Talk:Momentum map

Untitled
I see that the PlanetMath page 'momentum map' rather suggests this should be tied in with the coadjoint action. Charles Matthews 21:32, 13 January 2006 (UTC)

equivariance
My understanding is that the moment map is required to be equivariant with respect to the action of the group on the manifold and the coadjoint action of the group on the dual of its Lie algebra. Yggdrasil014 23:20, 22 January 2007 (UTC)


 * Yes: this is a translation of the fact that the map from the Lie algebra of G to the Lie algebra of functions under Poisson bracket is required to be a Lie algebra homomorphism. Geometry guy 10:21, 10 February 2007 (UTC)

No: I don't think it is required to be equivariant. If the group is compact then it can be chosen to be equivariant, but in general not and one has to modify the coadjoint action in order to make it equivariant (as shown by Souriau). Alternatively, one can modify the Poisson brackets so that map is a Lie algebra homomorphism. Simplifix (talk) 15:13, 12 February 2008 (UTC)

Clean-up?
I think this page needs a bit of a clean-up. I've made a few minor changes, but the main problem is that motivation and definition are mixed together in a confusing way. For me, a Hamiltonian group action is a group action on a symplectic manifold with a (equivariant) moment map, although I don't mind to define it using Poisson brackets instead, as long as the equivalence of the two formulations is made clear. I hope to come back to this, anyway. Geometry guy 10:26, 10 February 2007 (UTC)


 * Hi, Geometry guy. I'm glad you're working on the article; your edits have been helpful already. Some notes:


 * (1) The addition of the equivariant condition in a recent edit seems to be causing the confusion. (Also, much of the rest is a careless amalgam of Donaldson & Kronheimer and McDuff & Salamon.) You propose that we define the moment map, define equivariance, define Hamiltonian group action using it, and then elaborate on this in terms of the current definition? Sounds good.


 * (2) I think the new opening sentence focuses too much on conserved quantities. A symplectic topologist who uses the moment map to do a symplectic cut is probably not thinking about them; she's thinking more about algebraic geometry. Also, there is no description of conserved quantities in the article. (But the article would be better if there were one...please? :) Joshua R. Davis 15:44, 10 February 2007 (UTC)


 * Thanks. Yes, something like that was what I had in mind. I think for most symplectic geometers, the equivariance of the moment map is part of the definition, but I don't want to insist on that. I agree with you about conserved quantities (and it's no problem with me if you want to redo the opening paragraph), but hope that a physicist will be moved to contribute to this article at some point. Meanwhile, I'll do what I can... Geometry guy 15:57, 10 February 2007 (UTC)

I've still not got round to this, so I will add a tag. Geometry guy 21:23, 21 February 2007 (UTC)

Moment vs momentum
Which title would be preferred? As I understand it, geometers tend to use moment map, whereas physicists are more likely to say momentum map. I'm a geometer, but I always say "momentum map", because I believe the term "moment map" is a mistranslation of the term application moment coined in Souriau's original (French) article. The terms "moment" and "momentum" are not synonymous in English, and "momentum" is the relevant word in this context. Geometry guy 10:31, 10 February 2007 (UTC)


 * Thanks for pointing out the possible French origin. I don't care much about which we use --- especially since Momentum map already redirects here --- but there are a few points to keep in mind. Original research is not allowed; the terminology, the history, and the math itself should all be backed up with sources. We should be using whatever terminology is commonly used, whether its etymology is simple or not. Perhaps you could add a note about the etymology, if you can source it. My original choice of Moment map for the title arose from its use in Donaldson & Kronheimer and McDuff & Salamon, two standard references in geometry. But I'm not wedded to it. Joshua R. Davis 15:00, 10 February 2007 (UTC)


 * Good point. At the moment (excuse the pun), this article reads primarily like a symplectic geometry article (even though it is in wiki-project physics), in which case, I think "moment map" is probably still the prevalent term, although more and more geometers are using "momentum map". I'll add a reference to Souriau anyway. I'm happy to postpone the renaming issue: to some extent it depends on whether there is more physics input. Thanks for starting this article, by the way: it is a very important topic! Geometry guy 15:49, 10 February 2007 (UTC)


 * Now it's part of WikiProject:Mathematics as well. I labeled it "geometry" rather than "mathematical physics", because it enjoys geometric application independent of physics. Joshua R. Davis 15:47, 12 February 2007 (UTC)


 * Great! And I agree with your choice. Still, I think it is worthwhile to attract physicists to this article.... Geometry guy 23:37, 12 February 2007 (UTC)

Importance
I've changed the Math importance rating from low to mid. The moment map is far from "peripheral knowledge, possibly trivial"; it is a key tool in symplectic geometry and mathematical physics, and ties in to toric geometry as well --- which reminds me that we need to talk about that here. Joshua R. Davis 15:26, 1 March 2007 (UTC)


 * I fully support the uprating. Geometry guy 17:31, 1 March 2007 (UTC)

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 10:00, 10 November 2007 (UTC)

Convexity
One of the most beautiful properties of the moment(um) map is the convexity of its image (Atiyah, Guillemin-Sternberg, Kirwan theorems). I think this should be included in the article. Perhaps I'll have a go at writing something, unless someone else does it first. Simplifix (talk) 15:16, 12 February 2008 (UTC)


 * I agree that it should be discussed. Please do have a go. Joshua R. Davis (talk) 15:49, 12 February 2008 (UTC)

exponential map necessary?
In the "formal definition" section, wouldn't it be simpler to put
 * $$\, (R_x)_* \xi,$$

where $$R_x$$ is right translation by $$x$$, instead of
 * $$\left.\frac{d}{dt}\right|_{t = 0} \exp(t \xi) \cdot x?$$

I don't want to change it myself because I am not very familiar with this area. —Preceding unsigned comment added by 24.30.147.224 (talk) 02:25, 2 October 2008 (UTC)


 * x is an element in the manifold so R_x does not make sense (the manifold is not necessarily a Lie group).Eok20 (talk) 23:05, 17 May 2010 (UTC)

Curvature is a moment map
Ana Cannas da Silva's book ''Lec. on sym. geo.'' showed that(atribute to Atiyah & Bott) the space of connection on a principal bundle is a (infinite dimension) symplectic manifold, the group of gauge tran. act on this space Hamitonian, the curvature is the moment map!--刻意(Kèyì) 06:12, 23 March 2009 (UTC)

Assessment comment
Substituted at 00:20, 30 April 2016 (UTC)

Closed vs. exact
The definition of momentum map as stated uses exactness, not closedness. Panpach 11:39, 2. October 2018 (UTC)
 * The text was correct, but I have added text to clarify the relationship. Mgnbar (talk) 17:52, 2 October 2018 (UTC)
 * Unfortunately, the new text is redundant with the subsequent section on Hamiltonian actions. Mgnbar (talk) 20:16, 2 October 2018 (UTC)