Nets of real subspaces on homogeneous spaces and Algebraic Quantum Field Theory
Karl-Hermann Neeb
TL;DR
This work develops a geometric and representation-theoretic framework to connect unitary Lie group representations with nets of local observables in AQFT by translating operator-algebra nets into nets of real subspaces and exploiting modular theory. Central tools include standard subspaces and their modular data, the BW condition via Euler elements, and the use of crown domains and antiunitary representations to construct nets on causal homogeneous spaces. A key contribution is the Euler Element Theorem, which identifies exactly which infinitesimal generators can implement modular flows compatible with BW, and the BGL construction, which produces nets from Euler data on wedge regions. By linking second-quantized Weyl algebras with nets of real subspaces and establishing dualities via V and V', the notes provide a unifying picture for constructing and classifying nets on homogeneous spaces with causal structure, with implications for AQFT modelling on curved spacetimes and for understanding modular dynamics in operator algebras.
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.