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I recently ran across this Not Even Wrong blog post which has the following passage

The moment map however gives you much more [than Noether's theorem], with phase space providing structure that is not visible just from the action.

My reading of this, from the context around it, is that what he means by "Noether's theorem" refers to only the version arising from symmetries of an action.

However, I believe there is there is a separate statement of Noether's theorem in terms of moment maps à la

Geometric Mechanics and Symmetry From Finite to Infinite Dimensions By Darryl D. Holm et. al. (2009)

specifically

Theorem 8.7. (Noether's formula for cotangent bundles) Let $G$ act on $Q$, and by cotangent lifts on $T^{*} Q .$ Then, the momentum map $\mathbf{J}: T^{*} Q \rightarrow \mathfrak{g}^{*}$ is obtained via the formula $$ J_{\xi}(\mathbf{q}, \mathbf{p})=\left\langle\mathbf{p}, \xi_{Q}(\mathbf{q})\right\rangle $$ where, for every $\xi \in \mathfrak{g}$, the map $J_{\xi}: T^{*} Q \rightarrow \mathbb{R}$ satisfies $J_{\xi}(\mathbf{q}, \mathbf{p})=\langle\mathbf{J}(\mathbf{q}, \mathbf{p}), \xi\rangle .$

My understanding of this is that the moment map itself plays the role of the conserved quantity in this setup.

Is it fair to say that this second formulation does make use of the full information of momentum maps, and that Dr. Woit's passage just means that the Noether theorem in terms of actions does not give you everything the second formulation would?

I have not yet completely understood the theorem depicted above so I am looking for some conceptual toeholds.

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  • $\begingroup$ Posting images of text and equations is very strongly discouraged. Please use Mathjax for mathematics (it is the site standard) and text for (big surprise) text. Note that images cannot be searched by the site engine. $\endgroup$ – StephenG Apr 15 at 20:09
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    $\begingroup$ @StephenG Switched... $\endgroup$ – rschwieb Apr 15 at 20:15
  • $\begingroup$ Thanks for the super fast response. $\endgroup$ – StephenG Apr 15 at 20:16
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Woit appears to be talking about situations where you have a symmetry of phase space that doesn't arise from a symmetry of the configuration space variables. Noether's theorem in terms of symmetries of the action only deals with the latter types of symmetries, since the action is a functional of configuration space. Woit specifically mentions the case of the 3D harmonic oscillator. He notes that while the action has $SO(3)$ rotational symmetry leading to conserved quantities associated with angular momentum via Noether's theorem, the phase space actually admits a larger symmetry group $U(3)$. These additional symmetries involve transformations that mix momenta and positions, and hence are not present as symmetries of the action. These symmetries are sometimes called hidden symmetries due to not being manifest in the action.

Your example involving a cotangent bundle is within the more restrictive class of Noether symmetries, since it involves transformations that arise from an action on the configuration space $Q$. The hidden symmetries described above would arise from transformations that are not constant in the fiber direction of the cotangent bundle, i.e. those that don't arise as the cotangent lift of a transformation of $Q$. Hence even in this formulation, the theorem is missing the possibility of having additional hidden symmetries. However, the moment map in this case would be able to provide conserved quantities for the additional symmetries not arising from transformations of the configuration space.

I think more generally Woit was trying to say that there can be more information in the phase space of the theory beyond what is manifest in the configuration space.

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  • $\begingroup$ I think I understand what you're saying. So it does seem that Noether-type theorems are (traditionally at least) restricted to flows arising from symmetries of configuration space, no matter if you are using a Hamiltonian or Lagrangian. That is very enlightening. From what I've been reading (Baez's recent discussion on Noether's theorem) a configuration space flow can be traded up to a flow on phase space and further on "observable space," and back down again. It looks like the moment map is a phase space gadget that can't be pressed down into configuration space. $\endgroup$ – rschwieb Apr 16 at 13:19
  • $\begingroup$ But given a symmetry of the phase space (not necessarily arising from a symmetry on the original manifold), one can still extract conserved quantities using the same program, right? The crux of Woit's comment is just that certain formulations of Noether's theorem "miss" symmetries, not that Noether's program doesn't work for the missing symmetries, right? $\endgroup$ – rschwieb Jun 3 at 14:08

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