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

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 Σ 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 H ∩ Σ∼=S2.