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\begin{document}

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\section*{interacting field algebra of observables}

\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{causal_locality_of_interacting_field_quantum_observables}{Causal locality of interacting field quantum observables}\dotfill \pageref*{causal_locality_of_interacting_field_quantum_observables} \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}

In [[perturbative quantum field theory]] the [[algebra of observables]] of an [[interacting field theory]] constructed as a [[perturbation]] of the [[Wick algebra]] [[algebra of observables|of observables]] of a [[free field theory]] is called, for emphasis, the \emph{interacting field algebra of observables}, often just ``interacting field algebra'', for short.

In terms of [[causal perturbation theory]], the interacting field algebra is obtained from the free field [[Wick algebra]] of observables and the [[perturbative S-matrix]] by [[differentiation|differentiating]] \emph{[[Bogoliubov's formula]]}, yielding a \emph{[[Møller operator]]}.

More abstractly, the algebra of observables is the [[formal deformation quantization]] (specifically [[Fedosov deformation quantization]]) of the [[interacting field theory]] (\hyperlink{Collini16}{Collini 16}, \hyperlink{HawkinsRejzner16}{Hawkins-Rejzner 16}).

$\,$

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

\hypertarget{causal_locality_of_interacting_field_quantum_observables}{}\subsubsection*{{Causal locality of interacting field quantum observables}}\label{causal_locality_of_interacting_field_quantum_observables}

\begin{prop}
\label{CausalLocalityOfBogoliubovFormula}\hypertarget{CausalLocalityOfBogoliubovFormula}{}
\textbf{([[causal locality]])}

As the spacetime support varies, the algebras of [[interacting field quantum observables]] spanned via the Bogoliubov formula consistitute a [[causally local net of observables]], hence an instance of [[perturbative AQFT]].

\end{prop}
(\hyperlink{DuetschFredenhagen00}{D\"u{}tsch-Fredenhagen 00, section 3}, following \hyperlink{BrunettiFredenhagen99}{Brunetti-Fredenhagen 99, section 8}, \hyperlink{IlinSlavnov78}{Il'in-Slavnov 78})

For \textbf{[[proof]]} see \href{S-matrix#PerturbativeQuantumObservablesIsLocalnet}{this prop.} at \emph{[[S-matrix]]}.

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

\begin{itemize}%
\item [[Bogoliubov's formula]]


\item [[quantum Møller operator]]


\item [[interaction picture]]



\end{itemize}
[[!include products in pQFT -- table]]

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

The observation that the pertruabtive interacting field quantum observables form a [[causally local net of quantum observables]] 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}
then rediscovered 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})

\end{itemize}
and made more explicit 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})

\end{itemize}
The observation that these algebras are the [[formal deformation quantization]] of the interacting field theory is due to

\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}
[[!redirects interacting field algebras of observables]]

[[!redirects quantum observable on interacting fields]] [[!redirects quantum observables on interacting fields]]

[[!redirects interacting field algebra of quantum observables]] [[!redirects interacting field algebras of quantum observables]]

[[!redirects interacting quantum observable]] [[!redirects interacting quantum observables]]

[[!redirects interacting field quantum observable]] [[!redirects interacting field quantum observables]]

[[!redirects interacting field observable]] [[!redirects interacting field observables]]

[[!redirects interacting field algebra]] [[!redirects interacting field algebras]]

[[!redirects interacting observable algebra]] [[!redirects interacting observable algebras]]

[[!redirects interacting quantum field observable]] [[!redirects interacting quantum field observables]]

[[!redirects local interacting field observable]] [[!redirects local interacting field observables]]

[[!redirects interacting local observable]] [[!redirects interacting local observables]]



\end{document}