Crowned Lie groups and nets of real subspaces

TL;DR

This paper introduces complex crown domains for Lie groups and uses them to construct nets of real subspaces satisfying key quantum field theory conditions, unifying various existing approaches.

Contribution

It defines complex crown domains for Lie groups and demonstrates their use in constructing nets of real subspaces with important properties, linking representation theory and quantum field theory.

Findings

Constructed nets of real subspaces satisfying Reeh--Schlieder and Bisognano--Wichmann conditions.

Characterized existence of such nets via a regularity condition involving Euler elements.

Showed all antiunitary representations of the split oscillator group possess the necessary property.

Abstract

We introduce the notion of a complex crown domain for a connected Lie group $G$, and we use analytic extensions of orbit maps of antiunitary representations to these domains to construct nets of real subspaces on $G$ that are isotone, covariant and satisfy the Reeh--Schlieder and Bisognano--Wichmann conditions from Algebraic Quantum Field Theory. This provides a unifying perspective on various constructions of such nets.The representation theoretic properties of different crowns are discussed in some detail for the non-abelian $2$-dimensional Lie group ${\rm Aff}({\mathbb R})$. We also characterize the existence of nets with the above properties by a regularity condition in terms of an Euler element in the Lie algebra ${\mathfrak g}$ and show that all antiunitary representations of the split oscillator group have this property.