Algebraic Quantum Field Theory and Causal Symmetric Spaces

TL;DR

This paper reviews recent advances in the geometric and algebraic structures underlying Quantum Field Theory, focusing on causal symmetric spaces, modular groups, and von Neumann algebra nets, to deepen understanding of quantum causality.

Contribution

It introduces new methods to construct von Neumann algebra nets on causal symmetric spaces satisfying key quantum field theory conditions, based on Euler elements and 3-gradings.

Findings

Construction of nets satisfying Reeh--Schlieder and Bisognano-Wichmann conditions.

Analysis of wedge regions in causal symmetric spaces.

Application of Euler elements to quantum field theory geometry.

Abstract

In this article we review our recent work on the causal structure of symmetric spaces and related geometric aspects of Algebraic Quantum Field Theory. Motivated by some general results on modular groups related to nets of von Neumann algebras,we focus on Euler elements of the Lie algebra, i.e., elements whose adjoint action defines a 3-grading. We study the wedge regions they determine in corresponding causal symmetric spaces and describe some methods to construct nets of von Neumann algebras on causal symmetric spaces that satisfy abstract versions of the Reeh--Schlieder and the Bisognano-Wichmann condition.