Jason (jcreed) wrote,
Jason
jcreed

I was randomly doodling some grids earlier today and a discrete notion of Lorentz boosts sort of popped into my head. I wonder if it is completely old news to people who know about discrete models of physics, particularly because it could have easily been invented almost a hundred years ago; it only captures special relativity, not GR or quantum or anything like that. It goes like this:

Suppose you have a collection S of states, and a group action B of the multiplicative group Q\{0} on S. A tableau T is a map Z2 → S. Basically it's like a standard 1-D cellular automaton layout, except "time" proceeds diagonally up and right in the following sense: the laws of physics are represented by a function f : S x S → S, and a tableau T is valid at (x,y) if T(x,y)=f(T(x-1,y),T(x,y-1)). So the tableau is obeying the laws of physics if the state at every lattice point is computed in terms of its two "light-cone" neighbors in the past. The whole picture of the world is rotated 45 degrees from how it's usually presented, so the speed-of-light world-lines are vertical and horizontal. Naturally we say a tableau is valid full stop if it's valid at every point in the lattice.

Now let n be some integer, and let Bn : S → S be n acting on S according to the action B. The laws of physics respect a boost of coefficient n if:

for any valid tableau T, there exists a valid tableau U, such that for all x, y, T(nx,y)=BnU(x,ny)

Conversely we say laws of physics respect a boost of coefficient 1/n if

for any valid tableau T, there exists a valid tableau U, such that for all x, y, T(x,ny)=B1/nU(nx,y)

The intuition is that the laws of physics stay the same if we stretch out spacetime along one speed-of-light-axis, and shrink it the same factor along the other; of course we may have to boost the "momenta" of "particles" by some factor, but that's what group action B is there to take care of.

Question is, are there interesting S, B, f that are Lorentz invariant, i.e. respect all boosts? I almost succeeded at hacking up a world of noninteracting particles of rational "velocities" that plunked around the discrete world by Floyd-Steinberg-esque accumulation, but I got some weird fencepost errors and gave up.
Tags: math, physics
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