Algebraic Quantum Theory on Manifolds: A Haag-Kastler Setting for Quantum Geometry

TL;DR

This paper extends the algebraic quantum field theory framework to manifolds without metric, defining causality topologically and applying it to quantum geometry near horizons, broadening the scope of quantum gravity models.

Contribution

It develops a topological causal structure and extends the Haag-Kastler axioms to manifolds without metric, enabling quantum geometry analysis in a more general setting.

Findings

Defined C-causality using cones on manifolds

Constructed nets of C*-algebras on manifolds

Applied framework to quantum geometry near horizons

Abstract

Motivated by the invariance of current representations of quantum gravity under diffeomorphisms much more general than isometries, the Haag-Kastler setting is extended to manifolds without metric background structure. First, the causal structure on a differentiable manifold M of arbitrary dimension (d+1>2) can be defined in purely topological terms, via cones (C-causality). Then, the general structure of a net of C*-algebras on a manifold M and its causal properties required for an algebraic quantum field theory can be described as an extension of the Haag-Kastler axiomatic framework. An important application is given with quantum geometry on a spatial slice within the causally exterior region of a topological horizon H, resulting in a net of Weyl algebras for states with an infinite number of intersection points of edges and transversal (d-1)-faces within any neighbourhood of the spatial boundary S^2.