The well-known Noether's theorem is a vital tool in classical physics. But it assumes some hypothesis, many of which could be removed by a detailed look.

So my question is: In what directions has this theorem has been generalized in the context of Noncommutative geometry, Quantum Groups, Quantum Mechanics, etc? More specifically, the theorem (as found on the book Methods of Differential Geometry in Analytical mechanics, by Manuel de Léon & Paulo L. Rodrigues.) states, and I quote

"If $L$ admits a vector field $X$ on $M$, then $X^vL$ is a first integral of $L$, i.e., $\xi_L(X^vL) = 0$ where $\xi_L$ is the Euler-Lagrange vector field for $L$."

Here, $M$ is a smooth manifold and $L: TM \rightarrow \mathbb{R}$ is a regular Lagrange function.

I'm asking if there is any work in the direction of extending this type of statement into the context of, say, noncommutative manifolds and vector bundles thereof, or actions and coactions of a quantum group instead of an action of a Lie algebra, etc... Of course, an extension which respects quantum formalism.

Of course, one would naturally expect conservation of energy, momentum in QM, but do these follow from some form of Noether Theorem (NT)?

Also, what about more abstract symmetries? For instance, functional symmetries in cellular automata, or turing machines, or even further away from mainstream mathematics, what about semantic and syntactic symmetries in natural languages? Is is even possible to apply some NT-like theorem to these "symmetries"?

Thank you.

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    $\begingroup$ The directions of generalization of Noether's theorem(s) that you suggest seem rather arbitrary, with the exception of Quantum Mechanics. QM is an exception because it has strong parallels with Classical Mechanics, so that a version of both the statement and the proof of Noether's theorem(s) naturally suggests itself. I do not see that the same can be said for these other directions. I would say that the onus is on you to make it clear why there should even be generalizations like that, before something more definite could be said. $\endgroup$ – Igor Khavkine Mar 8 '15 at 16:59
  • $\begingroup$ For the various statements and generalizations of Noether's theorem(s) in their classical setting, take a look at the literature quoted in the answers to this question. $\endgroup$ – Igor Khavkine Mar 8 '15 at 17:00
  • $\begingroup$ Should I edit my question in order to adress this issue or comment right here with some answer? $\endgroup$ – Henrique Tyrrell Mar 8 '15 at 18:23
  • $\begingroup$ Something tells me that the additional clarifications you'd need to make won't fit in a comment. $\endgroup$ – Igor Khavkine Mar 8 '15 at 19:39

Noether's theorem was successfully generalized to the setting of quantum field theory in the fifties. This is called Ward-Takahashi identity.

  • $\begingroup$ Thank you very much. This completely answers the question in one of the three contexts. $\endgroup$ – Henrique Tyrrell Mar 8 '15 at 3:18

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