-(d/dt)2ψ = c2 ψ - (u2(d/dx)2 + (uv+vu)(d/dx)(d/dy) + v2(d/dy)2)ψ
and I was like oooooh that is so close to Klein-Gordon, except for that pesky uv+vu term. And then I was like "durr, WWPDD?" and I remembered that what you do in these situations is stare into the proverbial fireplace and replace scalars with vectors so that you can pick uv such that uv is not equal to vu, in fact such that uv = -vu and u2 = v2, so that that nasty crossterm vanishes and you get
-(d/dt)2ψ = c2 ψ - u2∇2ψ
and everything's hunky-dory and rotation- and Lorentz-invariant.
So I turned the crank backwards on my aforementioned intuition about continuous space vs. discrete space, the idea being that it should tell me something about doing little two-qubit interactions on a lattice, and I thought I got a nice plain ol' hexagonal one like so:
where you throw a term into the Hamiltonian operator for each edge. And I thought this would guarantee me asymptotic circular symmetry, but it didn't exactly:
first of all, you've got the hexagonal artifact going on there, and second of all, I scaled it horizontally by an eyeballed factor of a little less than 2 (and seriously, what naturally arising constants are a little less than 2? Pff) to make it look circular...ish, and third of all, I reckoned that Klein-Gordon was supposed to hold nicely on every fourth lattice site (kind of implying that I have Dirac spinors not Weyl spinors?) but in fact the pattern repeats with a unit cell that is 8 pixels big, so WTFFFF.
Anyway yeah, like the dog says, I have very little confidence in my methods here.