1 Answers
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 -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 ϵ {\displaystyle \epsilon } the particle of mass m {\displaystyle m} moves a distance ϵ c {\displaystyle \epsilon c} 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" of the space–time path is weighted by − i ϵ m c 2 / ℏ {\displaystyle -i\epsilon mc^{2}/\hbar } , 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 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.