Sum of squares in discrete physical spaces

Abstract

Discretization of three-dimensional physical spaces can induce, on observable physical quantities, effects which are not present in the continuum. Consider, as an example, the problem of the radiation spectrum of a blackbody, studied in introductory courses in quantum mechanics. One sees that the Rayleigh assumption of continuous and uniform frequency distribution of standing waves inside a cubic cavity with perfectly reflecting inner walls can be validated by a heuristic type of reasoning. However, by means of number theory, one sees that there might exist frequencies for which it is not possible to have standing waves inside the cavity. Nevertheless, within the same context, one can argue that a more general criterion can be adopted to validate the hypothesis of continuity of the observables which are expressed as the square root of the sum of three integers of a threedimensional space On x n y n z.