nLab
Borchers property

**algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical) ## Concepts **field theory**: classical, pre-quantum, quantum, perturbative quantum **Lagrangian field theory** * field bundle * Lagrangian density * Euler-Lagrange form, presymplectic current * locally variational field theory * covariant phase space **quantum mechanical system** * subsystem * quantum observable * operator algebra, C*-algebra, von Neumann algebra * local net of observables * causal locality * Haag-Kastler axioms * Wightman axioms * field net * conformal net * quantum states, * cyclic vector, separating vector * quasi-free state * vacuum state * Hadamard state **quantization** * covariant phase space * Peierls-Poisson bracket, causal propagator * causal perturbation theory, perturbative AQFT * Wick algebra * microcausal functionals * normal ordered product * S-matrix, scattering amplitude * causal additivity * time-ordered product, Feynman propagator * Feynman diagram, Feynman perturbation series * interacting field algebra ## Theorems ### States and observables * Kochen-Specker theorem * Gleason's theorem * Wigner theorem ### Operator algebra * GNS construction * modular theory * Fell's theorem * Stone-von Neumann theorem * Haag's theorem ### Local QFT * Reeh-Schlieder theorem * Bisognano-Wichmann theorem * PCT theorem * spin-statistics theorem * DHR superselection theory ### Perturbative QFT * main theorem of perturbative renormalization ### Euclidean QFT * Osterwalder-Schrader theorem (Wick rotation)

Contents

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 (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

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

Revised on June 30, 2010 10:42:23 by Tim van Beek (192.76.162.8)