nLab localized endomorphism

Idea

Given a local net of observables

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

a natural algebra endomorphism

$$\rho : \mathcal{A} \to \mathcal{A}$$

is called local or localized if outside of a bounded region of spacetime $X$ it is the identity.

Localized endomorphisms play a central role in DHR superselection theory.

Definition

Definition

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 $\mathcal{O}$.