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\begin{document}

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\section*{causally local net 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{basic_general_definition}{Basic general definition}\dotfill \pageref*{basic_general_definition} \linebreak
\noindent\hyperlink{extra_axioms}{Extra axioms}\dotfill \pageref*{extra_axioms} \linebreak
\noindent\hyperlink{EinsteinLocalities}{Einstein locality}\dotfill \pageref*{EinsteinLocalities} \linebreak
\noindent\hyperlink{StrongLocality}{Strong locality}\dotfill \pageref*{StrongLocality} \linebreak
\noindent\hyperlink{timeslice_axiom}{Time-slice axiom}\dotfill \pageref*{timeslice_axiom} \linebreak
\noindent\hyperlink{duality}{Duality}\dotfill \pageref*{duality} \linebreak
\noindent\hyperlink{positive_energy_condition}{Positive energy condition}\dotfill \pageref*{positive_energy_condition} \linebreak
\noindent\hyperlink{spectrum_condition}{Spectrum condition}\dotfill \pageref*{spectrum_condition} \linebreak
\noindent\hyperlink{special_cases_and_variants}{Special cases and variants}\dotfill \pageref*{special_cases_and_variants} \linebreak
\noindent\hyperlink{minkowski_nets__vacuum_representation}{Minkowski nets / Vacuum representation}\dotfill \pageref*{minkowski_nets__vacuum_representation} \linebreak
\noindent\hyperlink{conformal_nets}{Conformal nets}\dotfill \pageref*{conformal_nets} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{in_perturbation_theory}{In perturbation theory}\dotfill \pageref*{in_perturbation_theory} \linebreak


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

In [[quantum field theory]] (or [[classical field theory]]) a \emph{causally local net of observables} is a system (a [[co-presheaf]]) of [[algebras of observables]] assigned to regions of [[spacetime]], such that this satisfies a few basic properties, such as notably \emph{[[causal locality]]}, saying that observables whose spacetime [[support]] is [[spacelike]]-separated (graded-)commute with each other ([[Poisson bracket|Poisson]]-commute, in the case of classical field theory).

(Beware that the meaning of ``net'' here is vaguely similar to, but different from, the concept of \emph{[[net]]} (as in: generalized [[sequences]] of points) as used in [[topology]]. Better terminology might be ``causally local system of spacetime-localized observables''. But ``local net'' is traditional and has become standard.)

In the context of [[algebraic quantum field theory]] the structure of the local net of quantum observables is used as the very [[axiom|axiomatization]] of what a [[quantum field theory]] actually is (``[[Haag-Kastler axioms]]''). This may be thought of as a formalization of a spacetime-localized form of the [[Heisenberg picture]] of [[quantum physics]] (or rather the [[interaction picture]] in the case of [[perturbative AQFT]]); as opposed to the formalization of the [[Schrödinger picture]] in [[FQFT]], where instead the [[state]]-propagation is used as the basic axiom.

Traditionally local nets of observables are assumed to take values in [[C\emph{-algebras]], but the basic form of the axioms does not actually refer to [[topology|topological]] structure on the algebras, and makes sense more generally.}

In particular in [[perturbative quantum field theory]] made precise via [[causal perturbation theory]], the [[algebras of quantum observables]] are taken to be [[formal power series algebras]] (reflecting the [[infinitesimal]] nature of [[perturbation theory]]) and one \emph{derives} from [[causal additivity]] of the [[S-matrix]] that the perturbative [[quantum observables]] form a local net of formal power series algebras (see at \emph{\href{S-matrix#CausalLocality}{S-matrix -- Causal locality and Quantum observables}}). Accordingly, this infinitesimal/perturbative version of [[AQFT]] is called \emph{[[perturbative AQFT]]}.

Other variants may be considered. For [[AQFT on curved spacetimes]] one generalizes from observables associated with regions of [[Minkowski spacetime]] to observables associated with more general [[globally hyperbolic spacetimes]]. Combining this with perturbation theory is then called \emph{[[locally covariant perturbative AQFT]]}.

