Types of quantum field thories
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.
After the definition of objects and arrows we show several structures that the DHR category has.
The transportable endomorphisms are the objects of the DHR category .
It is straightforward to see that is a category:
Several structural properties follow immediatly from the definition:
is a algebroid.
Combining these two structures we get that is a star-category.
Since the arrows inherit a norm, we actually get
is a C-star-category.
Let be admissible representations and be their transportable endomorphisms localized in respectively. Choose a double cone that contains and . Since the local von Neumann algebra is not trivial, it contains a nontrivial projection , that is .
Thanks to the Borchers property there is a double cone containing the closure of , and partial isometries such that .
Now we set
It is possible to show that is unitarily equivalent to and that is a transportable (and therefore in particular a localized) endomorphism. So we will call a direct sum of and .
We first define the “tensor product”:
For endomorphisms we set .
For intertwiners and we define the tensor product via .
In the AQFT literature the tensor product of arrows is sometimes called the crossed product of intertwiners.
The tensor product as defined above turns into a monoidal category.
First: The tensor product of arrows is well defined, for any 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 , left and right unitor and the associator are the identities, that is, is strict.
Now to the braiding. The braiding is symmetric in dimensions only, this has a topological reason: The causal complement of a double cone is pathwise connected in dimensions only, but not in dimensions.
When we talk about 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 choose causally separated double cones and localized in and localized in . These endomorphisms are then called spectator endomorphisms of and .
Note that it is a theorem that causally disjoint transportable endomorphisms commute, therefore spectator endomorphisms always commute.
For transportable endomorphisms and spectator endomorphisms choose unitary interwiners and . Such unitaries are called transporters.
Obviously both spectator endomorphisms and transporters are not unique, in general.
For transportable endomorphisms , spectator endomorphisms 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.