algebraic quantum field theory (locally covariant perturbative, homotopical)
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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.