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

**Z**

^{2}→ 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 B

_{n}: 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)=B

_{n}U(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)=B

_{1/n}U(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.