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In this article by physicist Andrei Linde about quantum cosmology and inflation, when talking about constructing multiverse models, he indicates that cellular automata are not described by action principles:

(...) There are some additional steps that one may want to make. In our analysis we still assumed that any evolution must be described by some kind of action. Meanwhile there are some theories where equations of motion are known even though the action is unavailable. One may consider other models of evolution, based, e.g., on cellular automata.

In particular, he is referring to two equations in the paper (10 & 11) which are basically an action principle, and as such, it is the integral of some lagrangians.

However, can't cellular automata be described by lagrangians, as these papers indicate?

For example, in the first one of these papers, 't Hooft formulated the holographic principle with cellular automata and he uses hamiltonians (which can be equivalent to lagrangians) in the paper to develop this model. In the second one, he explores cellular automata models of fundamental physics and he described hamiltonians and lagrangians. There is also this one which also explores cellular automata models using lagrangians and hamiltonians.

So, couldn't cellular automata be described by lagrangians (or hamiltonians) after all?

vengaq
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1 Answers1

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No, in general cellular automata do not respect any conservation laws or symmetries, so they will not be representable by Lagrangians. Only a few rather special ones might.

For example, consider Game of Life. It has no conservation of energy (a pattern of active cells can decay into nothing), it has no conservation of momentum, and the dynamics is irreversible. This is not going to be representable.

Anders Sandberg
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