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)

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”.

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,

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 $VV^* = E, V^*V = \mathbb{1}$.

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.

- 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)