Is there a simple maximally general generalization of Noether's theorem to arbitrary dynamical systems?

Question

Noether's theorem informally states something like "symmetries in the dynamical law imply conserved quantities". However, the theorem is generally stated in terms of physics-specific classes of dynamical systems such as Lagrangian or Hamiltonian dynamics.

I'm not a physicist and I am curious whether the result generalizes to non-physical dynamical systems. E.g. we can ask whether cellular automata satisfy conservation laws. Moreover, I think if I had a more general formulation of Noether's theorem that didn't rely on a lot of physics-specific details, I'd understand it better even in a physics context.

The most general formulation of a dynamical system I know is a tuple $(T, X, \Phi)$ where $T$ is a monoid (representing the time domain, e.g. $\mathbb R$ for continuous time, or $\mathbb Z$ for discrete time), $X$ is a non-empty set (state space) and $\Phi$ is a function $\Phi :(T\times X)\to X$ (the dynamical law).

Is there a generalization of Noether's theorem that doesn't assume anything (or at least makes minimal assumptions) about the dynamical system? Possibly not even that state space and time are continuous, so that the result applies directly to e.g. cellular automata?

Answer 1

Here is a really simple version of Noether's theorem valid for completely arbitrary dynamical systems: if $X$ is a set being acted on by an arbitrary "time monoid" $M$, and $G$ is a group of symmetries of $X$ (meaning it commutes with the action of $M$), then consider the partition of this dynamical system into a disjoint union of orbits for the action of $G$.

Proposition: Time evolution leaves invariant what $G$-orbit a point $x \in X$ is in.

So, if you like, here the "conserved quantities" are given by the indicator functions of each orbit. This may not seem impressive but I claim it is the correct analogue of Noether's theorem in this extremely general setting. One reason I believe this is that we already get something much closer to Noether's theorem by assuming that $X$ is a vector space rather than a set.

Namely, let $V$ be a vector space, again being acted on by an arbitrary "time monoid" $M$, now with the condition that the elements of $M$ act by linear maps. Let $G$ be a group of invertible linear maps acting on $V$ commuting with the action of $M$.

Proposition: Time evolution leaves invariant what $G$-invariant subspaces a point $v \in V$ is in. In particular, if $v \in V$ lies in a subspace which is irreducible as a representation of $G$, then time evolution keeps $v$ in this representation.

This is close to the "quantum Noether's theorem," which I think is actually easier to understand than the classical theorem. In quantum mechanics $V$ is a Hilbert space of "wave functions" like $L^2(\mathbb{R}^3)$ and $G$ is, for example, a group like $SO(3)$ acting on $\mathbb{R}^3$ by rotations. This choice of $G$ recovers a version of conservation of angular momentum. There is a whole philosophy in physics about using irreducible representations in this way; you can see a bit more about it here.

To get the real quantum Noether's theorem from here requires Stone's theorem on one-parameter unitary groups, which allows us to convert one-parameter subgroups of $G$ (equivalently, elements of the Lie algebra $\mathfrak{g}$) into observables (equivalently, self-adjoint operators) commuting with time evolution, together with possibly Wigner's theorem to really justify restricting attention to the case that $G$ acts by unitary operators. From here the classical Noether's theorem is in some sense a classical limit of the quantum one.