Talk:Noether's theorem

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Help with this template Comments: edit – history – watch – refresh The article doesn't state the precise theorem. It article can be made much more accessible. Start with finite-dimensional Hamiltonian mechanics and include an example like conservation of linear and angular momentum. Importance should perhaps be "high". -- Jitse Niesen (talk) 19:11, 31 May 2007 (UTC)

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The article doesn't state the precise theorem. It article can be made much more accessible. Start with finite-dimensional Hamiltonian mechanics and include an example like conservation of linear and angular momentum. Importance should perhaps be "high". -- Jitse Niesen (talk) 19:11, 31 May 2007 (UTC)

1 Initial Discussion
2
3 Nina Byers' article
- 3.1 Notes
4 Conformal transformation
5 Phase and electric charge
6 Conserved Current
7 making it more comprehensible
8 Example 2 from article
- 8.1 Example 2: Conservation of linear momentum
9 WikiProject class rating
10 Simpler special-case proof?
11 Disagrees with Emmy Noether
12 Converse of Noether's theorem true?
13 Dali picture
14 Confusing nature?

[edit] Initial Discussion

This is fascinating. They should teach this in undergraduate physics: what a powerful idea!

They do teach it in undergrad physics, usually during the second term of mechanics. —Preceding unsigned comment added by 128.135.100.164 (talk) 23:52, 17 February 2008 (UTC)

________

It is actually a misleading idea because it suggests that there is such a thing as a general law of energy conservation in physics. There isn't, because the notion of energy can be strictly defined only in Newtonion Physics. Noether's theorem in fact assumes Newtonion Physics as it uses Langrangian functions which in turn contain potential energy functions which in turn can only be defined for conservative force fields, i.e. for Newtonian physics (for related aspects see my site http://www.physicsmyths.org.uk/conservation.htm .

"the notion of energy can be strictly defined only in Newtonion Physics". Eh? -- The Anome 18:22, 8 Mar 2004 (UTC)

You might also want to read this post from John Baez -- The Anome 13:52, 2 May 2004 (UTC)

Question: Noether or Nöther? -- Anon.

It's Noether. 199.17.230.76 18:31, 23 Oct 2004 (UTC)

Heh. Noether, or neither? :-) zowie 00:17, 27 May 2006 (UTC)

The article should cleanly state the theorem before starting to give several proofs. As it stands, the intro contains some general intuitive hints, but the statement of the theorem is nowhere to be found. 199.17.230.76 18:31, 23 Oct 2004 (UTC)

To every symmetry group transformation, there corresponds a conserved current.

How's this formulation? Ancheta Wis 07:46, 24 Oct 2004 (UTC)

Well, that's a slogan but not a theorem. It omits the assumptions, and doesn't refer to cleanly definined concepts. 199.17.230.81 18:50, 24 Oct 2004 (UTC)

The concepts are meaningful to a physicist, but not to a mathematician, I see. I suggest reading invariant, conservation law, law of physics etc. If those are insufficient for you, then there are mathematical reviews in the literature which should meet your viewpoint. I should warn you that even John von Neumann's mathematical reviews of quantum mechanics etc did not survive close scrutiny by others, so you may wish to point out where you see deficiencies, and then we can make this the basis of a to-do list whose objective is to rectify the deficiencies, point by point. Are you willing to concede the concept of an observer, or do we have to go farther back than that? Ancheta Wis 23:19, 24 Oct 2004 (UTC)

I share the concerns of 199.17.230.81. The problem is not (at least not for me) that the concept "symmetry group transformation" and "conserved current" are not clear, but that it is simply not true that to every symmetry corresponds a conservation law. As the first sentence states, the model needs to be based on an action principle, but I did not find an explanation on what this exactly means. -- Jitse Niesen 10:25, 25 Oct 2004 (UTC)

Well, one could try to state the theorem that Emmy Noether actually proved, rather than discussing what people assume it says.