Moreover, if [[gauge theory]] with nontrivial global [[gauge field]] configurations is to be considered ([[instantons]]) then one may show that one needs to consider some kind of [[homotopy theory|homotopy theoretic]] local nets of \emph{[[homotopical algebras]]}. See at \emph{[[homotopical algebraic quantum field theory]]}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In the literature there is a certain variance and flexibility of what precisely the axioms on a local net of observables are, though the core aspects are always the same: it is a [[copresheaf]] of ([[C-star algebra|C-star]] [[associative algebra|algebra]] s) on pieces of [[spacetime]] such that algebras assigned to [[causal locality|causally disconnected]] regions commute inside the algebra assigned to any joint neighbourhood.

Historically this was first formulated for [[Minkowski spacetime]] only, where it is known as the [[Haag-Kastler axioms]]. Later it was pointed out (\hyperlink{BrunettiFredenhagen}{BrunettiFredenhagen}) that the axioms easily and usefully generalize to arbitrary [[spacetime]]s.

We give the modern general formulation first, and then comment on its restriction to special situations.

\hypertarget{basic_general_definition}{}\subsubsection*{{Basic general definition}}\label{basic_general_definition}

\begin{defn}
\label{CatOfSpacetimeEmbeddings}\hypertarget{CatOfSpacetimeEmbeddings}{}
Write $LorSp$ for the [[category]] whose

\begin{itemize}%
\item [[object]]s are [[spacetime]] manifolds; ([[Lorentzian manifold]]s equipped with a time-orientation);


\item [[morphism]] are \emph{causal} [[isometry|isometric]] [[embedding]].



\end{itemize}
Here we say a morphism $f : X \hookrightarrow Y$ is a \textbf{causal embedding} if for every two points $x_1,x_2 \in X$ we have that $f(x_1)$ is in the [[future]] of $f(x_2)$ in $Y$ only if $x_1$ is in the future of $x_2$ in $X$.

\end{defn}
Write $Alg$ for a suitable [[category]] of [[associative algebra]]s. Usually this is taken to be the category of [[C-star algebra]]s or that of [[von Neumann algebra]]s. Write

\begin{displaymath}
Alg_{inc} \hookrightarrow Alg
\end{displaymath}
for the [[subcategory]] on the [[monomorphism]]s.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{causally local net of observables} is a [[functor]]

\begin{displaymath}
\mathcal{A} : LorSp \to Alg_{inc} \to Alg
\end{displaymath}
such that

\begin{itemize}%
\item ([[causal locality]]) whenever $X_1 \coprod X_2 \hookrightarrow X$ is a causal embedding, def. \ref{CatOfSpacetimeEmbeddings}, we have that $\mathcal{A}(X_1) \subset \mathcal{A}(X)$ commutes with $\mathcal{A}(X_2) \subset \mathcal{A}(X)$.

\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The [[local quantum field theory|locality]] [[axiom]] encodes the the physical property known as \emph{[[Einstein-causality]]} or \emph{micro-causality}, which states that physical effects do not propagate faster that the speed of light.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Many \emph{auxiliary} [[linear operator|operators]] in [[quantum field theory]] do \emph{not} satisfy causal locality: for instance operators associate to [[current]]s in [[gauge theory]]. The idea is that those operators that actually do qualify as \emph{[[observables]]} do satisfy the axiom, however, i.e. in particular those that are [[gauge symmetry|gauge invariant]].

\end{remark}
\hypertarget{extra_axioms}{}\subsubsection*{{Extra axioms}}\label{extra_axioms}

\hypertarget{EinsteinLocalities}{}\paragraph*{{Einstein locality}}\label{EinsteinLocalities}

Commutativity of spacelike separated observables can be argued to capture only part of [[causal locality]].

A natural stronger requirement is that [[spacelike]] separated regions of [[spacetime]] are literally \href{quantum%20mechanics#Subsystems}{independent quantum subsystems} of any larger region. By the formalization of \emph{independent subsystem} in [[quantum mechanics]] this means the following:

\begin{defn}
\label{EinsteinLocality}\hypertarget{EinsteinLocality}{}
A local net $\mathcal{A}$ satisfies \textbf{Einstein locality} if for every causal embedding $X_1 \coprod X_2 \to X$ the \href{http://ncatlab.org/nlab/show/quantum%20mechanics#Subsystems}{subsystems}

\begin{displaymath}
\mathcal{A}(X_1) \hookrightarrow \mathcal{A}(X)
\end{displaymath}
and

\begin{displaymath}
\mathcal{A}(X_2) \hookrightarrow \mathcal{A}(X)
\end{displaymath}
are \emph{independent} in that the algebra $\mathcal{A}(X_1) \vee \mathcal{A}(X_2) \in \mathcal{A}(X)$ which they generate is [[isomorphism|isomorphic]] to the [[tensor product]] $\mathcal{A}(X_1) \otimes \mathcal{A}(X_2)$.

