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\begin{document}

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\section*{DHR category}

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

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

[[!include AQFT and operator algebra contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\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{dhr_is_a_cstarcategory_with_a_direct_product}{DHR is a C-star-category with a direct product}\dotfill \pageref*{dhr_is_a_cstarcategory_with_a_direct_product} \linebreak
\noindent\hyperlink{dhr_is_a_symmetric_monoidal_category}{DHR is a symmetric monoidal category}\dotfill \pageref*{dhr_is_a_symmetric_monoidal_category} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[quantum field theory]] presented by a [[local net of observables]] ([[AQFT]])

\begin{displaymath}
\mathcal{A} : Open(X) \to Algebras
\end{displaymath}
a \emph{local endomorphism} is a [[natural transformation|natural]] [[associative algebra|algebra]] [[homomorphism]] $\rho : \mathcal{A} \to \mathcal{A}$ which is supported (nontrivial) on a [[compact space|compact]] region of [[spacetime]] $X$.

These \emph{local endomorphism} are physically interpreted as local [[charges]]. By the locality of the local net, one finds that local endomorphisms natural form a [[braided monoidal category]]. This is called the \emph{DHR category}.

The DHR category is thus built from data used in [[DHR superselection theory]] and is used to provide a simplified proof of the [[Doplicher-Roberts reconstruction theorem]].

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

After the definition of objects and arrows we show several structures that the DHR category has.

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

See [[DHR superselection theory]] and [[Haag-Kastler vacuum representation]] for the terminology used here.

\begin{defn}
\label{}\hypertarget{}{}
The transportable endomorphisms are the \textbf{[[objects]]} of the DHR category $\Delta$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
For two transportable endomorphisms the \textbf{set of [[intertwiners]]} are the \textbf{[[morphisms]]}.

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

\hypertarget{dhr_is_a_cstarcategory_with_a_direct_product}{}\subsubsection*{{DHR is a C-star-category with a direct product}}\label{dhr_is_a_cstarcategory_with_a_direct_product}

It is straightforward to see that $\Delta$ is a [[category]]:

The [[identity]] morphism for each object in $\Delta$ is given by the identiy in $\mathcal{A}$. The [[composition]] of arrows is simply the composition of [[intertwiners]]:

From

\begin{displaymath}
R \rho_1 = \rho_2 R
\end{displaymath}
\begin{displaymath}
T \rho_2 = \rho_3 T
\end{displaymath}
follows

\begin{displaymath}
T R \rho_1 = \rho_3 T R
\end{displaymath}
Several structural properties follow immediatly from the definition:

\begin{lemma}
\label{}\hypertarget{}{}
$\Delta$ is a $\mathbb{C}-$[[algebroid]].

\end{lemma}
\begin{ulemma}
$\Delta$ is a [[dagger-category]] since, if $R$ is an [[intertwiner]] of the pair $(\rho_1, \rho_2)$, then $R^*$ is obviously an [[intertwiner]] of the pair $(\rho_2, \rho_1)$.

\end{ulemma}
Combining these two structures we get that $\Delta$ is a [[star-category]].

Since the arrows inherit a norm, we actually get

\begin{lemma}
\label{}\hypertarget{}{}
$\Delta$ is a [[C-star-category]].

\end{lemma}
\begin{prop}
\label{}\hypertarget{}{}
It is possible to introduce a finite [[direct product]] in $\Delta$, if the net satisfies the [[Borchers property]].

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
The [[Haag-Kastler vacuum representation]] that we talk about here satisfies the [[Borchers property]].

\end{remark}
\begin{proof}
Let $\pi_1, \pi_2$ be admissible representations and $\rho_1, \rho_2$ be their transportable [[localized endomorphism|endomorphisms localized]] in $K_1, K_2$ respectively. Choose a double cone $K_0 \in \mathcal{J}_0$ that contains $K_1$ and $K_2$. Since the local von Neumann algebra $\mathcal{M}(K_0)$ is not trivial, it contains a nontrivial projection $E$, that is $0 \lt E \lt \mathbb{1}$.

Thanks to the Borchers property there is a double cone $K$ containing the closure of $K_0$, and partial isometries $W_1, W_2 \in \mathcal{M}(K)$ such that $W_1 W_1^* = E, W_2 W_2^* = \mathbb{1} - E$.

