What model represents Dirac-equation solutions as discrete sums on a checkerboard?

The Feynman checkerboard represents Dirac-equation solutions as discrete sums on a checkerboard. It is a sum-over-paths model for a free relativistic spin-½ particle in one spatial dimension.

In the model, a particle moves left or right at each small time step, producing zigzag paths across a two-dimensional spacetime lattice. Every change in direction receives a complex weight related to the particle’s mass, and the weighted paths are summed. In the continuum limit, that sum produces a propagator satisfying the one-dimensional Dirac equation.

The model is not the same as the Ising model, although later work connected the checkerboard construction with Ising-like ideas. Nor is it a general four-dimensional simulation of all relativistic quantum mechanics: its original setting is 1+1-dimensional spacetime.

Feynman developed the idea in the 1940s while working on his spacetime approach to quantum mechanics. It was published later, including in the 1965 book Quantum Mechanics and Path Integrals, coauthored with Albert Hibbs. The model is notable for linking path integrals, spin, chirality, and discrete quantum phases.