Short-Distance Cutoffs in Curved Space

TL;DR

The paper explores how space-time with certain differentiability and finiteness properties can be reconstructed from discrete samples, linking sampling theory with space-time structure.

Contribution

It introduces a framework where space-time fields can be fully reconstructed from discrete points, bridging sampling theory and curved space-time geometry.

Findings

Space-time may be reconstructed from dense discrete samples.

Sampling criteria could be based on Planck-scale spacing.

Mathematical foundation from sampling theory applied to space-time.

Abstract

It is shown that space-time may possess the differentiability properties of manifolds as well as the ultraviolet finiteness properties of lattices. Namely, if a field's amplitudes are given on any sufficiently dense set of discrete points this could already determine the field's amplitudes at all other points of the manifold. The criterion for when samples are sufficiently densely spaced could be that they are apart on average not more than at a Planck distance. The underlying mathematics is that of classes of functions that can be reconstructed completely from discrete samples. The discipline is called sampling theory and is at the heart of information theory. Sampling theory establishes the link between continuous and discrete forms of information and is used in ubiquitous applications from scientific data taking to digital audio.