AQFT and operator algebra
homotopical algebraic quantum field theory
The DHR superselection theory is about superselection sectors in the Haag-Kastler approach to AQFT. As such, it has to state one or more conditions on representations of the given Haag-Kastler net that specify the representations of the quasi-local algebra that are deemed physically admissible.
It is named after Sergio Doplicher, Rudolf Haag and John Roberts.
In the following we consider the theory starting with a Haag-Kastler vacuum representation on Minkowski spacetime.
The DHR condition says that all expectation values (of all observables) should approach the vacuum expectation values, uniformly, when the region of measurement is moved away from the origin. This is one possible abstraction of the situation in a high energy particle collider, where the registered events are localized both in time and in space.
The DHR condition excludes long range forces like electromagnetism from consideration, because, by Stokes' theorem, the electric charge in a finite region can be measured by the flux of the field strength through a sphere of arbitrary large radius. The DHR analysis is of interest nevertheless, because it has reached a certain maturity and is therefore an excellent object to study: see for example the Doplicher-Roberts reconstruction theorem.
After the definition of admissible representations we collect some notions that will enable us to state a description of all admissible representations using intrinsic properties of the quasi-local algebra. The central concept that we define is that of a transportable endomorphism, these form the objects of a category, the DHR category.
We start with a Haag-Kastler vacuum representation that we assume to be irreducible and Haag dual.
Let $\pi_0$ be the vacuum representation from now on, $K$ denote double cones and $\mathcal{J}_0$ be the causal index set of double cones, just as $\mathcal{O}$ are bounded open sets and $\mathcal{J}$ the causal index set of bounded open sets.
Recall that the C-star algebra
is called the quasi-local algebra of the given net. We will in the same way associate a C-star algebra to every open region $\mathcal{O}_0$ via
A representation $\pi$ of the local algebra $\mathcal{A}$ is called (DHR) admissible if $\pi | \mathcal{A}(\mathcal{K}^{\perp})$ is unitarily equivalent to $\pi_0 | \mathcal{A}(\mathcal{K}^{\perp})$ for all $K \in \mathcal{J}_0$, that is, for all double cones $K$.
Note that the equivalence of the representations is required for the causal complement of all double cones, not a special one, so that the characterization that โthe representations can not be distinguished at space-like infinityโ is misleading.
Recall that a mapping
is called a unital endomorphism if $\rho$ is linear, multiplicative ($\rho(AB) = \rho(A) \rho(B)$) and $\rho(\mathbb{1})=1$.
We will drop โunitarilyโ from now on.
For any representation $\pi$ and endomorphism $\rho$ the composition $\pi \circ \rho$ is another representation.
Conversely, under the assumption of Haag duality every DHR admissible representation $\rho$ comes from an endomorphism this way. For let $A \in \mathcal{A}(K)$ and $B \in \mathcal{A}(K^\pert)$ be any two causally unrelated localized obserbales. Then for $\rho : \mathcal{A} \to \mathcal{B}(\mathcal{H})$ a DHR representation, we have on the one hand
and on the other, due to the causal structure
This says that $\rho(A) \in \pi_0(\mathcal{A}(K'))'$ and hence by Haag duality $\cdots \in \pi_0(\mathcal{A}(K))''$. Under the assumption that the local net of observables takes values in von Neumann algebras this in turn in $\cdots = \pi_0(\mathcal{A}(K))$ and so $\rho$ indeed factors as an endomrphism of $\mathcal{A}$ followed by the vaccum representation.
For this reason one can hope to gain some insights into the representations by studying the endomorphisms, while the set of endomorphisms has certainly more structure than that of representations: For example, endomorphisms may have inverses, and endomorphisms form a monoid by composition.
First we define โunitarily equivalentโ and โintertwinerโ analog to the definition for representations of C-star algebras:
Two endomorphisms $\rho_1, \rho_2$ are unitarily equivalent if there is a unitary operator $U \in \mathcal{A}$ such that $\rho_1 = ad(U) \rho_2 = U \rho_2 U^{-1}$. The endomorphisms $ad(U)$ with $U \in \mathcal{A}$ are called inner automorphisms (of $\mathcal{A}$). An element $R \in \mathcal{A}$ such that $R \rho_1 = \rho_2 R$, not necessarily unitary, is called an intertwiner or intertwining operator.
The DHR selection criterion selects representations that โlook like the vaccumโ on the causal complement of elements of our index set (in this case the double cones). The analog for endomorphims is this:
An endomorphim $\rho$ is localized or localizable if there is a bounded open set $\mathcal{O} \in \mathcal{J}$ such that $\rho$ is the identity on the algebra of the causal complement $\mathcal{A}(\mathcal{O}^{\perp})$. Such an endomorphism is localized in $\mathcal{O}$.
We have to restrict the notion of โunitarily equivalentโ to a โlocalizedโ version accordingly:
An element $A \in \mathcal{A}$ is a local operator if there is a double cone K such that $A \in \mathcal{M}(K)$. In particular, a local unitary operator $U$ is a local unitary, and two endomorphisms are locally unitarily equivalent if there is a unitary as in the definition of unitarily equivalent that is a local operator.
For an endomorphism $\rho$ let $\hat \rho$ be the equivalence class with respect to locally unitarily equivalence. Then we can define:
A localized endomorphism is transportable if for every bounded open set $\mathcal{O} \in \mathcal{J}$ there is a $\rho_0 \in \hat \rho$ that is localized in $\mathcal{O}$.
Transportable endomorphisms naturally form the objects of a category, the DHR category.
transportable endomorphisms are compatible with the net structure
Let $\rho$ be a transportable endomorphism that is localized in the double cone $K_0$, then for all double cones $K \supset K_0$, $\rho$ maps $\mathcal{M}(K)$ to itself.
intrinsic characterization of admissible representations
A representation $\pi$ of $\mathcal{A}$ is admissible iff there is a transportable endomorphism $\rho$ such that $\pi$ is unitarily equivalent to $\pi_0 \circ \rho$.
Reference: This is theorem 2.1.3 in the book by Baumgรคrtel.
stability of admissible representations
(i) finite direct sums of admissible representations are admissible.
(ii) subrepresentations of admissible representations are admissible.
For transportable endomorphisms we get even more:
products The product, i.e. the concatenation, of transportable endomorphisms is a transportable endomorphism.
product of causally disjoint localized endomorphisms is commutative
Let $\rho_1, \rho_2$ be transportable endomorphisms localized in $K_1, K_2$ respectivley with $K_1 \perp K_2$, then $\rho_1 \rho_2 = \rho_2 \rho_1$.
Note that the product of equivalenc classes is well defined, and $\hat \rho_1 \hat \rho_2 = \widehat{\rho_1\rho_2}$. Therefore the theorem above implies that the product of equivalence classes is commutative.
The DHR superselection theory has a reformulation in terms of cohomology of local nets of observables. For the moment, see there for more.
see AQFT ff.
Of particular relevance (besides the original work of Doplicher and Roberts) are
Hellmut Baumgรคrtel: Operator algebraic Methods in Quantum Field Theory. A series of lectures. Akademie Verlag 1995 (ZMATH entry)
Hans Halvorson, Michael Mรผger, Algebraic Quantum Field Theory (arXiv:math-ph/0602036)