Umbral Deformations on Discrete Spacetime

TL;DR

This paper explores how a minimal length scale implies spacetime discreteness and introduces umbral deformation as a systematic method to discretize continuum physics, analyzing effects on oscillations and wave propagation.

Contribution

It provides a framework for applying umbral deformation to discretize continuum physics at Planck-scale lengths, with functional examples and analysis of wave phenomena.

Findings

Discrete oscillations are characterized by the deformation.

Wave propagation exhibits broad features influenced by discreteness.

Functional methods effectively implement the umbral deformation.

Abstract

Given a minimum measurable length underlying spacetime, the latter may be effectively regarded as discrete, at scales of order the Planck length. A systematic discretization of continuum physics may be effected most efficiently through the umbral deformation. General functionals yielding such deformations at the level of solutions are furnished and illustrated, and broad features of discrete oscillations and wave propagation are outlined.