Extensions
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 J. A. Wheeler, that antiparticles are equivalent to particles moving backwards in time. His notes contain several sketches of chessboard paths with added spacetime loops. The first extension of the model to explicitly contain such loops was the ‘Spiral Model' in which chessboard paths were allowed to spiral in spacetime. 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. Subsequently the roles of Zitterbewegung, antiparticles and the Dirac Sea in the Chessboard model have been elucidated and the implications for the Schrödinger equation considered through the non-relativistic limit.
Further extensions of the original 2-dimensional spacetime model include features such as improved summation rules and generalized lattices.There has been no consensus on an optimal extension of the Chessboard model to a fully four-dimensional space-time. Two distinct classes of extensions exist, those working with a fixed underlying lattice and those that embed the two dimensional case in higher dimension. 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. In the latter extensions the fixed speed property is maintained at the expense of variable directions at each step.
Read more about this topic: Feynman Checkerboard
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