Borchers property

**algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical)
Introduction
## Concepts
**field theory**: classical, pre-quantum, quantum, perturbative quantum
**Lagrangian field theory**
* field (physics)
* field bundle
* field history
* space of field histories
* Lagrangian density
* Euler-Lagrange form, presymplectic current
* Euler-Lagrange equations of motion
* locally variational field theory
* covariant phase space
* Peierls-Poisson bracket
* advanced and retarded propagator,
* causal propagator
**quantization**
* geometric quantization of symplectic groupoids
* algebraic deformation quantization, star algebra
**quantum mechanical system**, **quantum probability**
* subsystem
* observables
* field observables
* 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, expectation value
* pure state
wave function
collapse of the wave function/conditional expectation value
* mixed state, density matrix
* space of quantum states
* vacuum state
* quasi-free state,
* Hadamard state
* Wightman propagator
* 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
* gauge 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
* interacting field algebra
* Bogoliubov's formula
* quantum Møller operator
* adiabatic limit
* infrared divergence
* interacting vacuum
**renormalization**
* ("re-")normalization scheme
* extension of distributions
* ("re"-)normalization condition
* quantum anomaly
* renormalization group
* interaction vertex redefinition
* Stückelberg-Petermann renormalization group
* renormalization group flow/running coupling constants
* effective quantum field theory
* UV cutoff
* counterterms
* relative effective action
* Wilsonian RG, Polchinski flow equation
## Theorems
{#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

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)

Last revised on June 30, 2010 at 10:42:23. See the history of this page for a list of all contributions to it.