nLab Borchers property

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

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

• Hans-Jürgen Borchers, A remark on a theorem of B. Misra, Comm. Math. Phys. 4 5 (1967) 315-323 [euclid]