From local nets to Euler elements

Various aspects of the geometric setting of Algebraic Quantum Field Theory (AQFT) models related to representations of the Poincaré group can be studied for general Lie groups, whose Lie algebra contains an Euler element, i.e., ad h is diagonalizable with eigenvalues in $\{-1,0,1\}$. This has been explored by the authors and their collaborators during recent years. A key property in this construction is the Bisognano–Wichmann property (thermal property for wedge region algebras) concerning the geometric implementation of modular groups of local algebras.

In the present paper we prove that under a natural regularity condition, geometrically implemented modular groups arising from the Bisognano–Wichmann property are always generated by Euler elements. We also show the converse, namely that in presence of Euler elements and the Bisognano–Wichmann property, regularity and localizability hold in a quite general setting. Lastly we show that, in this generalized AQFT, in the vacuum representation, under analogous assumptions (regularity and Bisognano–Wichmann), the von Neumann algebras associated to wedge regions are type III₁ factors, a property that is well-known in the AQFT context.