Charles Matthews 12:12, 25 Oct 2004 (UTC)

Great idea! Perhaps we can have three sections: statement, proof, controversy about physical implications. -- The Anome 12:14, 25 Oct 2004 (UTC)

There is even an English translation on the Web:

http://www.physics.ucla.edu/~cwp/articles/noether.trans/english/mort186.html

Charles Matthews 16:08, 25 Oct 2004 (UTC)

[edit]

This article seems to be written for a mathematics textbook, not for an encyclopedia. The following quote illustrates the problem: "But if you think about it, any two conserved currents differ by a divergenceless vector field". Umm, yeah. Maybe I'm being unreasonable expecting that an encyclopedia article should be comprehensible to someone who studied mathematics up to science degree level? Metamatic 16:17, 16 Dec 2004 (UTC)

That's the personal style of one contributor. I deprecate it. Charles Matthews 22:11, 16 Dec 2004 (UTC)

So do I - it may be a relatively unpleasant style even for a math textbook. But it is counterproductive to criticize it if you don't offer a better version than the person who wrote it. Is the beginning acceptable? When we have time, we can add a meaningful and comprehensible treatment of the mathematical issues, too. --Lumidek 22:58, 16 Dec 2004 (UTC)

Criticism may encourage someone else to attempt to translate it into English. And I would offer a better version if I could actually understand it to start with. Metamatic 23:01, 2005 May 27 (UTC)

You might start reading at Divergence#Properties, working backward from the goal of understanding the article, up through the links until an article is comprehensible from your POV. Then start reading forward from that page, drilling down through the links, translating the statements to yourself (not just parroting the content of the article) until you can state something consequential in English (but it is probably best to restrict the statement to the Talk page, until you attain consensus on the content of your statement). When you finally get to something you want you say in Wikipedia, then Be Bold. Ancheta Wis 15:48, 28 May 2005 (UTC)

Indeed, it would be prudent if someone can verify that the proof sections as currently written are not copyright violations of an actual published textbook passage. It reads rather like one. If it really is the case, at the very least we should have a proper citation of the source text to avoid plagiarism concerns. 24.16.32.174 10:31, 24 May 2006 (UTC)

I doubt the contributions by User:Phys from 2004, which are still probably extant in the article and have the chatty flavour, are anything to worry about. Charles Matthews 12:01, 24 May 2006 (UTC)

For what it's worth, here's a proposed revision which I think greatly minimize the "chattiness", eliminates pointless weasel words, unnecessary fluff, and improves organization (eg. previously the "Application" and "Explanation" sections are all jumbled together). I'm still concern about plagiarism though (due to lack of citation to original sources of the presented proof), as well as whether there might be a proof that might be more accessible. (I personally have no hope of understanding it even with my college engineering background, but fortunately I'm just copyediting here.) 24.16.32.174 12:13, 24 May 2006 (UTC)

I am sorry to re-open the debate on this page. I studied Noether's Theorem as part of a course on Calculus of Variations last year for my MSc in Maths. It seems to me that the main problem with this article (which has great potential) is that it is not clear that Noether's Theorem is part of Calculus of variations and can only be understood as part of that subject. Noether's Theorem is just not comprehensible (or applicable) if problems are not formulated in terms of a Variational principle. So I would strongly recommend that the introduction be re-written to make this clear. If everyone else doesn't have any problems with this I will take on the task of the re-write of the Intro. Wilmot1 13:48, 18 October 2007 (UTC)

[edit] Nina Byers' article

This is my reading of Nina Byers' article, whose reference I added to the page yesterday; I started with the lead sentence of the variational principle article, and augmented it based on the Byers article: Ancheta Wis 20:20, 30 Jan 2005 (UTC)

A variational principle is a principle in physics which is expressed in terms of the calculus of variations. According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint¹. These expressions are also called Hermitian. Thus such an expression describes an invariant under a Hermitian transformation. Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. On July 16, 1918, before a scientific organization in Goettingen, Klein read a paper written by Emmy Noether, because she was not allowed to present the paper before the scientific organization herself. In particular, in what is referred to in physics as Noether's theorem, this paper identified the conditions under which the Poincaré group of transformations (what is now called a gauge group) for General Relativity define conservation laws. (The relationship of these invariants (the symmetries under a group of transformations) and what are now called conserved currents, depends on a variational principle, or action principle.) Noether's papers made the requirements for the conservation laws precise.

