From local nets to Euler elements

TL;DR

This paper demonstrates that in a generalized algebraic quantum field theory setting, modular groups linked to wedge regions are generated by Euler elements, establishing a connection between geometric properties and algebraic structures.

Contribution

It proves that modular groups satisfying the Bisognano-Wichmann property are generated by Euler elements and shows the converse, linking regularity, localizability, and the type of von Neumann algebras.

Findings

Modular groups are generated by Euler elements under regularity.

The converse holds: Euler elements imply regularity and localizability.

Wedge region algebras are type III_1 factors in vacuum representation.

Abstract

Various aspects of the geometric setting of Algebraic Quantum Field Theory (AQFT) models related to representations of the Poincar\'e 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 III_1 factors, a property that is well-known in the AQFT context.