nLab Borchers property

Contents

Context

Algebraic Quantum Field Theory

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

Idea

In algebraic quantum field theory, the Borchers property is a property of local nets in the Haag-Kastler approach to quantum field theory that follows, by a theorem of Borchers, from three physically motivated axioms on local nets. If every local algebra of the net is a type III factor, then the Borchers property is also a direct consequence.

The Borchers property is therefore often used as an “intermediate technical assumption”.

Sometimes this property is also abbreviated as “property B”.

Definition

Let (𝒪)\mathcal{M}(\mathcal{O}) be a net of von Neumann algebras indexed by bounded open subsets of Minkowski spacetime, for more details see Haag-Kastler vacuum representation,

Definition

The net (𝒪)\mathcal{M}(\mathcal{O}) satisfies the Borchers property if for every double cones K 1,K 2K_1, K_2 with K¯ 1K 2\bar K_1 \subseteq K_2 and E(𝒦 1)E \in \mathcal{M}(\mathcal{K}_1) a nonzero projection, EE is (Murray-von Neumann) equivalent to the identity with respect to (𝒦 2)\mathcal{M}(\mathcal{K}_2). That is, there is a partial isometry V(𝒦 2)V \in \mathcal{M}(\mathcal{K}_2) such that VV *=E,V *V=𝟙VV^* = E, V^*V = \mathbb{1}.

Properties

Proposition

A net satisfying causality, the spectrum condition and weak additivity (see Haag-Kastler vacuum representation for definitions) satisfies the Borchers property.

Remark

In particular. the local algebras of nets that satisfy the Borchers property cannot be finite where finite is meant in the sense of the Murray-von Neumann classification of factors.

References

Last revised on March 30, 2023 at 07:18:07. See the history of this page for a list of all contributions to it.