Endomorphisms, Intertwiners, and Tensor Products

Localized endomorphisms put charges, charge-changing operators, and fusion inside one concrete C*-tensor category. Objects are transportable endomorphisms of the observable algebra; arrows are intertwiners; tensor product is composition. This is more information than a fusion table: localization regions, transporter choices, arrow spaces, and their coherent action on observables are essential to the construction.

Required background. Superselection Sectors and DHR Reconstruction supplies the selected endomorphisms; Representations, Intertwiners, Invariants, and Tensor Decomposition supplies the representation-theoretic model for arrows and decomposition.

Helpful background. Monoidal, Rigid, and Braided Language explains coherence, duals, and braiding abstractly.

Localized endomorphisms and their arrows

Let $\mathfrak A$ be the vacuum quasilocal algebra. An object $\rho$ is localized in a double cone $O$ if $\rho(A)=A$ for $A\in\mathfrak A(O')$, and transportable if an equivalent representative can be localized in every double cone. For two objects, define

$$\operatorname{Hom}(\rho,\sigma) =\{T\in\mathfrak A:T\rho(A)=\sigma(A)T\ \forall A\in\mathfrak A\}.$$

If $T$ is unitary, $\sigma=\operatorname{Ad}T\circ\rho$ and the objects represent the same charge type. Nonunitary isometries and projections encode subobjects and reducible sectors. Composition and adjoint of operators make $\operatorname{Hom}(\rho,\sigma)$ into a Banach space with $T^*T\in\operatorname{End}(\rho)$ and $TT^*\in\operatorname{End}(\sigma)$. For irreducible $\rho$, Schur’s lemma gives $\operatorname{End}(\rho)=\mathbb C1$.

These definitions have a local consequence. If $\rho$ and $\sigma$ are both localized in $O$, Haag duality places every intertwiner between them in $\mathfrak A(O)$, after a harmless enlargement of $O$. The arrow is therefore an observable local charge converter, not an arbitrary map between Hilbert spaces. See Halvorson and Müger 2006, Definitions 8.1–8.5 and §§8.1–8.2, pp. 60–77.

Composition is fusion

For objects, set $\rho\otimes\sigma=\rho\circ\sigma$. It remains localized in any double cone containing localization regions for suitable transported representatives. Associativity is literal:

$$(\rho\sigma)\tau=\rho(\sigma\tau).$$

For $S\in\operatorname{Hom}(\rho,\rho')$ and $T\in\operatorname{Hom}(\sigma,\sigma')$, define

$$S\otimes T=S\rho(T)=\rho'(T)S.$$

The equality is exactly the intertwining relation for $S$. A direct calculation gives

$$(S_2S_1)\otimes(T_2T_1) =(S_2\otimes T_2)(S_1\otimes T_1),$$

so the construction is bifunctorial. Because $\rho$ acts on the arrow $T$, the product is not ordinary operator multiplication. Omitting this action is a common and consequential error.

Transporters provide the operational construction. If $U\in\operatorname{Hom}(\rho,\widetilde\rho)$ moves $\rho$ from $O$ to $\widetilde O$, then $\widetilde\rho=\operatorname{Ad}U\circ\rho$. Products such as $V\sigma(U)$, rather than merely $VU$, transport composites. Independence of auxiliary localization choices is proved with locality and the intertwining relations; it is not automatic from group-like charge labels.

Charge-one fusion in the complex scalar net

Let $\rho_1$ be a charge-one endomorphism of the $U(1)$-fixed observable net, localized in $O_1$. Choose a unitary transporter $U$ so that $\widetilde\rho_1=\operatorname{Ad}U\circ\rho_1$ is localized in a second double cone $O_2$. The two-charge object is

$$\rho_2=\rho_1\widetilde\rho_1.$$

If $V$ transports $\rho_1$ to another representative $\widehat\rho_1$, then $\rho_1(V)$ is an intertwiner from $\rho_1\widetilde\rho_1$ to $\rho_1\widehat\rho_1$. Indeed,

$$\rho_1(V)\rho_1\widetilde\rho_1(A) =\rho_1\widehat\rho_1(A)\rho_1(V).$$

Thus transporter functoriality makes the composition independent, up to equivalence, of where the second unit of charge was placed. The resulting charge-addition law is the local-algebraic version of Fusion, Junctions, and Endpoints.

