A Generalized Shannon Sampling Theorem, Fields at the Planck Scale as Bandlimited Signals

TL;DR

This paper extends the Shannon sampling theorem to fields defined on unsharp space-time coordinates, showing they are continuous with finite degrees of freedom, akin to bandlimited signals, bridging quantum gravity models and information theory.

Contribution

It generalizes the Shannon sampling theorem to fields on unsharp coordinates, revealing their finite degrees of freedom and connection to quantum gravity models.

Findings

Fields are continuous but have finite degrees of freedom.

Unsharpness in coordinates is analogous to optical and electronic signal unsharpness.

The classical Shannon sampling theorem is recovered as a special case.

Abstract

It has been shown that space-time coordinates can exhibit only very few types of short-distance structures, if described by linear operators: they can be continuous, discrete or "unsharp" in one of two ways. In the literature, various quantum gravity models of space-time at short distances point towards one of these two types of unsharpness. Here, we investigate the properties of fields over such unsharp coordinates. We find that these fields are continuous - but possess only a finite density of degrees of freedom, similar to fields on lattices. We observe that this type of unsharpness is technically the same as the aperture induced unsharpness of optical images. It is also of the same type as the unsharpness of the time-resolution of bandlimited electronic signals. Indeed, as a special case we recover the Shannon sampling theorem of information theory.