I will formulate my question in the classical case, where things are simplest.

Usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the configuration space $M$ which fix the action functional $S:P\rightarrow \mathbb{R}$, where $P$ is the space of time evolutions, ie. differentiable paths in $M$. The idea is that, given some initial configuration $(x_0,v_0)\in TM$, there is a path in $P$ passing through $x_0$ with velocity $v_0$ and minimizing $S$ among all such paths. I will assume that this path is unique, which is almost always the case. Thus, if a diffeomorphism fixes $S$, it commutes with determining this path. One says that the physics is unchanged by taking the diffeomorphism.

Now here's the question: are there other diffeomorphisms which leave the physics unaltered? All one needs to do is ensure that the structure of the critical points of $S$ are unchanged by the diffeomorphism.

I'll be more particular. Write $P_{x_0,v_0}$ as the set of paths in $M$ passing through $x_0$ with velocity $v_0$. A diffeomorphism $\phi:M\rightarrow M$ is a symmetry of the theory $S:P\rightarrow \mathbb{R}$ iff for each $(x_0,v_0) \in TM$, $\gamma \in P_{x_0,v_0}$ is a critical point of $S|_{P_{x_0,v_0}}$ iff $\phi \circ \gamma$ is a critical point of $S|_{P_{\phi (x_0),\phi^* (v_0)}}$.

It is not obvious to me that this implies $S = S \circ \phi^{-1}$, where $\phi^{-1}$ is the induced map by postcomposition on $P$. If there are such symmetries, what can we say about Noether's theorem?

A perhaps analogous situation in the Hamiltonian formalism is in the correspondence between Hamiltonian flows and infinitesimal canonical transformations. Here, a vector field $X$ can be shown to be an infinitesimal canonical transformation iff its contraction with the Hamiltonian 2-form is closed. This contraction can be written as $df$ for some function $f$ (and hence $X$ as the Hamiltonian flow of $f$) in general iff $H^1(M)=0$. Is this analogous? What is the connection? It's been pointed out that this obstruction does not depend on the Hamiltonian, so is likely unrelated.

Thanks!

PS. If someone has more graffitichismo, tag away.

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