Hilbert had derived the same equation as the Einstein equation for General Relativity within a period of the same few weeks as Einstein, in November 1915. The chief difficulty, which concerned David Hilbert, was that the conservation of energy does not hold for a region subject to a gravitational field. (Byers' commentary² notes that sometimes the objects which are needed to define conserved quantities are not tensors, but pseudotensors.³) Hilbert's unified theory remained uncelebrated because of this difficulty. Noether's theorem remains right in line with current developments in physics to this day.

[edit] Notes

Note 1: Cornelius Lanczos, The Variational Principles of Mechanics (Dover Publications, New York, 1986). ISBN 0-486-65067-7.
Note 2: E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws by Nina Byers
Note 3: A pseudotensor changes its sign under inversion by some transformation matrix. See note.

I propose adding this to the Variational principle article, with links to Noether's theorem. Ancheta Wis 09:55, 1 Feb 2005 (UTC)

Well, I would hope to have this clarified. I don't understand what it means for an expression to be self-adjoint, for example: that terminology usually applies to an operator. I also don't understand what is being said about pseudotensors, though that might be my ignorance. The comment about Hilbert's work seems excessive. Charles Matthews 19:13, 28 May 2005 (UTC)

1) Based on my background, to me, a matrix, an operator, or other form (including some expression) can be Hermitian. 2) I refer to Byers' last paragraph of part III in her article. 3) I will strike the Hilbert statement in the Variational principle article, based on your comment. Ancheta Wis 21:13, 28 May 2005 (UTC)

[edit] Conformal transformation

How is the $\phi^4\,$ example a conformal transformation? It would seem that it is merely a scale transformation. A general conformal transformation admits special conformal transformations as well. Even the non-interacting case is only conformally invariant in 2D. Furthermore, couple it to gravity, find the stress energy tensor $T^{\mu\nu} = \frac{1}{\sqrt{g}} \frac{\delta S}{\delta g_{\mu\nu}}\,$ and find that it is not traceless. Is there an improvement term?--Lionelbrits 03:47, 25 March 2007 (UTC)

[edit] Phase and electric charge

The noether current associated with phase is a charge carrying EM current, not charge itself -- and I don't think either can be thought of a conjugate vraiables. The article needs amending to reduce the over-emphasis on conjugate variables. --Michael C. Price ^talk 13:42, 2 June 2007 (UTC)

[edit] Conserved Current

Under "Mathematical statement of the theorem" I followed the link to "Conserved current" and there it appears to be a thing which is conjugate to a variable which has a differentiable symmetry, or something that fits this theorem. This seems to be a bit tautologous. Is it a current which is non-divergent? I will be adding an appeal to the conserved current page. Please forgive any clumsiness in my editing; this is the first time I have felt obliged to add anything to this very fine project so please, experienced users, if I have erred please let me know gently... Toospaice 04:55, 22 June 2007 (UTC)

[edit] making it more comprehensible

Is there a good reason why this article throws around a lot of terminology, gives a hyper-vague 'mathematical statement', and then spends most of the energy on the highly technical proof? As Jitse Niesen remarked in rating the article, it would have been more logical and more comprehensible to start with a simple finite-dimensional hamiltonian case, stating the theorem precisely, illustrate it by examples, and then discuss a variational formulation. Also, I know that the following approach was tried at other articles where proof distracts too much from the subject: move the proof into a separate (more technical) article. If there is a strong agreement on keeping the present structure, I would feel inclined to write a separate, gentler article of a type Noether theorem (symplectic geometry). Opinions? Arcfrk 04:14, 26 June 2007 (UTC)

[edit] Example 2 from article

I've removed the following example:

[edit] Example 2: Conservation of linear momentum

Still considering 1-dimensional time, let

$\mathcal{S}[\vec{x}]\,$	$=\int \mathrm{d}t \mathcal{L}[\vec{x}(t),\dot{\vec{x}}(t)]$
	$=\int \mathrm{d}t \left [\sum^N_{\alpha=1} \frac{m_\alpha}{2}(\dot{\vec{x}}_\alpha)^2 -\sum_{\alpha<\beta} V_{\alpha\beta}(\vec{x}_\beta-\vec{x}_\alpha)\right]$

i.e. N Newtonian particles where the potential only depends pairwise upon the relative displacement.