Now we set

\begin{displaymath}
\rho := W_1 \rho_1 W_1^* + W_2 \rho_2 W_2^*
\end{displaymath}
It is possible to show that $\pi_0 \rho$ is unitarily equivalent to $\pi_1 \oplus \pi_2$ and that $\rho$ is a transportable (and therefore in particular a localized) endomorphism. So we will call $\rho$ a \textbf{direct sum} of $\rho_1$ and $\rho_2$.

\end{proof}
\hypertarget{dhr_is_a_symmetric_monoidal_category}{}\subsubsection*{{DHR is a symmetric monoidal category}}\label{dhr_is_a_symmetric_monoidal_category}

We first define the ``[[tensor product]]'':

\begin{defn}
\label{}\hypertarget{}{}
For endomorphisms we set $\rho_1 \otimes \rho_2 := \rho_1 \rho_2$.

For intertwiners $S \in Hom(\rho, \rho^{\prime})$ and $T \in Hom(\sigma, \sigma^{\prime})$ we define the tensor product via $S \otimes T := S \rho(T)$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In the [[AQFT]] literature the tensor product of arrows is sometimes called the \emph{crossed product} of intertwiners.

\end{remark}
\begin{lemma}
\label{}\hypertarget{}{}
The tensor product as defined above turns $\Delta$ into a [[monoidal category]].

\end{lemma}
\begin{proof}
First: The tensor product of arrows is well defined, for any $A \in \mathcal{A}$ we have:

\begin{displaymath}
S \otimes T (\rho_1 \otimes \rho_2) (A) 

= (S \rho(T)) \rho (\sigma(A)) 

= S \rho(T \sigma(A))

= \rho^{\prime} (T \sigma(A)) S

= \rho^{\prime} (\sigma^{\prime}(A) T) S

= \rho^{\prime} \sigma^{\prime}(A) \rho^{\prime}(T) S

= \rho^{\prime} \sigma^{\prime}(A) S \rho(T)
\end{displaymath}
which shows that the tensor product of intertwiners is an intertwiner for the tensor product of endomorphisms. The unit object is the identity endomorphism $\mathbb{1} \in \mathcal{A}$, left and right unitor and the associator are the identities, that is, $\Delta$ is \textbf{strict}.

\end{proof}
Now to the braiding. The braiding is [[symmetric monoidal category|symmetric]] in $d \ge 3$ dimensions only, this has a topological reason: The causal complement of a double cone is pathwise connected in $d \ge 3$ dimensions only, but not in $d \le 2$ dimensions.

\begin{remark}
\label{}\hypertarget{}{}
When we talk about $d = 1$ dimensions we mean one space dimension and zero time dimensions, so that a double cone in this context is simply an open interval, and spacelike separated double cones are simply disjoint open intervals.

To define the braiding we will need the following concepts:

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
For transportable endomorphisms $\rho, \sigma$ choose causally separated double cones $K_1 \perp K_2$ and $\rho_0 \in \hat \rho$ localized in $K_1$ and $\sigma_0 \in \hat \sigma$ localized in $K_2$. These endomorphisms $\rho_0, \sigma_0$ are then called \textbf{spectator endomorphisms} of $\rho$ and $\sigma$.

\end{defn}
Note that it is a theorem that causally disjoint transportable endomorphisms commute, therefore spectator endomorphisms always commute.

\begin{defn}
\label{}\hypertarget{}{}
For transportable endomorphisms $\rho, \sigma$ and spectator endomorphisms $\rho_0, \sigma_0$ choose unitary interwiners $U \in Hom(\rho, \rho_0)$ and $V \in \Hom(\sigma, \sigma_0)$. Such unitaries are called \textbf{transporters}.

\end{defn}
Obviously both spectator endomorphisms and transporters are not unique, in general.

\begin{defn}
\label{}\hypertarget{}{}
For transportable endomorphisms $\rho, \sigma$, spectator endomorphisms $\rho_0, \sigma_0$ and transporters U, V define the \textbf{permutator} or \textbf{permutation symmetry} via

\begin{displaymath}
\epsilon(\rho, \sigma) := (V^* \otimes U^*) (U \otimes V)
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The permutators are well defined and independent of the choice of spectator endomorphisms and transporters.

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

See at \emph{[[DHR superselection theory]]}.

[[!redirects DHR categories]]



\end{document}