nLab
Borchers property

Idea

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

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

• H.-J. Borchers: A remark on a theorem of B. Misra (available online from project euclid here)