Lorentz and Galilei invariance on lattices

Abstract

We show that the algebraic aspects of Lie symmetries and generalized symmetries in nonrelativistic and relativistic quantum mechanics can be preserved in linear lattice theories. The mathematical tool for symmetry preserving discretizations on regular lattices is the umbral calculus.

Discretization of partial differential equations preserving their physical symmetries

“…One conclusion is that different discrete physical systems require different approaches. It was shown elsewhere [12,13] that symmetries of linear theories can be studied in terms of commuting difference operators on fixed lattices. The results of this and related papers [1-10, 14, 15, 17, 21] indicate that at least for nonlinear discrete phenomena, the lattice should be considered as a dynamical one evolving together with the solution and described by an invariant system of difference equations.…”

“…The P∆S given by (12) and (14) is no longer invariant under the infinite-dimensional conformal group. The symmetry group of the lattice is reduced to dilations and translations of t and x.…”

On Appell sequences of polynomials of Bernoulli and Euler type

“…In other words, the Bernoulli-type polynomials of order p considered here correspond to a class of formal power series G(t) representing suitable difference delta operators of order p (denoted by p ). These delta operators have been introduced in [28,29], where a version of Rota's operator approach, based on the theory of representations of the Heisenberg-Weyl algebra, has been outlined. In this paper we wish to further illustrate the connection between Rota's approach and the theory of Appell and Sheffer polynomials.…”

Umbral Deformations on Discrete Space–time

“…Remarkably, even as something like a fundamental spacetime lattice of spacing a = O(l P lanck ) is thus likely to underlie conventional physics, continuous symmetries, such as Galilei or Lorentz invariance, can actually survive unbroken such a deformation into discreteness, in a nonlocal, umbral realization [2].…”