The Feynman Checkerboard Wikipedia article states: "There has been no consensus on an optimal extension of the Chessboard model to a fully four-dimensional space-time."
Why is it hard to extend it to more than 1+1 dimensions?
Answer
There's an interesting paper on the subject, written at a lower level than some, on my website here.
It notes that there are issues with the particle in 3 dimensions apparently being superluminal with a speed of at least $\sqrt{3}\;c$.
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