Time-slicing quantum spacetimes

TL;DR

This paper develops a framework for quantum Riemannian geometry on curved spacetimes with a foliation into quantum spatial slices, providing solutions for fuzzy spheres and discrete models like Z_n-FLRW.

Contribution

It introduces a general construction for quantum Levi-Civita connections on quantum spatial slices and solves it for specific models including fuzzy spheres and Z_n-FLRW.

Findings

Provides a simple form for evolving quantum metrics with a first order ODE.

Derives fuzzy versions of classical (pseudo)-Riemannian manifolds.

Fully solves for rotationally invariant spacetimes with discrete angular directions.

Abstract

For quantum field theory on curved spacetimes, a critical role is played by their foliation into spacelike time-slices at each value $t$ of a coordinate time, with corresponding metric in ADM form. We provide a general construction for the spacetime quantum Levi-Civita connection when each spatial slice is replaced by a quantum Riemannian geometry. This is then fully solved for a class of spatial algebras including fuzzy spheres and for any time-dependent spatial quantum metric, shift 1-form and lapse function. The result takes a particularly simple form if the spatial metric evolves in time according to a first order ODE which, in the case of a fuzzy sphere, requires the spatial metric to rotate in time according to the value at each $t$ of the shift vector. As an application, our results provide in principle fuzzy versions of most (pseudo)-Riemannian manifolds. We also fully solve the case of rotationally invariant spacetimes with angular directions replaced by a discrete circle, including a new $\Bbb Z_n$-FLRW model.