Well, the system I am running is the same as before:
|L,n> -> c|L,n-1> + d|R,n-1>
|R,n> -> d|L,n+1> - c|R,n+1>
where c and d are constants such that c^2 + d^2 = 1.
The top picture has c=d=1/sqrt(2), and the bottom one has c set to 1-epsilon for epsilon I think like 1/50 or so.
Why is the banding interesting? Well, the important quantity in special relativity is the Minkowski metric x^2 + y^2 + z^2
- t^2. (minus sign in red for emphasis. It is hard to italicize a minus sign :P) Or in my 1-d case here, x^2 - t^2. The 'surfaces of constant value' for this metric are hyperbolae whose arms bend down to meet the edges of the light-cone. The banding is (possibly) exiting to me since it suggests the Minkowski metric can arise naturally out of a simple discrete model. But like I said, maybe they're not really exactly hyperbolic-shaped, and maybe it's not therefore as exciting.
For comparison, there definitely is a super easy toy model from which the
Euclidean metric asymptotically arises: just imagine a classical particle doing a discrete random walk in N dimensions, independently in each direction. Its probability distribution in one dimension is a Gaussian ~ e^(-x^2), and independent dimensions multiply, so you get something ~ e^(-x^2)e(-y^2)(e-z^2), and exponentials turn multiplication into addition, so this is e^(-(x^2+y^2+z^2)) = e^(-r^2), and staring you in the face is a function that depends only on the distance r from the origin.
If I wanted to sound woo-y, I would say that maybe the fundamental reason that physics just *has* to be rotation invariant (as opposed to some other weird transformation on 3-dimensional space) is because of the central limit theorem. No matter what lattice you start with, you're going to end up with the 2-norm on a large scale, and the familiar notion of rotation that we all know and love that preserves it.
Edited at 2011-06-28 12:12 pm (UTC)
