Feynman Checkerboard

The Feynman Checkerboard or Relativistic Chessboard model was Richard Feynman’s sum-over-paths formulation of the kernel for a free spin ½ 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 the particle of mass moves a distance ( being the speed of light) to the left or right. 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 vice versa) of the spacetime path is weighted by (with denoting the reduced Planck's constant), in the limit of vanishing 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 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.