nLab
DHR category

Idea

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

$$𝒜:\mathrm{Open}(X)\to \mathrm{Algebras}$$\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 $({\rho }_1,{\rho }_2)$, 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

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 $ℳ(K_0)$ is not trivial, it contains a nontrivial projection $E$, that is $0<E<𝟙$.

Thanks to the Borchers property there is a double cone $K$ containing the closure of $K_0$, and partial isometries $W_1,W_2\in ℳ(K)$ 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 $S\in \mathrm{Hom}(\rho ,{\rho }^{\prime })$ and $T\in \mathrm{Hom}(\sigma ,{\sigma }^{\prime })$ we define the tensor product via $S\otimes T:=S\rho (T)$.

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:

$$S\otimes T({\rho }_1\otimes {\rho }_2)(A)=(S\rho (T))\rho (\sigma (A))=S\rho (T\sigma (A))={\rho }^{\prime }(T\sigma (A))S={\rho }^{\prime }({\sigma }^{\prime }(A)T)S={\rho }^{\prime }{\sigma }^{\prime }(A){\rho }^{\prime }(T)S={\rho }^{\prime }{\sigma }^{\prime }(A)S\rho (T)$$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 ${\rho }_0\in \hat{\rho }$ 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.

Definition

For transportable endomorphisms $\rho ,\sigma$ and spectator endomorphisms ${\rho }_0,{\sigma }_0$ choose unitary interwiners $U\in \mathrm{Hom}(\rho ,{\rho }_0)$ and $V\in Hom(\sigma ,{\sigma }_0)$. Such unitaries are called transporters.

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

Definition

For transportable endomorphisms $\rho ,\sigma$, spectator endomorphisms ${\rho }_0,{\sigma }_0$ and transporters U, V define the permutator or permutation symmetry via

$$ϵ(\rho ,\sigma ):=(V^{*}\otimes U^{*})(U\otimes V)$$\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.