I was taking a look at the 1994 PhD thesis of Mark Andrew Smith on Cellular Automata Methods in Mathematical Physics. I could only find one subsequent paper by Smith, on polymer simulation in 1999 with B. Ostrovsky. I assume he is no longer active. The only other work I found was some apparently self-published work by Canadian engineers in 1999, Tom Ostoma and Mike Trushyk. Like Smith they didn’t publish anything after 1999. It doesn’t seem to be an actively pursued field. The only reason I could find for this lack of pursuit was a comment on the Math Stack Exchange website by Willie Wong stating that
One of the reasons that it may be difficult to model Minkowski space based on cellular automata is that there are no “non-trivial” finite sub-groups of O(3,1), where non-trivial means that it doesn’t just reduce to just a finite sub group of O(3) via conjugation. So while cellular automata can be manifestly be homogeneous and isotropic (so admits a discrete O(3) symmetry), it becomes conceptually difficult to imagine some cellular automata capturing Lorentz symmetry.
Suppose we are trying to model physical space with a cubic square array of automata so that each cell has 26 neighbors. Suppose we imagine a center cell with coordinates 0,0,0 and all other cells have coordinates x,y,z offset from 0,0,0.
Consider the propagation of a ray of light emanating from a cell x,y,z in a direction indicated by x-y plane angle thetaXY and y-z plan angle thetaYZ with respect to x,y,z as the origin for purposes of describing the angle, with intensity I.
Suppose we are doing a simulation of N cells of space. For a given cell x,y,z, N-1 other cells may originate a light ray whose angle is such that it passes through x,y,z. At any given time step, any one of a cell’s 26 neighbors thus has the burden of transmitting N-1 light ray descriptions (x,y,z,thetaXY,thetaYZ,I) received indirectly from other cells on to this cell. It is not immediately clear that these descriptions can be combined in the sense of say “Fetch & Add” in an Ultracomputer.
What is the amount of information necessary to hold within a cell and to pass between cells in a single clock cycle, to represent the flow of light in a physical simulation in a cellular automata in a cubic array with N cells?
The light issue occurred to me because when I use my eyes, while not quite a single cell source, I can still move my head and see information which has some to me from very far and very near distances, and this information is instantly available to me wherever I move my head, which means that every neighboring cell needs to carry information about light travelling from any distance. This appears at first glance to be an infinite amount of information. Also, if one were powering a cell in a cellular automata and each cell needed to hold an infinite amount of information, it would also seem to need an infinite amount of energy to process or represent that information. It all gets very confusing at that point.
I like the idea of using cellular automata to represent physical space with fidelity to laws of relativity. In particular I watched the movie Interstellar, where one of the plotlines was the idea that wormhole travel, while quick for the traveler, would still entail the same time dilation effect as if the traveler did not use a wormhole. It would be fun to use cellular automata simulations to model this effect.