The net is called dual if every index is dual i.e. satisfies duality.

A weaker concept is that of essentially dual:

Define an extension $(\hat M_k)$ of the net $(M_k)$ via

$\hat M_i = (\bigcup_{j \perp i} M_j)'$

This extension is not necessarily a causal net anymore. If it is, then it is dual by definition. The net $(M_k)$ is essentially dual, if the extended net $(\hat M_k)$ is dual, which is true iff $(\hat M_k)$ is causal.

The net $(M_k)$ is called maximal if there is no proper extension which satisfies the causality condition?.

Examples

A Haag-Kastler vacuum representation satisfies Haag duality if every double cone aka diamond is a dual index. The reason for this relaxation is that full duality of every index is often too restrictive, so that the less restrictive Haag duality plays an important role in the theory.

Let $J_0$ be the index set of diamonds, a Haag-Kastler vacuum representation is essentially Haag dual if the net $M(J_0)$ (that is the original net restricted to diamonds as indices) is essentially dual.

Revised on July 9, 2011 13:42:19
by Urs Schreiber
(89.204.137.65)