Todd Rowland
Wolfram Research
Maryland
Registered: Oct 2003
Posts: 115 
I was talking to Anthony Martin about this problem at the Midwest NKS 2005 conference. It is a good problem, worth pursuing from a NKS perspective.
Let me suggest some ways to get started.
First of all, one might as well start in 2D, (or possibly start in 1D), just as E+M is usually taught. The most obvious thing to try is to borrow from the fluids model, the basic idea that each cell on a hexagonal grid contains at most one particle going in a given direction (not unlike the Pauli principle). One might start by insisting on conservation of momentum and conservation of energy, and enumerate all rules satisfying those properties. The search for a purely particle based system is smaller because noncolliding particles pass through each other, while I'd expect a E+M model to relax this requirement.
One would need a test, and the first might be with a system just consisting of photons. Only later, would one try adding photonelectron interactions and calculating whether the photons were properly scattered by an electron, and so on.
Feynman's book is a nice reference, and one could also consult p.1060. It is worth pointing out that these sorts of particle models don't take into account phase, which is a property which would ideally be a consequence of a model, rather than something built into it.
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