nLab
separating vector

Context

Functional analysis

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

Definition

Let \mathcal{M} be a von Neumann algebra acting on a Hilbert space \mathcal{H}.

A vector xx \in \mathcal{H} is a separating vector if M(x)=0M(x) = 0 implies M=0M = 0 for all MM \in \mathcal{M}.

Properties

The notions of separating vector is dual to that of cyclic vector with respect to the commutant \mathcal{M}', that is a vector is cyclic for \mathcal{M} iff it is separating for \mathcal{M}'.

Applications

In the context of AQFT separating vectors appear as vacuum states . See Reeh-Schlieder theorem.

Revised on November 30, 2010 11:43:54 by Urs Schreiber (131.211.232.96)