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.



What is the current state of generalizations Noether's theorem?
Noether's theorem was successfully generalized to the setting of quantum field theory in the fifties. This is called [Ward-Takahashi identity](http://en.wikipedia.org/wiki/Ward%E2%80%93Takahashi_identity).