For $\vec{Q}$ , let's consider the generator of Galilean transformations (i.e. a change in the frame of reference). In other words,

$Q_i[x^j_\alpha(t)]=t \delta^j_i.$

Note that

$Q_i[\mathcal{L}]=\sum_\alpha m_\alpha \dot{x}_\alpha^i-\sum_{\alpha<\beta}\partial_i V_{\alpha\beta}(\vec{x}_\beta-\vec{x}_\alpha)(t-t)$

$=\sum_\alpha m_\alpha \dot{x}_\alpha^i.$

This has the form of $\frac{\mathrm{d}}{\mathrm{d}t}\sum_\alpha m_\alpha x^i_\alpha$ so we can set

$\vec{f}=\sum_\alpha m_\alpha \vec{x}_\alpha.$

Then,

$\vec{j}=\sum_\alpha \left(\frac{\partial}{\partial \dot{\vec{x}}_\alpha}\mathcal{L}\right)\cdot\vec{Q}[\vec{x}_\alpha]-\vec{f}$

$=\sum_\alpha (m_\alpha \dot{\vec{x}}_\alpha t-m_\alpha \vec{x})$

$=\vec{P}t-M\vec{x}_{CM}$

where $\vec{P}$ is the total momentum, M is the total mass and $\vec{x}_{CM}$ is the center of mass. Noether's theorem states:

$\dot{\vec{j}} = 0 \Rightarrow {\vec{P}}-M \dot{\vec{x}}_{CM} = 0$ .

This example is confused. The conservation of momentum comes from spatial translation invariance. This example uses momentum translation invariance. A boost (change of reference frame) is a translation in momentum and the conserved quantity is the displacement of the center of mass. I may fix this if I find the time but I more than welcome someone else to do this. Alfred Centauri 00:22, 30 July 2007 (UTC)

Hi, I've restored the example, unchanged, to the article on the grounds that all the examples need a drastic rewrite and I don't see that example #2 is any worse than the rest. Look at example #1, for example: Modelling a one dimensional particle and it starts blathering on about curved Riemannian space and metrics. urgh! I can only think that it was a cut-and-paste job from somewhere else.--Michael C. Price ^talk 07:39, 30 July 2007 (UTC)

Seeing how long the article has stood without much improvement, I'm inclined to remove it all and start afresh. What do you guys think about that? Unfortunately, I don't know that much of Noether's theorem in a PDE setting, which makes it hard to write about that and impossible to try and fix the examples. Or perhaps you think that it's better if I write a bit about the ODE setting and leave the examples as they are? -- Jitse Niesen (talk) 08:33, 30 July 2007 (UTC)

I favour leaving the examples in until we have better material available. Bad though the examples are, they are of some help. I was intending to read the article completely -- if it is readable -- before deciding on a course of action. In the meantime do add any ODE (ordinary DE?) stuff you know. I like the suggestion mentioned in the previous section, which you've made, of starting from some simple examples before moving on to the more generic proofs. --Michael C. Price ^talk 09:05, 30 July 2007 (UTC)

I've changed my mind, having reached the point where example 1 has reduced to

$\frac{\partial L}{\partial t} = \frac{\mathrm{d}L}{\mathrm{d}t}$

which is just plain silly. I've only ever seen Noether's theorem applied non-trivially to the field theory case, anyway. I would say, blast (or tuck away in a more technical article) everything from section 3 (Proof) onwards and crib what we need from the excellent Baez and Byers links in the reference section. They can't copyright the laws of physics, so we should be okay.--Michael C. Price ^talk 15:45, 30 July 2007 (UTC)

<sigh> Apparently you guys didn't understand... Example 2 is not an example of conservation of momentum. If you are going to put it back in, at least change the title of the example. Alfred Centauri 12:56, 30 July 2007 (UTC)

Okay, I'll leave you to update example 2. --Michael C. Price ^talk 14:33, 30 July 2007 (UTC)

[edit] 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)

[edit] Simpler special-case proof?

