nLab
localized endomorphism

Idea

Given a local net of observables

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

a natural algebra endomorphism

$$\rho :𝒜\to 𝒜$$\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 $𝒪\in 𝒥$ such that $\rho$ is the identity on the algebra of the causal complement $𝒜({𝒪}^{\perp })$. Such an endomorphism is localized in $𝒪$.