Induced representations of Poincare group on the lattice: spin 1/2 and 1 case

TL;DR

This paper investigates how to construct and analyze the discrete Poincare group on a lattice, focusing on spin 1/2 and 1 cases, using the Wigner-Mackey method to understand their representations.

Contribution

It extends the Wigner-Mackey construction to the lattice setting, exploring the properties of the discrete Poincare group and its irreducible representations for specific spins.

Findings

Constructed the discrete Poincare group on the lattice.

Analyzed orbit conditions and irreducibility of representations.

Discussed the asymptotic limit of the constructed representations.

Abstract

Following standard methods we explore the construction of the discrete Poincare group, the semidirect product of discrete translations and integral Lorentz transformations, using the Wigner-Mackey construction restricted to the momentum and position space on the lattice. The orbit condition, irreducibility and assimptotic limit are discussed.