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Given a quantum field theory presented by a local net of observables (AQFT)
a local endomorphism is a natural algebra homomorphism $\rho : \mathcal{A} \to \mathcal{A}$ 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.
After the definition of objects and arrows we show several structures that the DHR category has.
See DHR superselection theory and Haag-Kastler vacuum representation for the terminology used here.
The transportable endomorphisms are the objects of the DHR category $\Delta$.
For two transportable endomorphisms the set of intertwiners are the morphisms.
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
follows
Several structural properties follow immediatly from the definition:
$\Delta$ is a $\mathbb{C}-$algebroid.
$\Delta$ 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 $(\rho_2, \rho_1)$.
Combining these two structures we get that $\Delta$ is a star-category.
Since the arrows inherit a norm, we actually get
$\Delta$ is a C-star-category.
It is possible to introduce a finite direct product in $\Delta$, if the net satisfies the Borchers property.
The Haag-Kastler vacuum representation that we talk about here satisfies the Borchers property.
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 $K_0 \in \mathcal{J}_0$ 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
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 $\rho_2$.
We first define the “tensor product”:
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 $S \otimes T := S \rho(T)$.
In the AQFT literature the tensor product of arrows is sometimes called the crossed product of intertwiners.
The tensor product as defined above turns $\Delta$ into a monoidal category.
First: The tensor product of arrows is well defined, for any $A \in \mathcal{A}$ we have:
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 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.
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:
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 spectator endomorphisms of $\rho$ and $\sigma$.
Note that it is a theorem that causally disjoint transportable endomorphisms commute, therefore spectator endomorphisms always commute.
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 transporters.
Obviously both spectator endomorphisms and transporters are not unique, in general.
For transportable endomorphisms $\rho, \sigma$, spectator endomorphisms $\rho_0, \sigma_0$ and transporters U, V define the permutator or permutation symmetry via
The permutators are well defined and independent of the choice of spectator endomorphisms and transporters.
See at DHR superselection theory.
Last revised on December 2, 2011 at 10:22:11. See the history of this page for a list of all contributions to it.