Spin on a 4D Feynman Checkerboard

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.

Extends in several ways our earlier paper, arXiv:hep-th/0310166

References in corpus (1)

• Causal cells: spacetime polytopes with null hyperfaces