Spin on a 4D Feynman Checkerboard

TL;DR

This paper presents a discretized path integral formulation for the Weyl equation on a 4D lattice, avoiding fermion doubling and incorporating mass terms for chiral and Majorana particles.

Contribution

It introduces a novel lattice scheme for Weyl fermions that prevents fermion doubling and includes mass terms through amplitude modifications.

Findings

Fermion doubling is avoided in this discretization.

Path integral amplitude depends on steps, bends, and chirality.

Mass terms are incorporated via specific amplitude factors.

Abstract

We discretize the Weyl equation for a massless, spin-1/2 particle on a time-diagonal, hypercubic spacetime lattice with null faces. The amplitude for a step of right-handed chirality is proportional to the spin projection operator in the step direction, while for left-handed it is the orthogonal projector. Iteration yields a path integral for the retarded propagator, with matrix path amplitude proportional to the product of projection operators. This assigns the amplitude $i^{\pm T}\, {3}^{-B/2}\,2^{-N}$ to a path with $N$ steps, $B$ bends, and $T$ right-handed minus left-handed bends, where the sign corresponds to the chirality. Fermion doubling does not occur in this discrete scheme. A Dirac mass $m$ introduces the amplitude $i\epsilon m$ to flip chirality in any given time step $\epsilon$, and a Majorana mass similarly introduces a charge conjugation amplitude.