\end{defn}
This appears as (\hyperlink{BrunettiFredenhagen}{BrunettiFredenhagen, 5.3.1, axiom 4}).

\begin{remark}
\label{}\hypertarget{}{}
A local net is Einstein local precisely if it is a [[monoidal functor]]

\begin{displaymath}
\mathcal{A} : (LorSp, \coprod) \to (Alg, \otimes)
  \,.
\end{displaymath}
\end{remark}
This appears as (\hyperlink{BrunettiFredenhagen}{BrunettiFredenhagen, 5.3.1, theorem 1}).

\begin{remark}
\label{}\hypertarget{}{}
Einstein locality implies [[causal locality]], but is stronger.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Other properties implied by Einstein locality are sometimes extracted as separate axioms. For instance the condition that for $X_1 \coprod X_2 \to X$ a causal embedding, we have

\begin{displaymath}
\mathcal{A}(X_1) \cap \mathcal{A}(X_2) = \mathbb{C}
  \,.
\end{displaymath}
\end{remark}
\hypertarget{StrongLocality}{}\paragraph*{{Strong locality}}\label{StrongLocality}

In (\hyperlink{Nuiten11}{Nuiten 11}) the following variant of [[causal locality]] was considered and shown to be equivalent to a [[descent]] condition for the system of [[Bohr toposes]] associated with a local net of observables

\begin{defn}
\label{StrongLocality}\hypertarget{StrongLocality}{}
\textbf{([[strong causal locality]])}

A net of observables is \textbf{[[strong causal locality|strongly (causally) local]]} if it is microlocal (i.e. [[causal locality|causally local]]) in that algebras $A_1 = A(O_1)$ and $A_2 = A(O_2)$ associated with spacelike separated regions commute with each other, and in addition for all commutative subalgebras $C_1 \subset A_1$ and $C_2 \subset A_2$ the algebra $C_1 \vee C_2 \subset A(O_1 \vee O_2)$ satisfies

\begin{enumerate}%
\item $(C_1 \vee C_2) \cap A_1 = C_1$


\item $(C_1 \vee C_2) \cap A_2 = C_2$.



\end{enumerate}
\end{defn}
This is (\hyperlink{Nuiten11}{Nuiten 11, def. 14}).

\begin{remark}
\label{}\hypertarget{}{}
It is clear that Einstein locality implies strong locality, def. \ref{EinsteinLocality}

\begin{displaymath}
Einstein\;locality \;\;\Rightarrow  \;\; Strong\;locality
  \,.
\end{displaymath}
In fact strong locality is strictly weaker than Einstein locality in that there are strongly locally embedded subalgebras which are not Einstein locally embedded. More discussion of this is in (\hyperlink{Wolters13}{Wolters 13, section 6.3.3}).

\end{remark}
\hypertarget{timeslice_axiom}{}\paragraph*{{Time-slice axiom}}\label{timeslice_axiom}

\begin{defn}
\label{}\hypertarget{}{}
A local net is said to satisfy the \textbf{[[time slice axiom]]} if whenever

\begin{displaymath}
i : X_1 \to X_2
\end{displaymath}
is a causal embedding of [[globally hyperbolic]] [[spacetime]]s such that $X_1$ contains a [[Cauchy surface]] of $X_2$, then

\begin{displaymath}
\mathcal{A}(i) : \mathcal{A}(X_1) \stackrel{\simeq}{\to}
   \mathcal{A}(X_2)
\end{displaymath}
is an [[isomorphism]].

