Mathematical Puzzles from Causal Set Quantum Gravity

Abstract

I will discuss some of the mathematical puzzles that arise from the causal set approach to quantum gravity. In this approach, any causal continuum spacetime is said to be emergent from an underlying ensemble of locally finite posets which represents a discretisation of the causal structure. If the discrete substructure is to capture continuum geometry to sufficient accuracy, then it must be "approximately" close to it. How can we quantify this closeness? This discreteness, while also preserving local Lorentz invariance, leads to a fundamental non-locality. This is not only an obstacle to a “traditional” initial value formulation, but also to the geometric interpretation of entanglement entropy. Is there an analytic way to quantify the remanent Planckian non-locality? These questions, as well as others arising from the quantum dynamics of causal, may be of potential interest to mathematicians, in particular Geometers and Combinatorists.