# From local nets to Euler elements

December 19, 2023

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.

Keywords:

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