Feynman checkerboard

The Feynman checkerboard, or relativistic chessboard model, was Richard Feynman's sum-over-paths formulation of the kernel for a free spin-⁠1/2⁠ particle moving in one spatial dimension. It provides a representation of solutions of the Dirac equation in (1+1)-dimensional spacetime as discrete sums.

The model can be visualised by considering relativistic random walks on a two-dimensional spacetime checkerboard. At each discrete timestep ${\displaystyle \epsilon }$ the particle of mass ${\displaystyle m}$ moves a distance ${\displaystyle \epsilon c}$ to the left or right (${\displaystyle c}$ being the speed of light). For such a discrete motion, the Feynman path integral reduces to a sum over the possible paths. Feynman demonstrated that if each "turn" (change of moving from left to right or conversely) of the space–time path is weighted by ${\displaystyle -i\epsilon mc^{2}/\hbar }$ (with ${\displaystyle \hbar }$ denoting the reduced Planck constant), in the limit of infinitely small checkerboard squares the sum of all weighted paths yields a propagator that satisfies the one-dimensional Dirac equation. As a result, helicity (the one-dimensional equivalent of spin) is obtained from a simple cellular-automata-type rule.

The checkerboard model is important because it connects aspects of spin and chirality with propagation in spacetime^[1] and is the only sum-over-path formulation in which quantum phase is discrete at the level of the paths, taking only values corresponding to the 4th roots of unity.