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\section*{DHR superselection theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{aqft}{}\paragraph*{{AQFT}}\label{aqft}

[[!include AQFT and operator algebra contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{abstract}{Abstract}\dotfill \pageref*{abstract} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_terms_of_local_net_cohomology}{In terms of local net cohomology}\dotfill \pageref*{in_terms_of_local_net_cohomology} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The DHR superselection theory is about [[superselection sectors]] in the [[Haag-Kastler approach]] to [[AQFT]]. As such, it has to state one or more conditions on representations of the given [[Haag-Kastler net]] that specify the representations of the quasi-local algebra that are deemed physically admissible.

It is named after [[Sergio Doplicher]], [[Rudolf Haag]] and [[John Roberts]].

In the following we consider the theory starting with a [[Haag-Kastler vacuum representation]] on [[Minkowski spacetime]].

The DHR condition says that \textbf{all expectation values (of all observables) should approach the vacuum expectation values, uniformly, when the region of measurement is moved away from the origin.} This is one possible abstraction of the situation in a high energy particle collider, where the registered events are localized both in time and in space.

The DHR condition excludes long range forces like electromagnetism from consideration, because, by [[Stokes' theorem]], the electric charge in a finite region can be measured by the flux of the field strength through a sphere of arbitrary large radius. The DHR analysis is of interest nevertheless, because it has reached a certain maturity and is therefore an excellent object to study: see for example the [[Doplicher-Roberts reconstruction theorem]].

\hypertarget{abstract}{}\subsection*{{Abstract}}\label{abstract}

After the definition of admissible representations we collect some notions that will enable us to state a description of all admissible representations using intrinsic properties of the quasi-local algebra. The central concept that we define is that of a \textbf{transportable endomorphism}, these form the objects of a [[category]], the [[DHR category]].

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

We start with a [[Haag-Kastler vacuum representation]] that we assume to be irreducible and [[Haag dual]].

Let $\pi_0$ be the vacuum representation from now on, $K$ denote double cones and $\mathcal{J}_0$ be the [[causal index set]] of double cones, just as $\mathcal{O}$ are bounded open sets and $\mathcal{J}$ the [[causal index set]] of bounded open sets.

Recall that the [[C-star algebra]]

\begin{displaymath}
\mathcal{A} := clo_{\| \cdot \|} \bigl( \bigcup_{\mathcal{O}\in\mathcal{J}}\mathcal{M}(\mathcal{O}) \bigr)
\end{displaymath}
is called the quasi-local algebra of the given net. We will in the same way associate a [[C-star algebra]] to every open region $\mathcal{O}_0$ via

\begin{displaymath}
\mathcal{A}(\mathcal{O}_0) := clo_{\| \cdot \|} \bigl( \bigcup_{\mathcal{O}_0 \supseteq \mathcal{O}\in\mathcal{J}}\mathcal{M}(\mathcal{O}) \bigr)
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
A representation $\pi$ of the local algebra $\mathcal{A}$ is called (DHR) \textbf{admissible} if $\pi | \mathcal{A}(\mathcal{K}^{\perp})$ is unitarily equivalent to $\pi_0 | \mathcal{A}(\mathcal{K}^{\perp})$ for all $K \in \mathcal{J}_0$, that is, for all double cones $K$.

\end{defn}
Note that the equivalence of the representations is required for the causal complement of \emph{all} double cones, not a special one, so that the characterization that ``the representations can not be distinguished at space-like infinity'' is misleading.

\begin{defn}
\label{}\hypertarget{}{}
Recall that a mapping

\begin{displaymath}
\rho: \mathcal{A} \to \mathcal{A}
\end{displaymath}
is called a \textbf{unital endomorphism} if $\rho$ is linear, multiplicative ($\rho(AB) = \rho(A) \rho(B)$) and $\rho(\mathbb{1})=1$.

\end{defn}
We will drop ``unitarily'' from now on.

\begin{remark}
\label{RepresentationsFromEndomorphismsAndViceVersa}\hypertarget{RepresentationsFromEndomorphismsAndViceVersa}{}
For any representation $\pi$ and endomorphism $\rho$ the composition $\pi \circ \rho$ is another representation.

Conversely, under the assumption of [[Haag duality]] every DHR admissible representation $\rho$ comes from an endomorphism this way. For let $A \in \mathcal{A}(K)$ and $B \in \mathcal{A}(K^\pert)$ be any two causally unrelated localized obserbales. Then for $\rho : \mathcal{A} \to \mathcal{B}(\mathcal{H})$ a DHR representation, we have on the one hand

\begin{displaymath}
\rho(A B) = \rho(A) \pi_0(B)
\end{displaymath}
and on the other, due to the causal structure

\begin{displaymath}
\rho(A B) = \rho(B A) = \pi_0(B) \rho(A)
  \,.
\end{displaymath}
This says that $\rho(A) \in \pi_0(\mathcal{A}(K'))'$ and hence by [[Haag duality]] $\cdots \in \pi_0(\mathcal{A}(K))''$. Under the assumption that the [[local net of observables]] takes values in [[von Neumann algebras]] this in turn in $\cdots = \pi_0(\mathcal{A}(K))$ and so $\rho$ indeed factors as an endomrphism of $\mathcal{A}$ followed by the vaccum representation.

