Covariant homogeneous nets of standard subspaces

TL;DR

This paper generalizes the algebraic construction of quantum field theories from wedge regions using Lie group symmetries, introducing new models and classifying supporting Lie algebras without relying on specific spacetime structures.

Contribution

It introduces a framework for constructing covariant homogeneous nets of standard subspaces from general Lie groups, extending the BGL construction beyond traditional spacetime settings.

Findings

Classified Lie algebras supporting abstract wedge regions

Developed a generalized net construction for a broad class of Lie groups

Provided new models of quantum field theories based on these generalized nets

Abstract

Rindler wedges are fundamental localization regions in AQFT. They are determined by the one-parameter group of boost symmetries fixing the wedge. The algebraic canonical construction of the free field provided by Brunetti-Guido-Longo (BGL) arises from the wedge-boost identification, the BW property and the PCT Theorem. In this paper we generalize this picture in the following way. Firstly, given a $\mathbb Z_2$-graded Lie group we define a (twisted-)local poset of abstract wedge regions. We classify (semisimple) Lie algebras supporting abstract wedges and study special wedge configurations. This allows us to exhibit an analog of the Haag-Kastler one-particle net axioms for such general Lie groups without referring to any specific spacetime. This set of axioms supports a first quantization net obtained by generalizing the BGL construction. The construction is possible for a large family of Lie groups and provides several new models. We further comment on orthogonal wedges and extension of symmetries.