Relativistic Cellular Automata - MathOverflow

Relativistic Cellular Automata

2010-11-16T08:42:14Z

Cellular automata provide interesting models of physics: Google Scholar gives more than 25,000 results when searching for "cellular automata" physics.

Google Scholar still gives more than 2,000 results when searching for "quantum cellular automata".

But it gives only 1 (one!) result when searching for "relativistic cellular automata", i.e. cellular automata with a (discrete) Minkoswki space-time instead of an Euclidean one.

How can this be understood?

Why does the concept of QCA seem more promising than that of RCA?

Are there conceptual or technical barriers for a thorough treatment of RCA?

Answer by Willie Wong for Relativistic Cellular Automata

2010-11-16T12:10:45Z

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.

Answer by Hans Stricker for Relativistic Cellular Automata

2010-11-17T09:57:30Z

I asked this very same question at physics.stackexchange, too (do the policies of MO have anything against this?), and got an interesting hint, which I leave to your attention:

Cellular automata methods in mathematical physics

Answer by Peter Shor for Relativistic Cellular Automata

2010-11-17T15:32:43Z

What Willie's answer shows is that, for some non-trivial Lorentz-translatable cellular automaton, every cell would need an infinite number of neighbors, a contradiction. There's a way of getting around this, though. You could make each cell correspond to a point in space-time and also a boost (a boost is essentially a velocity in the Lorentz group). Then, cells would interact with cells both close to them in space-time and also close in boost. I don't know whether anybody has considered cellular automata like this.

In order for this to have a correspondence to realistic quantum field theories, it would have to be the case that when two particles interact at a high boost, the interaction strength goes to 0 as the boost goes to infinity. I don't know whether this is true, although the thought experiment of considering particles falling into a black hole through a sea of Hawking radiation makes it seem like it might be.