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[Jun. 27th, 2011|08:41 pm]

I spent like five minutes making another view of that QCA-like thing I mentioned earlier, plotting born rule probability as brightness across time and space:

Sadly it doesn't exhibit the banding that real legit lorentz-invariant propagators do. It has some banding, true, but it wibbles in the wrong direction.
Oh wait no! If I adjust the parameters a bit, I do see some nice properly oriented banding.

Maybe I was just losing it in the Moiré before. Dunno if it's actually properly hyperbolic or not.
And as just one further thought, surely this picture is not Lorentz *invariant* because you can see a band of roughly constant value down the center vertically, which is a dead giveaway of a privileged rest frame. But maybe you can make it Lorentz *covariant* if you apply some transformation on the relative amplitudes of the 'L' and 'R' particles at the same time?

[User Picture]From: gwillen
2011-06-28 03:18 am (UTC)
Man, I no longer have any idea what you're talking about, but it sounds pretty interesting. You should try to explain it to me so I can shake my head and remain puzzles. ;-)
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[User Picture]From: jcreed
2011-06-28 11:35 am (UTC)
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)
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[User Picture]From: gaspaheangea
2011-06-29 01:59 am (UTC)
It kind of looks like a square rainbow. You may want to look a t the Airy functions
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[User Picture]From: jcreed
2011-06-29 02:25 am (UTC)
Yeah, nibot suggested that too back on http://jcreed.livejournal.com/1522191.html so maybe yinz are on to something. I don't really see how to make it work yet, but thanks for the suggestion!
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