nLab
separating vector

Contents

Context

Functional analysis

AQFT

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

Introduction

Concepts

field theory:

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.

Last revised on November 30, 2010 at 11:43:54. See the history of this page for a list of all contributions to it.