Relativistic Cellular Automata

Check out Mark Smith's PhD thesis titled Cellular automata methods in mathematical physics, specifically Chapter 4: Lorentz Invariance in Cellular Automata.

The conclusion part of the chapter:

Symmetry is an important aspect of physical laws, and it is therefore desirable to identify analogous symmetry in CA rules. Furthermore, the most important symmetry groups in physics are the Lorentz group and its relatives. While there is a substantial difference between the manifest existence of a preferred frame in CA and the lack of a preferred frame demanded by special relativity, there are still some interesting connections. In particular, CA have a well-defined speed of light which imposes a causal structure on their evolution, much as a Minkowski metric imposes a causal structure on spacetime. To the extent that these structures can be made to coincide between the CA and continuum cases, it makes sense to look for Lorentz invariant CA.

The diffusion of massless particles in one spatial dimension provides a good example of a Lorentz invariant process that can be expressed in alternative mathematical forms. A corresponding set of linear partial differential equations can be derived with a simple transport argument and then shown to be Lorentz invariant. A CA formulation of the process is also Lorentz invariant in the limit of low particle density and small lattice spacing. The equations can be solved with standard techniques, and the analytic solution provides a check on the results of the simulation. Generalization to higher dimensions seems to be difficult because of anisotropy of CA lattices, though it is still plausible that symmetry may emerge in complex, high-density systems. The model and analyses presented here can be used as a benchmark for further studies of symmetry in physical laws using CA.

In cellular automata I do know there is explicit dependence on step/time. In quantum mechanics (and other many other theories) it is natural to write local evolution with respect to time.

On the contrary, in 'pure' relativity, time is not that different from position. And thus there is no such natural interpretation like 'the next step is the next time'.

However, there are relativity-based equations (e.g. Dirac Equation, Maxwell Equation) in which time can be taken to be an independent variable. And for sure there are more papers on the topic than one, e.g.:

• Iwo Bialynicki-Birula, Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata, Phys. Rev. D 49, 6920–6927 (1994)

Furthermore, some relativistic aspects are easily implemented, like $c=\hbox{(pixel size)}/\hbox{(time step)}$. One cellular automaton-like thing is so-called Feynman Checkerboard, which bases on the assumption that every particle always travels at $c$ but also sometimes gets bounced (it turns that effective mass depends on the amplitude of bouncing).

• http://en.wikipedia.org/wiki/Feynman_checkerboard
• Feynman and Hibbs, Quantum Mechanics and Path Integrals, New York: McGraw-Hill, Problem 2-6, pp. 34-36, 1965 (the original textbook, when it appears as a problem for the reader)

I asked this very same question at mathoverflow, too (do the policies of PSE have anything against this?), and got one further answer, which I leave to your attention:

There are no "non-trivial" finite sub-groups of $O(3,1)$.