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 (physics) * field bundle * field history * Lagrangian density * Euler-Lagrange form, presymplectic current * locally variational field theory * covariant phase space * Peierls-Poisson bracket, causal propagator **quantization** * geometric quantization * algebraic deformation quantization **quantum mechanical system** * subsystem * observables * field observable * local observables * polynomial observables * microcausal observables * operator algebra, C*-algebra, von Neumann algebra * local net of observables * causal locality * Haag-Kastler axioms * Wightman axioms * field net * conformal net * state on a star-algebra, quantum probability * pure state, mixed state, density matrix * space of quantum states * vacuum state * quasi-free state, * Hadamard state * picture of quantum mechanics **free field quantization** * star algebra, Moyal deformation quantization * Wick algebra * canonical commutation relations, Weyl relations * normal ordered product * Fock space **gauge theories** * gauge symmetry * BRST complex, BV-BRST formalism * local BV-BRST complex * BV-operator * quantum master equation * master Ward identity * quantum anomaly **interacting field quantization** * causal perturbation theory, perturbative AQFT * interaction * S-matrix, scattering amplitude * causal additivity * time-ordered product, Feynman propagator * Feynman diagram, Feynman perturbation series * effective action * vacuum stability * renormalization * renormalization scheme * extension of distributions * renormalization condition * gauge anomaly, quantum anomaly * renormalization group * Stückelberg-Petermann renormalization group * Gell-Mann-Low renormalization cocycle * interacting field algebra * Bogoliubov's formula * adiabatic limit * infrared divergence * interacting vacuum ## Theorems ### States and observables * order-theoretic structure in quantum mechanics * Alfsen-Shultz theorem * Harding-Döring-Hamhalter theorem * Kochen-Specker theorem * Bell's theorem * Fell's theorem * Gleason's theorem * Wigner theorem * Bub-Clifton theorem * Kadison-Singer problem ### Operator algebra * Wick's theorem * GNS construction * cyclic vector, separating vector * 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 * Osterwalder-Schrader theorem (Wick rotation) ### Perturbative QFT * Schwinger-Dyson equation * main theorem of perturbative renormalization

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)