nLab
localized endomorphism

Context

AQFT

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

Given a local net of observables

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

a natural algebra endomorphism

ρ:𝒜𝒜 \rho : \mathcal{A} \to \mathcal{A}

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

Localized endomorphisms play a central role in DHR superselection theory.

Definition

Definition

An endomorphim ρ\rho is localized or localizable if there is a bounded open set 𝒪𝒥\mathcal{O} \in \mathcal{J} such that ρ\rho is the identity on the algebra of the causal complement 𝒜(𝒪 )\mathcal{A}(\mathcal{O}^{\perp}). Such an endomorphism is localized in 𝒪\mathcal{O}.

Created on December 1, 2011 12:37:21 by Urs Schreiber (134.76.83.9)