The Poincare Group of Discrete Minkowskian Space-Time

TL;DR

This paper explores the structure of the Poincare group in a discrete Minkowskian space-time lattice, identifying its representations and implications for lattice field theories, especially for massless particles and helicity states.

Contribution

It characterizes the lattice Poincare and Lorentz groups, their representations, and shows how Wigner's method applies to discrete space-time, enabling descriptions of massless fields with arbitrary helicity.

Findings

Lattice Lorentz group has irreducible projective representations for all helicities.

Unitary representations describe lattice free fields of zero mass and arbitrary helicity.

No representations with nonzero invariant mass are found.

Abstract

The lattice of integral points of 4-dimensional Minkowski space, together with the inherited indefinite distance function, is considered as a model for discrete space-time. The Lorentz and Poincare groups of this discrete space-time are identified as subgroups of the corresponding Lie groups. The lattice Lorentz group has irreducible projective (including linear) representations which are restrictions of (all) finite-dimensional irreducible projective representations of the Lorentz Lie group and hence can be used to describe all integral and half-odd-integral helicity. The (4-torus) momentum space has a well-defined ``light cone'' of null points and there are orbits of the lattice Lorentz group lying entirely in the torus light cone and having the lattice euclidean group of the plane as little group. Wigner's method for the Poincare Lie group can then be adapted to show, in the first instance, that the lattice Poincare group has unitary representations describing lattice free fields of zero mass and an arbitrary Lorentz helicity, in particular chiral fermions. There are no representations with a nonzero invariant mass.