Lorentz and Galilei Invariance on Lattices

TL;DR

This paper demonstrates how Lie and generalized symmetries in quantum mechanics can be maintained in lattice models using umbral calculus, bridging continuous symmetries with discrete frameworks.

Contribution

It introduces a method to preserve algebraic symmetries in lattice quantum theories through umbral calculus, a novel approach for symmetry-preserving discretizations.

Findings

Symmetries are preserved in lattice quantum models.

Umbral calculus effectively discretizes continuous symmetries.

The approach applies to both nonrelativistic and relativistic cases.

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.