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Bisognano-Wichmann theorem
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## Definitions
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quantum mechanical system
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observable
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operator algebra
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C*-algebra
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von Neumann algebra
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subsystem
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cyclic vector ,
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Haag-Kastler axioms
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## Theorems
### States and observables
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Kochen-Specker theorem
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Wigner theorem
### Operator algebra
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GNS construction
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modular theory
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Fell's theorem
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Haag's theorem
### Local QFT
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Bisognano-Wichmann theorem
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PCT theorem
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spin-statistics theorem
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DHR superselection theory
### Euclidean QFT
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Osterwalder-Schrader theorem
Contents
Idea
In the Haag-Kastler approach to quantum field theory the central object is a local net of operator algebras . The modular theory says that the local algebras have an associated modular group and a modular conjugation (see modular theory ). The result of Bisognano-Wichmann that this page is about, describes the relation of these to the Poincare group .
Abstract
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Definition
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The Theorem
Assume a Wightman field theory of a scalar neutral field is such that the smeared field operators generate the local algebras, see Wightman axioms .
Then:
The modular group of the algebra associated with a wedge and the vacuum vector coincides with the unitary representation of the group of Lorentz boosts which maps the wedge onto itself.
The modular conjugation of the wedge W is given by the formula
$J_W = \Theta U(R_W(\pi))$
Here $\Theta$ denotes the PCT-operator of the Wightman field theory and $U(R_W(\pi))$ is the unitary representation of the rotation which leaves the characteristic two-plane of the wedge invariant. The angle of rotation is $\pi$ .
The theory fulfills wedge duality, that is the commutant of the algebra associated to a wedge is the algebra associated to the causal complement of the wedge.
References
The original work is:
Bisognano, J. and Wichmann, E.H.: On the duality condition for a Hermitian scalar field , J. Math. Phys. 16 (1975), 985-1007.
There are a lot of secondary references, one is for example:
Daniele Guido: Modular Theory for the von Neumann Algebras of Local Quantum Physics (arXiv )
Revised on June 21, 2010 13:55:15
by
Urs Schreiber
(134.100.32.213)