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DHR category

Context

AQFT

Physics

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theory (physics), model (physics)

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Contents

Idea

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

𝒜:Open(X)Algebras \mathcal{A} : Open(X) \to Algebras

a local endomorphism is a natural algebra homomorphism ρ:𝒜𝒜\rho : \mathcal{A} \to \mathcal{A} which is supported (nontrivial) on a compact region of spacetime XX.

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 Δ\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 Δ\Delta is a category:

The identity morphism for each object in Δ\Delta is given by the identiy in 𝒜\mathcal{A}. The composition of arrows is simply the composition of intertwiners:

From

Rρ 1=ρ 2R R \rho_1 = \rho_2 R
Tρ 2=ρ 3T T \rho_2 = \rho_3 T

follows

TRρ 1=ρ 3TR T R \rho_1 = \rho_3 T R

Several structural properties follow immediatly from the definition:

Lemma

Δ\Delta is a \mathbb{C}-algebroid.

Lemma

Δ\Delta is a dagger-category since, if RR is an intertwiner of the pair (ρ 1,ρ 2)(\rho_1, \rho_2), then R *R^* is obviously an intertwiner of the pair (ρ 2,ρ 1)(\rho_2, \rho_1).

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

Since the arrows inherit a norm, we actually get

Lemma

Δ\Delta is a C-star-category.

Proposition

It is possible to introduce a finite direct product in Δ\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 π 1,π 2\pi_1, \pi_2 be admissible representations and ρ 1,ρ 2\rho_1, \rho_2 be their transportable endomorphisms localized in K 1,K 2K_1, K_2 respectively. Choose a double cone K 0𝒥 0K_0 \in \mathcal{J}_0 that contains K 1K_1 and K 2K_2. Since the local von Neumann algebra (K 0)\mathcal{M}(K_0) is not trivial, it contains a nontrivial projection EE, that is 0<E<𝟙0 \lt E \lt \mathbb{1}.

Thanks to the Borchers property there is a double cone KK containing the closure of K 0K_0, and partial isometries W 1,W 2(K)W_1, W_2 \in \mathcal{M}(K) such that W 1W 1 *=E,W 2W 2 *=𝟙EW_1 W_1^* = E, W_2 W_2^* = \mathbb{1} - E.

Now we set

ρ:=W 1ρ 1W 1 *+W 2ρ 2W 2 * \rho := W_1 \rho_1 W_1^* + W_2 \rho_2 W_2^*

It is possible to show that π 0ρ\pi_0 \rho is unitarily equivalent to π 1π 2\pi_1 \oplus \pi_2 and that ρ\rho is a transportable (and therefore in particular a localized) endomorphism. So we will call ρ\rho a direct sum of ρ 1\rho_1 and ρ 2\rho_2.

DHR is a symmetric monoidal category

We first define the “tensor product”:

Definition

For endomorphisms we set ρ 1ρ 2:=ρ 1ρ 2\rho_1 \otimes \rho_2 := \rho_1 \rho_2.

For intertwiners SHom(ρ,ρ )S \in Hom(\rho, \rho^{\prime}) and THom(σ,σ )T \in Hom(\sigma, \sigma^{\prime}) we define the tensor product via ST:=Sρ(T)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 Δ\Delta into a monoidal category.

Proof

First: The tensor product of arrows is well defined, for any A𝒜A \in \mathcal{A} we have:

ST(ρ 1ρ 2)(A)=(Sρ(T))ρ(σ(A))=Sρ(Tσ(A))=ρ (Tσ(A))S=ρ (σ (A)T)S=ρ σ (A)ρ (T)S=ρ σ (A)Sρ(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 𝟙𝒜\mathbb{1} \in \mathcal{A}, left and right unitor and the associator are the identities, that is, Δ\Delta is strict.

Now to the braiding. The braiding is symmetric in d3d \ge 3 dimensions only, this has a topological reason: The causal complement of a double cone is pathwise connected in d3d \ge 3 dimensions only, but not in d2d \le 2 dimensions.

Remark

When we talk about d=1d = 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 ρ,σ\rho, \sigma choose causally separated double cones K 1K 2K_1 \perp K_2 and ρ 0ρ^\rho_0 \in \hat \rho localized in K 1K_1 and σ 0σ^\sigma_0 \in \hat \sigma localized in K 2K_2. These endomorphisms ρ 0,σ 0\rho_0, \sigma_0 are then called spectator endomorphisms of ρ\rho and σ\sigma.

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

Definition

For transportable endomorphisms ρ,σ\rho, \sigma and spectator endomorphisms ρ 0,σ 0\rho_0, \sigma_0 choose unitary interwiners UHom(ρ,ρ 0)U \in Hom(\rho, \rho_0) and VHom(σ,σ 0)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 ρ,σ\rho, \sigma, spectator endomorphisms ρ 0,σ 0\rho_0, \sigma_0 and transporters U, V define the permutator or permutation symmetry via

ϵ(ρ,σ):=(V *U *)(UV) \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)