# Contents

## Idea

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

$𝒜:\mathrm{Open}\left(X\right)\to \mathrm{Algebras}$\mathcal{A} : Open(X) \to Algebras

a local endomorphism is a natural algebra homomorphism $\rho :𝒜\to 𝒜$ which is supported (nontrivial) on a compact region of spacetime $X$.

These 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 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.

## Abstract

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

## Definition

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

###### Definition

The transportable endomorphisms are the objects of the DHR category $\Delta$.

###### Definition

For two transportable endomorphisms the set of intertwiners are the morphisms.

## Properties

### DHR is a C-star-category 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 $𝒜$. The composition of arrows is simply the composition of intertwiners:

From

$R{\rho }_{1}={\rho }_{2}R$R \rho_1 = \rho_2 R
$T{\rho }_{2}={\rho }_{3}T$T \rho_2 = \rho_3 T

follows

$TR{\rho }_{1}={\rho }_{3}TR$T R \rho_1 = \rho_3 T R

Several structural properties follow immediatly from the definition:

###### Lemma

$\Delta$ is a $ℂ-$algebroid.

###### Lemma

$\Delta$ is a dagger-category since, if $R$ is an intertwiner of the pair $\left({\rho }_{1},{\rho }_{2}\right)$, then ${R}^{*}$ is obviously an intertwiner of the pair $\left({\rho }_{2},{\rho }_{1}\right)$.

Combining these two structures we get that $\Delta$ is a star-category.

Since the arrows inherit a norm, we actually get

###### Lemma

$\Delta$ is a C-star-category.

###### Proposition

It is possible to introduce a finite direct product in $\Delta$, if the net satisfies the Borchers property.

###### Remark

The Haag-Kastler vacuum representation that we talk about here satisfies the Borchers property.

###### Sketch of the Proof

Let ${\pi }_{1},{\pi }_{2}$ be admissible representations and ${\rho }_{1},{\rho }_{2}$ be their transportable endomorphisms localized in ${K}_{1},{K}_{2}$ respectively. Choose a double cone ${K}_{0}\in {𝒥}_{0}$ that contains ${K}_{1}$ and ${K}_{2}$. Since the local von Neumann algebra $ℳ\left({K}_{0}\right)$ is not trivial, it contains a nontrivial projection $E$, that is $0.

Thanks to the Borchers property there is a double cone $K$ containing the closure of ${K}_{0}$, and partial isometries ${W}_{1},{W}_{2}\in ℳ\left(K\right)$ such that ${W}_{1}{W}_{1}^{*}=E,{W}_{2}{W}_{2}^{*}=𝟙-E$.

Now we set

$\rho :={W}_{1}{\rho }_{1}{W}_{1}^{*}+{W}_{2}{\rho }_{2}{W}_{2}^{*}$\rho := W_1 \rho_1 W_1^* + W_2 \rho_2 W_2^*

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 direct sum of ${\rho }_{1}$ and ${\rho }_{2}$.

### DHR is a symmetric monoidal category

We first define the “tensor product”:

###### Definition

For endomorphisms we set ${\rho }_{1}\otimes {\rho }_{2}:={\rho }_{1}{\rho }_{2}$.

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

###### Remark

In the AQFT literature the tensor product of arrows is sometimes called the crossed product of intertwiners.

###### Lemma

The tensor product as defined above turns $\Delta$ into a monoidal category.

###### Proof

First: The tensor product of arrows is well defined, for any $A\in 𝒜$ we have:

$S\otimes T\left({\rho }_{1}\otimes {\rho }_{2}\right)\left(A\right)=\left(S\rho \left(T\right)\right)\rho \left(\sigma \left(A\right)\right)=S\rho \left(T\sigma \left(A\right)\right)={\rho }^{\prime }\left(T\sigma \left(A\right)\right)S={\rho }^{\prime }\left({\sigma }^{\prime }\left(A\right)T\right)S={\rho }^{\prime }{\sigma }^{\prime }\left(A\right){\rho }^{\prime }\left(T\right)S={\rho }^{\prime }{\sigma }^{\prime }\left(A\right)S\rho \left(T\right)$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)

which shows that the tensor product of intertwiners is an intertwiner for the tensor product of endomorphisms. The unit object is the identity endomorphism $𝟙\in 𝒜$, left and right unitor and the associator are the identities, that is, $\Delta$ is strict.

Now to the braiding. The braiding is 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.

###### Remark

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:

###### Definition

For transportable endomorphisms $\rho ,\sigma$ choose causally separated double cones ${K}_{1}\perp {K}_{2}$ and ${\rho }_{0}\in \stackrel{^}{\rho }$ localized in ${K}_{1}$ and ${\sigma }_{0}\in \stackrel{^}{\sigma }$ localized in ${K}_{2}$. These endomorphisms ${\rho }_{0},{\sigma }_{0}$ are then called spectator endomorphisms of $\rho$ and $\sigma$.

Note that it is a theorem that causally disjoint transportable endomorphisms commute, therefore spectator endomorphisms always commute.

###### Definition

For transportable endomorphisms $\rho ,\sigma$ and spectator endomorphisms ${\rho }_{0},{\sigma }_{0}$ choose unitary interwiners $U\in \mathrm{Hom}\left(\rho ,{\rho }_{0}\right)$ and $V\in Hom\left(\sigma ,{\sigma }_{0}\right)$. Such unitaries are called transporters.

Obviously both spectator endomorphisms and transporters are not unique, in general.

###### Definition

For transportable endomorphisms $\rho ,\sigma$, spectator endomorphisms ${\rho }_{0},{\sigma }_{0}$ and transporters U, V define the permutator or permutation symmetry via

$ϵ\left(\rho ,\sigma \right):=\left({V}^{*}\otimes {U}^{*}\right)\left(U\otimes V\right)$\epsilon(\rho, \sigma) := (V^* \otimes U^*) (U \otimes V)
###### Proposition

The permutators are well defined and independent of the choice of spectator endomorphisms and transporters.

## References

See at DHR superselection theory.

Revised on December 2, 2011 10:22:11 by Urs Schreiber (89.204.137.152)