Continuous space-time symmetries of the lattice Dirac equation

Abstract

We show that the (2 + 1)-dimensional Dirac equation on a square lattice is invariant under a nonlinear representation of the Poincaré group. We construct the generators explicitly and show that they reduce in the continuum limit to the usual linear representation of the Poincaré algebra. We also discuss the fourfold degeneracy of the lattice Dirac theory, and show how it can be removed by diagonalizing certain discrete transformations that commute with the Poincaré generators. The extension of our work to 3 + 1 dimensions and to interacting theories is briefly discussed.