\end{remark}
For this reason one can hope to gain some insights into the representations by studying the endomorphisms, while the set of endomorphisms has certainly more structure than that of representations: For example, endomorphisms may have inverses, and endomorphisms form a [[monoid]] by composition.

First we define ``unitarily equivalent'' and ``intertwiner'' analog to the definition for [[representations of C-star algebras]]:

\begin{defn}
\label{}\hypertarget{}{}
Two endomorphisms $\rho_1, \rho_2$ are \textbf{unitarily equivalent} if there is a [[unitary operator]] $U \in \mathcal{A}$ such that $\rho_1 = ad(U) \rho_2 = U \rho_2 U^{-1}$. The endomorphisms $ad(U)$ with $U \in \mathcal{A}$ are called \textbf{inner automorphisms} (of $\mathcal{A}$). An element $R \in \mathcal{A}$ such that $R \rho_1 = \rho_2 R$, not necessarily unitary, is called an \textbf{intertwiner} or \textbf{intertwining operator}.

\end{defn}
The DHR selection criterion selects representations that ``look like the vaccum'' on the causal complement of elements of our index set (in this case the double cones). The analog for endomorphims is this:

\begin{defn}
\label{}\hypertarget{}{}
An endomorphim $\rho$ is \textbf{[[localized endomorphism|localized]]} or \textbf{localizable} if there is a bounded open set $\mathcal{O} \in \mathcal{J}$ such that $\rho$ is the identity on the algebra of the causal complement $\mathcal{A}(\mathcal{O}^{\perp})$. Such an endomorphism is \textbf{localized in $\mathcal{O}$}.

\end{defn}
We have to restrict the notion of ``unitarily equivalent'' to a ``localized'' version accordingly:

\begin{defn}
\label{}\hypertarget{}{}
An element $A \in \mathcal{A}$ is a \textbf{local operator} if there is a double cone K such that $A \in \mathcal{M}(K)$. In particular, a local unitary operator $U$ is a \textbf{local unitary}, and two endomorphisms are \textbf{locally unitarily equivalent} if there is a unitary as in the definition of unitarily equivalent that is a local operator.

\end{defn}
For an endomorphism $\rho$ let $\hat \rho$ be the equivalence class with respect to locally unitarily equivalence. Then we can define:

\begin{defn}
\label{}\hypertarget{}{}
A [[localized endomorphism]] is \textbf{transportable} if for every bounded open set $\mathcal{O} \in \mathcal{J}$ there is a $\rho_0 \in \hat \rho$ that is localized in $\mathcal{O}$.

\end{defn}
Transportable endomorphisms naturally form the objects of a category, the [[DHR category]].

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

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

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{transportable endomorphisms are compatible with the net structure}

Let $\rho$ be a transportable endomorphism that is localized in the double cone $K_0$, then for all double cones $K \supset K_0$, $\rho$ maps $\mathcal{M}(K)$ to itself.

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{intrinsic characterization of admissible representations}

A representation $\pi$ of $\mathcal{A}$ is admissible iff there is a transportable endomorphism $\rho$ such that $\pi$ is unitarily equivalent to $\pi_0 \circ \rho$.

\end{theorem}
Reference: This is theorem 2.1.3 in the book by Baumg\"a{}rtel.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{stability of admissible representations}

(i) finite direct sums of admissible representations are admissible.

(ii) subrepresentations of admissible representations are admissible.

\end{theorem}
For transportable endomorphisms we get even more:

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{products} The product, i.e. the concatenation, of transportable endomorphisms is a transportable endomorphism.

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{product of causally disjoint localized endomorphisms is commutative}

Let $\rho_1, \rho_2$ be transportable endomorphisms localized in $K_1, K_2$ respectivley with $K_1 \perp K_2$, then $\rho_1 \rho_2 = \rho_2 \rho_1$.

\end{theorem}
Note that the product of equivalenc classes is well defined, and $\hat \rho_1 \hat \rho_2 = \widehat{\rho_1\rho_2}$. Therefore the theorem above implies that the product of equivalence classes is commutative.

\hypertarget{in_terms_of_local_net_cohomology}{}\subsubsection*{{In terms of local net cohomology}}\label{in_terms_of_local_net_cohomology}

The DHR superselection theory has a reformulation in terms of [[cohomology of local net of observables|cohomology of local nets of observables]]. For the moment, see there for more.

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

see [[AQFT]] ff.

Of particular relevance (besides the original work of Doplicher and Roberts) are

\begin{itemize}%
\item Hellmut Baumg\"a{}rtel: \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})


\item [[Hans Halvorson]], [[Michael Müger]], \emph{Algebraic Quantum Field Theory} (\href{http://arxiv.org/abs/math-ph/0602036}{arXiv:math-ph/0602036})



\end{itemize}
Discussion in the context of [[holographic entanglement entropy]]:

\begin{itemize}%
\item Horacio Casini, Marina Huerta, Javier M. Magan, Diego Pontello, \emph{Entanglement entropy and superselection sectors I. Global symmetries} (\href{https://arxiv.org/abs/1905.10487}{arXiv:1905.10487})

\end{itemize}
[[!redirects DHR analysis]] [[!redirects Doplicher-Haag-Roberts superselection theory]] [[!redirects Doplicher-Haag-Roberts superselection sector]] [[!redirects Doplicher-Haag-Roberts superselection sectors]]



\end{document}