Discrete reflection groups and induced representations of Poincare group on the lattice

TL;DR

This paper explores discretizing physical models by using discrete reflection groups to define Lorentz transformations and Poincaré group representations on a lattice, ensuring covariance of fundamental wave equations.

Contribution

It introduces a method to discretize Lorentz and Poincaré symmetries using integral reflection groups, advancing lattice formulations of relativistic quantum mechanics.

Findings

Calculated integral Lorentz transformations via discrete reflection groups.

Established covariance of Klein-Gordon and Dirac equations on the lattice.

Defined unitary Poincaré group representations on discrete spaces.

Abstract

We continue the program, presented in previous Symposia, of discretizing physical models. In particular we calculate the integral Lorentz transformations with the help of discrete reflection groups, and use them for the covariance of Klein-Gordon and Dirac wave equation on the lattice. Finally we define the unitary representation of Poincar group on discrete momentum and configuration space, induced by integral representations of its closed subgroup.