Wedge domains in compactly causal symmetric spaces

TL;DR

This paper introduces wedge domains in compactly causal symmetric spaces, generalizing Minkowski space wedges, and explores their geometric properties and quantum field theory analogs, including modular flow and the Bisognano--Wichmann property.

Contribution

It defines and characterizes wedge domains in symmetric spaces using modular flow and geometric conditions, extending quantum field theory concepts beyond Minkowski space.

Findings

Wedge domains coincide under multiple geometric characterizations.

Existence of covariant nets with modular groups matching the modular flow.

Geometric realization of abstract wedge spaces in symmetric spaces.

Abstract

Motivated by construction in Algebraic Quantum Field Theory we introduce wedge domains in compactly causal symmetric spaces M=G/H, which includes in particular anti de Sitter space in all dimensions and its coverings. Our wedge domains generalize Rindler wedges in Minkowski space. The key geometric structure we use is the modular flow on M defined by an Euler element in the Lie algebra of G. Our main geometric result asserts that three seemingly different characterizations of these domains coincide: the positivity domain of the modular vector field; the domain specified by a KMS like analytic extension condition for the modular flow; and the domain specified by a polar decomposition in terms of certain cones. In the second half of the article we show that our wedge domains share important properties with wedge domains in Minkowski space. If G is semisimple, there exist unitary representations of G and isotone covariant nets of real subspaces defined for any open subset of M, which assign to connected components of the wedge domains a standard subspace whose modular group corresponds to the modular flow on M. This corresponds to the Bisognano--Wichmann property in Quantum Field Theory. We also show that the set of G-translates of the connected components of the wedge domain provides a geometric realization of the abstract wedge space introduced by the first author and V. Morinelli.