Compact phase space, cosmological constant, discrete time

TL;DR

This paper investigates the quantization of geometry with a cosmological constant using discretized constant-curvature simplices, revealing a compact phase space and discrete time and space in 2+1 dimensions, with potential implications for 3+1 dimensions.

Contribution

It introduces a model where both intrinsic and extrinsic geometries are discrete, leading to a finite-dimensional Hilbert space and suggesting discretized time in quantum gravity.

Findings

Phase space is compact with finite-dimensional Hilbert space.

Both intrinsic and extrinsic geometries are discrete.

Results may extend to 3+1 dimensions.

Abstract

We study the quantization of geometry in the presence of a cosmological constant, using a discretiza- tion with constant-curvature simplices. Phase space turns out to be compact and the Hilbert space finite dimensional for each link. Not only the intrinsic, but also the extrinsic geometry turns out to be discrete, pointing to discreetness of time, in addition to space. We work in 2+1 dimensions, but these results may be relevant also for the physical 3+1 case.