\end{defn}
\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

See [[dual net of von Neumann algebras]]

\hypertarget{positive_energy_condition}{}\paragraph*{{Positive energy condition}}\label{positive_energy_condition}

(\ldots{})

\hypertarget{spectrum_condition}{}\paragraph*{{Spectrum condition}}\label{spectrum_condition}

(\ldots{})

\hypertarget{special_cases_and_variants}{}\subsubsection*{{Special cases and variants}}\label{special_cases_and_variants}

\hypertarget{minkowski_nets__vacuum_representation}{}\paragraph*{{Minkowski nets / Vacuum representation}}\label{minkowski_nets__vacuum_representation}

\begin{itemize}%
\item [[Haag-Kastler vacuum representation]]

\end{itemize}
\hypertarget{conformal_nets}{}\paragraph*{{Conformal nets}}\label{conformal_nets}

The notion of local net in the context of [[conformal field theory]] is a [[conformal net]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

see for instance (\hyperlink{BarGinoux11}{B\"a{}r-Ginoux 11})

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

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

\begin{itemize}%
\item [[local quantum field theory]], [[local prequantum field theory]]


\item [[causal locality]], [[Einstein locality]]


\item [[AQFT]]

\begin{itemize}%
\item [[Haag-Kastler axioms]]


\item \textbf{local net of observables}

\begin{itemize}%
\item [[net of C-star systems]] ([[global gauge group]], [[actions]])


\item [[field net]]


\item [[cohomology of local net of observables]]


\item [[factorization algebra of observables]], [[chiral algebra]], [[topological chiral homology]], [[blob complex]].



\end{itemize}


\end{itemize}

\item [[FQFT]]



\end{itemize}


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

For more details see the references at [[AQFT]].

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The axioms of local nets on general spacetimes were first articulated in

\begin{itemize}%
\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], \emph{Quantum Field Theory on Curved Backgrounds} (\href{http://arxiv.org/abs/0901.2063}{arXiv:0901.2063})

\end{itemize}
A comprehensive review, with plenty of background information, is in

\begin{itemize}%
\item [[Christian Bär]], [[Klaus Fredenhagen]], (eds.) \emph{Quantum field theory on curved spacetime} , Lecture notes in physics, Springer (2009)

\end{itemize}
Discussion of Einstein locality of a net of observables equivalently as a [[descent]] condition on the system of [[Bohr toposes]] induced by the [[algebras of observables]] is in

\begin{itemize}%
\item [[Joost Nuiten]], \emph{[[schreiber:bachelor thesis Nuiten|Bohrification of local nets of observables]]}, Proceedings of \href{http://qpl.science.ru.nl/}{QPL 2011}, \href{http://rvg.web.cse.unsw.edu.au/eptcs/content.cgi?QPL2011}{EPTCS 95, 2012}, pp. 211-218 (\href{http://arxiv.org/abs/1109.1397}{arXiv:1109.1397})

\end{itemize}
A review of this with some further discussion is in section 6 of

\begin{itemize}%
\item Sander Wolters, \emph{Quantum toposophy}, PhD Thesis 2013 (\href{http://www.math.ru.nl/~landsman/Wolters.pdf}{pdf})

\end{itemize}
Discussion of properly co-stacky nets of local observables in [[gauge theory]] is in

\begin{itemize}%
\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})

\end{itemize}
An [[operad]] for local nets of observables in [[AQFT]] is considered in

\begin{itemize}%
\item [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], \emph{Operads for algebraic quantum field theory} (\href{https://arxiv.org/abs/1709.08657}{arXiv:1709.08657})

\end{itemize}
The [[Boardman-Vogt resolution]] of the [[operad]] for [[local nets of observables]] (\hyperlink{BeniniSchenkelWoike17}{Benini-Schenkel-Woike 17}), lifting it to [[homotopy AQFT]], is considered in

\begin{itemize}%
\item [[Donald Yau]], \emph{Homotopical Quantum Field Theory} (\href{https://arxiv.org/abs/1802.08101}{arXiv:1802.08101})

\end{itemize}
\hypertarget{examples_2}{}\subsubsection*{{Examples}}\label{examples_2}

Discussion of standard example includes

\begin{itemize}%
\item [[Christian Bär]], N. Ginoux, \emph{Classical and quantum fields on lorentzian manifolds}, in \emph{Global Differential Geometry}, Springer Proceedings in Math. 17, Springer-Verlag (2011), 359-400 (\href{http://arxiv.org/abs/1104.1158}{arXiv:1104.1158})

\end{itemize}
\hypertarget{in_perturbation_theory}{}\subsubsection*{{In perturbation theory}}\label{in_perturbation_theory}

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) (\href{http://inspirehep.net/record/135575}{spire}, \href{http://dx.doi.org/10.1007/BF01035870}{doi})

\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).

For more on this see at \emph{[[S-matrix]]} and at \emph{[[pAQFT]]}.

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