A group theoretical approach to causal structures and positive energy on spacetimes

TL;DR

This paper develops a group theoretical framework linking spacetime symmetries with causality, horizons, and positive energy, enabling analysis of supergravity and string backgrounds through symmetry algebras.

Contribution

It introduces a novel approach connecting Adjoint-invariant convex cones in symmetry algebras to static metrics and positive energy representations, advancing the understanding of causal and horizon structures.

Findings

Existence of static metrics linked to invariant convex cones in symmetry algebras.

Positive energy representations correspond to Adjoint-invariant cones.

Classification of quantum observables via complexified universal algebra.

Abstract

This article presents a precise description of the interplay between the symmetries of a quantum or classical theory with spacetime interpretation, and some of its physical properties relating to causality, horizons and positive energy. Our major result is that the existence of static metrics on spacetimes and that of positive energy representations of symmetry groups, are equivalent to the existence of particular Adjoint-invariant convex cones in the symmetry algebras. This can be used to study backgrounds of supergravity and string theories through their symmetry groups. Our formalism is based on Segal's approach to infinitesimal causal structures on manifolds. The Adjoint action in the symmetry group is shown to correspond to changes of inertial frames in the spacetime, whereas Adjoint-invariance encodes invariance under changes of observers. This allows us to give a group theoretical description of the horizon structure of spacetimes, and also to lift causal structures to the Hilbert spaces of quantum theories. Among other results, by setting up the Dirac procedure for the complexified universal algebra, we classify the physically inequivalent observables of quantum theories. We illustrate this by finding the different Hamiltonians for stationary observers in AdS_2.