Abstract
We propose a simple counting problem involving chains of rectangles on a planar lattice. The boundaries of the chains form a type of random walk with a finite inner scale. With orientation neglected, the continuum limit of the walk densities obeys the Telegraph equation, a form of diffusion equation with a finite signal velocity. Taking into account the orientation of the rectangles, the same continuum limit yields the Dirac equation. This provides an interesting context in which the Dirac equation is phenomenological rather than fundamental.
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A talk by G. N. Ord given at “Lattices and Trajectories: A Symposium of Mathematical Chemistry in honour of Ray Kapral and Stu Whittington”, Fields Institute, May 2007.
Preprint submitted to Elsevier—13 March 2008.
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Ord, G.N. Counting oriented rectangles and the propagation of waves. J Math Chem 45, 65–71 (2009). https://doi.org/10.1007/s10910-008-9368-5
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DOI: https://doi.org/10.1007/s10910-008-9368-5