Precise conclusion and excluded converse

Given a Haag-dual vacuum net and the category of transportable localized endomorphisms, composition and the arrow formula above produce a strict C*-tensor category with simple unit; suitable infiniteness assumptions give direct sums and subobjects. If spacelike transport is available, locality further constructs exchange operators. These are structural conclusions about a concrete category acting on $\mathfrak A$.

The converse is false. A set of labels with an associative fusion rule need not arise from localized endomorphisms of any net. Even a fully abstract tensor category does not specify which morphism acts on which local algebra. One must still construct localization, transporters, arrow spaces, and a coherent embedding into $\operatorname{End}(\mathfrak A)$.

Adversarial failure: incompatible placements

Suppose one writes $a\times a=a^2$ for a charge label and chooses two purported representatives in regions $O_1,O_2$, but provides no transporter between them. If the representations induce different actions on an algebra in the causal complement, they are not equivalent localized copies of one charge. Their composites can then depend on placement. Even when unitary maps exist on Hilbert space, a map that fails $U\rho(A)=\widetilde\rho(A)U$ for all $A$ is not an intertwiner. Fusion labels alone conceal this failure; the explicit arrow equation exposes it.

Independent checks

Check first that $S\rho(T)=\rho'(T)S$ before using the tensor product of arrows. Check second that a proposed transporter preserves localization: $\operatorname{Ad}U\circ\rho$ must act identically on the new causal exterior. Check third that transporting constituents in different orders gives equivalent composites. Finally, decompose a reducible product using projections in its endomorphism algebra and verify that the multiplicities agree with the claimed fusion rule.

Exercises

1. Tensor-product arrow. Prove that $S\rho(T)$ intertwines $\rho\sigma$ with $\rho'\sigma'$.

Solution

For $A\in\mathfrak A$, $S\rho(T)\rho\sigma(A)=S\rho(T\sigma(A))=S\rho(\sigma'(A)T)=S\rho(\sigma'(A))\rho(T)=\rho'\sigma'(A)S\rho(T)$, using first that $T$ intertwines $\sigma$ and $\sigma'$, then that $S$ intertwines $\rho$ and $\rho'$.

2. Locality of a composite. If $\rho$ and $\sigma$ are localized in the same double cone $O$, show that $\rho\sigma$ is localized in $O$.

Solution

For $A\in\mathfrak A(O')$, $\sigma(A)=A$ and then $\rho\sigma(A)=\rho(A)=A$. Thus the composite acts trivially on the exterior.

3. Why ordinary multiplication fails. Let $S\in(\rho,\rho')$ and $T\in(\sigma,\sigma')$. Explain why $ST$ need not intertwine $\rho\sigma$ with $\rho'\sigma'$.

Solution

The relation for $T$ concerns $T\sigma(A)$, whereas the composite contains $\rho\sigma(A)$. Unless $T$ commutes with the range of $\rho$, $ST\rho\sigma(A)$ cannot be rearranged into $\rho'\sigma'(A)ST$. Applying $\rho$ to $T$ supplies exactly the missing covariance.

References

• Doplicher, Sergio, Rudolf Haag, and John E. Roberts. “Local Observables and Particle Statistics II.” Communications in Mathematical Physics 35 (1974): 49–85. DOI.
• Halvorson, Hans, and Michael Müger. “Algebraic Quantum Field Theory.” In Handbook of the Philosophy of Science, Vol. 2: Philosophy of Physics, edited by Jeremy Butterfield and John Earman, 731–922. Amsterdam: Elsevier, 2007. Open PDF, 2006 preprint.