRejzner14}{D\"u{}tsch-Fredenhagen-Keller-Rejzner 14}). This way the freedom in [[extension of distributions]] is identified with the traditional [[renormalization]] freedom in [[perturbative quantum field theory]], see at \emph{[[main theorem of perturbative renormalization]]} for more on this point.

(It may be argued, vividly so in \hyperlink{Scharf95}{Scharf 95, p. 181-182}, that the notorious ``infinities'' that ``plague'' quantum field theory in other approaches are nothing but the result of incorrectly dealing with the issue of [[extension of distributions]].)

In fact the [[interacting field algebra]] induced by the [[S-matrix]] constructed via causal perturbation theory this way is a model for [[Fedosov's formal deformation quantization]] of the given [[local Lagrangian density]] (\hyperlink{Collini16}{Collini 16}, \hyperlink{HawkinsRejzner16}{Hawkins-Rejzner 16}), thus justifying the method from first principles of (perturbative) [[quantization]].

The key axiom imposed on the [[S-matrix]] in causal perturbation theory is [[causal locality]], whence the name of the approach. This axiom asks that if adiabatic switching functions $g$ and $h$ have [[spacelike]] separated [[supports]] then the S-matrix \emph{factors}

\begin{displaymath}
S_{g + h} = S_g S_h
  \,.
\end{displaymath}
The [[time-ordered products]] $T(..)$ of [[field (physics)|fields]] are handled by splitting of [[distributions]] using tools from [[microlocal analysis]]. This is the mathematically rigorous step that takes care of what in other approaches are the ``[[ultraviolet divergences]]''.

Originally the idea was that in the end the [[limit of a sequence|limit]] $g \to 1$ had to be taken, removing the adiabatic switching of the the compact support of the interaction. In general this limit does not exist (``[[infrared divergences]]'', e.g. \hyperlink{AAS10}{AAS 10, section 6}).

But in (\hyperlink{IlinSlavnov78}{Il'in-Slavnov 78}) it was observed that in fact the [[algebra of observables]] on any [[bounded subsets|bounded]] region may be computed with a $g$ whose [[compact support]] contains the [[causal closure]] of that region. Later in (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00}) it was pointed out that this means that causal perturbation theory in fact serves to construct the [[causally local net of observables]] of the [[perturbative quantum field theory]] (see \href{S-matrix#PerturbativeQuantumObservablesIsLocalnet}{this prop} for details), as in the [[axioms]] for [[AQFT]] and in fact as in the axioms of [[AQFT on curved spacetimes]], but with values in [[formal power series algebras]] (as befits a [[perturbation theory]]) instead of [[C\emph{-algebras]] (suitable for [[non-perturbative quantum field theory]]).}

(This takes care of the ``[[algebraic adiabatic limit]]'', which defines the [[quantum observables]]. It does not yet in itself define the ``[[weak adiabatic limit]]'' for the would-be [[vacuum]] [[quantum state]].)

This way causal perturbation theory leads to a unification of [[AQFT]] [[AQFT on curved spacetimes]] with [[perturbative quantum field theory]]. This unification is now known as \textbf{[[locally covariant perturbative quantum field theory]]}, see there for more.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{MainTheoremOfRenormalization}{}\subsubsection*{{Main theorem of renormalization}}\label{MainTheoremOfRenormalization}

A central result in causal perturbation theory is called the \emph{[[main theorem of perturbative renormalization theory]]}. This says that any two \emph{renormalization schemes}, hence any two solutions to the inductive construction of the [[S-matrix]] $V \mapsto S(V)$, as indicated above, for [[interaction]] terms $V$, are related by a unique natural transformation $Z \colon V \to V'$

\begin{displaymath}
S' = S \circ Z
  \,.
\end{displaymath}
The collection of these operations $Z$ forms a [[group]] called the \emph{[[Stückelberg-Petermann renormalization group]]}.

This is a mathematical reflection of the idea that renormalization is about regarding a [[perturbative quantum field theory]] with [[interaction]] $V$ as a [[effective field theory]] at some energy scale and then discovering that at higher energy there is a more fundamental interaction $Z(V)$ which effectively looks like $V$ at lower energy.

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

\begin{itemize}%
\item [[locally covariant perturbative quantum field theory]]

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

Lecture notes in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[geometry of physics -- perturbative quantum field theory]]}

\end{itemize}
The method is due to

\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}
whence sometimes called the \emph{St\"u{}ckelberg-Bogoliubov-Epstein-Glaser method}.

