Wedge domains in non-compactly causal symmetric spaces

TL;DR

This paper explores wedge domains in non-compactly causal symmetric spaces, establishing their equivalence and connectedness, and extending these structures to complexified spaces with implications for causal and algebraic quantum field theory.

Contribution

It introduces three types of wedge domains in non-compactly causal symmetric spaces and proves their equivalence and connectedness, extending the analysis to complexified spaces.

Findings

Different wedge domains in G/H agree in their connected component.

Wedge domains extend naturally to complexified symmetric spaces.

The intersection of wedge domains with G/H matches the wedge domains in G/H.

Abstract

This article is part of an ongoing project aiming at the connections between causal structures on homogeneous spaces, Algebraic Quantum Field Theory (AQFT), modular theory of operator algebras and unitary representations of Lie groups. In this article we concentrate on non-compactly causal symmetric space $G/H$. This class contains the de Sitter space but also other spaces with invariant partial ordering. The central ingredient is an Euler element h in the Lie algebra of \fg. We define three different kinds of wedge domains depending on h and the causal structure on G/H. Our main result is that the connected component containing the base point eH of those seemingly different domains all agree. Furthermore we discuss the connectedness of those wedge domains. We show that each of those spaces has a natural extension to a non-compactly causal symmetric space of the form G_\C/G^c where G^c is certain real form of the complexification G_\$ of G. As G_\C/G^c is non-compactly causal it also comes with the three types of wedge domains. Our results says that the intersection of those domains with $G/H$ agrees with the wedge domains in G/H.