Nets of real subspaces on homogeneous spaces and Algebraic Quantum Field Theory

TL;DR

This paper explores the relationship between Lie group representations and nets of real subspaces in the context of Algebraic Quantum Field Theory, emphasizing geometric and algebraic structures.

Contribution

It introduces a novel translation of AQFT axioms into nets of real subspaces, linking geometric, algebraic, and representation-theoretic perspectives.

Findings

Established a correspondence between Lie group representations and nets of real subspaces.

Provided a geometric framework for understanding AQFT axioms.

Enhanced the mathematical foundation of algebraic quantum field theory.

Abstract

In these notes, we describe an interesting connection between unitary representations of Lie groups and nets of local algebras, as they appear in Algebraic Quantum Field Theory (AQFT). It is based on first translating the axioms for nets of operator algebras parameterized by regions in a space-time manifold into those for nets of real subspaces, and then study this structure from a perspective based on geometry and representation theory of Lie groups.