Exact solutions of the Dirac equation and induced representations of the Poincare group on the lattice

TL;DR

This paper derives the structure of the Dirac field on a lattice using discrete differential geometry and Lorentz transformations, revealing reducibility in Poincare group representations which relates to fermion doubling issues.

Contribution

It introduces a group theoretical approach to analyze the Dirac field on the lattice and demonstrates the reducibility of Poincare group representations in this context.

Findings

Poincare group representations on the lattice are reducible

The structure of the Dirac field is derived from discrete differential geometry

Provides insights into fermion doubling problem

Abstract

We deduce the structure of the Dirac field on the lattice from the discrete version of differential geometry and from the representation of the integral Lorentz transformations. The analysis of the induced representations of the Poincare group on the lattice reveals that they are reducible, a result that can be considered a group theoretical approach to the problem of fermion doubling.