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:

1. 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.

2. The modular conjugation of the wedge W is given by the formula

   $$J_W=\Theta U(R_W(\pi ))$$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 $\pi$.

3. 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)