The expression of causal perturbation theory in terms of [[Feynman diagram]] techniques is due to

\begin{itemize}%
\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}
That causal perturbation theory (in the generality of curved spacetimes, see \hyperlink{ReferencesOnCurvedSpacetimes}{below}) is equivalently the ([[Fedosov deformation quantization|Fedosov]]-)[[formal deformation quantization]] of the interacting Lagrangian density was shown for the [[scalar field]] [[phi{\tt \symbol{94}}4 theory]] in

\begin{itemize}%
\item [[Giovanni Collini]], \emph{Fedosov Quantization and Perturbative Quantum Field Theory} (\href{https://arxiv.org/abs/1603.09626}{arXiv:1603.09626})

\end{itemize}
and for the interacting scalar field in the toy example of regular non-local interactions in

\begin{itemize}%
\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}
\hypertarget{on_minkowski_spacetime}{}\subsubsection*{{On Minkowski spacetime}}\label{on_minkowski_spacetime}

Causal perturbation theory has been worked out in detail for the example of [[quantum electrodynamics]] on [[Minkowski spacetime]] in

\begin{itemize}%
\item [[Günter Scharf]], \emph{[[Finite Quantum Electrodynamics -- The Causal Approach]]}, Berlin: Springer-Verlag, 1995, 2nd edition

\end{itemize}
and for [[quantum chromodynamics]] and [[perturbative quantum gravity]] on [[Minkowski spacetime]] in

\begin{itemize}%
\item [[Günter Scharf]], \emph{[[Quantum Gauge Theories -- A True Ghost Story]]}, Wiley 2001

\end{itemize}
Non-technical exposition of this includes

\begin{itemize}%
\item [[Arnold Neumaier]], \emph{\href{https://www.physicsforums.com/insights/causal-perturbation-theory/}{Causal Perturbation Theory}} (2015)


\item [[Arnold Neumaier]], \emph{\href{https://www.mat.univie.ac.at/~neum/physfaq/topics/causalQFT.html}{Action-based quantum field theory and causal perturbation theory}} (one section in \emph{\href{http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html}{A theoretical physics FAQ}})



\end{itemize}
Technical review includes

\begin{itemize}%
\item Andreas Aste, Cyrill von Arx, [[Günter Scharf]], \emph{Regularization in quantum field theory from the causal point of view}, Prog. Part. Nucl. Phys.64:61-119, 2010 (\href{https://arxiv.org/abs/0906.1952}{arXiv:0906.1952})


\item Christian P\"o{}selt, \emph{The method of Epstein and Glaser I}, 2002 (\href{http://wwwthep.physik.uni-mainz.de/~scheck/Poeselt.pdf}{pdf})



\end{itemize}
\hypertarget{ReferencesOnCurvedSpacetimes}{}\subsubsection*{{On curved spacetimes}}\label{ReferencesOnCurvedSpacetimes}

The generalization of causal perturbation theory to [[quantum field theory on curved spacetimes]] is developed 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}, Communications in Mathematical Physics, 223(2):289--326, 2001 (\href{https://arxiv.org/abs/gr-qc/0103074}{arXiv:gr-qc/0103074})


\item [[Stefan Hollands]], [[Robert Wald]], \emph{Existence of local covariant time ordered products of quantum fields in curved spacetime}, Communications in Mathematical Physics, 231(2):309--345, 2002 (\href{https://arxiv.org/abs/gr-qc/0111108}{arXiv:gr-qc/0111108})


\item [[Stefan Hollands]], [[Robert Wald]], \emph{On the renormalization group in curved spacetime}, Communications in Mathematical Physics, 237(1-2):123--160, 2003 (\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}, Reviews in Mathematical Physics, 17(03):227--311, 2005 (\href{https://arxiv.org/abs/gr-qc/0404074}{arXi:gr-qc/0404074})


\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}
Review is in

\begin{itemize}%
\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})

\end{itemize}
\hypertarget{local_covariance}{}\subsubsection*{{Local covariance}}\label{local_covariance}

The observation that the method of causal perturbation theory naturally leads to [[locally covariant perturbative quantum field theory]] is due to

\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 was re-discovered and then popularized in

\begin{itemize}%
\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], section 8 of \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})

\end{itemize}
Review includes

\begin{itemize}%
\item [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], section 4.1 of \emph{Perturbative Construction of Models of Algebraic Quantum Field Theory} (\href{https://arxiv.org/abs/1503.07814}{arXiv:1503.07814})

\end{itemize}
For more on this see the references at \emph{[[locally covariant perturbative quantum field theory]]}.

[[!redirects causal perturbation theories]]



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