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.
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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
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.
The original work is:
There are a lot of secondary references, one is for example:
Last revised on June 21, 2010 at 13:55:15. See the history of this page for a list of all contributions to it.