A Covariant Information-Density Cutoff in Curved Space-Time

In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible perfectly from discrete samples. In this Letter, sampling theory is generalized to pseudo-Riemannian manifolds. This provides a new set of mathematical tools for the study of space-time at the Planck scale: theories formulated on a differentiable space-time manifold can be completely equivalent to lattice theories. There is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity.Comment: 4 pages, RevTe

Short-Distance Cutoffs in Curved Space

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.Comment: Talk presented at 10th Marcel Grossmann Meeting, Rio de Janeiro, July 20-26, 200