Cones and causal structures on topological and differentiable manifolds

TL;DR

This paper develops a general framework for causal structures on manifolds, extending classical concepts to topological and differentiable settings, with potential applications in quantum field theory and quantum gravity.

Contribution

It introduces cone structures and causality notions on manifolds of dimension greater than two, generalizing standard space-time causal concepts to topological and differentiable manifolds.

Findings

Defined local cone structures via homeomorphisms or diffeomorphisms.

Introduced C-causality and a topological causal complement.

Proposed tools for quantum field theory and quantum gravity formulations.

Abstract

General definitions for causal structures on manifolds of dimension d+1>2 are presented for the topological category and for any differentiable one. Locally, these are given as cone structures via local (pointwise) homeomorphic or diffeomorphic abstraction from the standard null cone variety in R^{d+1}. Weak and strong local cone (LC) structures refer to the cone itself or a manifold thickening of the cone respectively. After introducing cone (C-)causality, a causal complement with reasonable duality properties can be defined. The most common causal concepts of space-times are generalized to the present topological setting. A new notion of precausality precludes inner boundaries within future/past cones. LC-structures, C-causality, a topological causal complement, and precausality may be useful tools in conformal and background independent formulations of (algebraic) quantum field theory and quantum gravity.