Feynman checkerboard

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.

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.

[3] The model was not included with the original path-integral article[2] because a suitable generalization to a four-dimensional spacetime had not been found.

[12] Although Feynman did not live to publish extensions to the chessboard model, it is evident from his archived notes that he was interested in establishing a link between the 4th roots of unity (used as statistical weights in chessboard paths) and his discovery, with John Archibald Wheeler, that antiparticles are equivalent to particles moving backwards in time.

Unlike the chessboard case, causality had to be implemented explicitly to avoid divergences, however with this restriction the Dirac equation emerged as a continuum limit.

[14] Subsequently, the roles of zitterbewegung, antiparticles and the Dirac sea in the chessboard model have been elucidated,[15] and the implications for the Schrödinger equation considered through the non-relativistic limit.

[16] Further extensions of the original 2-dimensional spacetime model include features such as improved summation rules[17] and generalized lattices.

Two distinct classes of extensions exist, those working with a fixed underlying lattice[19][20] and those that embed the two-dimensional case in higher dimension.

[21][22] The advantage of the former is that the sum-over-paths is closer to the non-relativistic case, however the simple picture of a single directionally independent speed of light is lost.