What do people think of prefacing the proof section by a simple proof of a special case amenable to techniques of freshman-level calculus? The proof for time-invariant symmetries of configuration spaces built on R^n, or even just R, for example. The proof for the very general case currently presented is accessible only to specialists, I fear. PerVognsen (talk) 09:35, 23 November 2007 (UTC)

[edit] Disagrees with Emmy Noether

According to the article on Emmy Noether, Noether's theorem provides a one-to-one correspondence between conserved currents and symmetries. This article doesn't show how to get from a conserved quantity to a symmetry. Also, it doesn't assign a unique conserved current to each symmetry, since there's so much latitute in choosing f, which is introduced without explanation during the proof. It's not even clear that such an f can always be chosen, but when it can, any divergence free vector field can be added to f, which will give a complete different conserved current. Frankly, this article doesn't make any sense, especially the proof. Maybe we should explain what a "functional derivation" is, and why it can be applied to numbers, functions from M to T, and functions from M to R. I'm not sure whether it even makes sense to think of Q[] as a derivation since it's being applied to functions like phi which take values in an arbitrary manifold, not a ring. 128.208.87.93 (talk) 02:29, 6 March 2008 (UTC)

I am surprised. A symmetry characterizes that which remains unchanged under a transformation. A conserved quantity remains unchanged by definition. What is it that you do not see? --Ancheta Wis (talk) 11:18, 6 March 2008 (UTC)

Historically, this arose from the Erlangen program. In fact, Felix Klein presented her paper in Göttingen because she was not allowed to do so, July 16th, 1918. The Emmy Noether article needs work, by the way. --Ancheta Wis (talk) 11:33, 6 March 2008 (UTC)

The Erlangen program article has a table which lists 3 columns of transformations (dilation, reflection, translation) but which is missing a fourth column, the Inversion transformation, whose invariants are an example of a hidden symmetry. (These sorts of transformations are studied in physics right now) Roger Penrose has publicized some aspects of this fourth column, for example in his lectures about the Weyl curvature hypothesis. --Ancheta Wis (talk) 02:45, 7 March 2008 (UTC)

The whole point is that there isn't a unique conserved current associated with each symmetry. Take translations for example; we have so many equally valid conserved stress-energy tensors out there, each only differing by a surface term. However, the integral of any conserved current from this family over a spatial cross section will give the same Noether charge, which is really what matters. AnonyScientist (talk) 08:49, 19 April 2008 (UTC)

[edit] Converse of Noether's theorem true?

The Goldstein reference (p. 594) states that the converse of Noether's theorem is not true, that some conservation laws cannot be derived from Noether's theorem. He cites the conserved quantities associated with soliton solutions, as appear in the Sine-Gordon equation and the Korteweg-de Vries equation. I'm not sure if this is a fair example, but please see the discussion at the Laplace-Runge-Lenz vector page as well. There is a way to derive its conservation from the Noether theorem, but some physicists seem to view the required "symmetry" transformation as cheating. ;) I'm willing to believe that Goldstein was mistaken, but I think we need a reference to the scientific literature before we can include it in the article. Willow (talk) 21:04, 18 April 2008 (UTC)

About solitonic superselection sectors, in quantum physics, there is a symmetry which multiplies each sector with a phase which is proportional to the topological number. However, classically, the Noether charge, which corresponds to the topological number only generates the trivial transformation despite the fact that it's not constant. The problem is, the topological sectors are disjoint and the topological charge is constant for each sector, even though it's not constant between sectors. So you're right, there is a classical counterexample in a weak sense. AnonyScientist (talk) 07:08, 19 April 2008 (UTC)

[edit] Dali picture

While a very famous painting n'all, why on earth is there one of Dali's paintings on this article? What has "warping clocks" got to do with invariance under transformation? Deamon138 (talk) 10:03, 3 May 2008 (UTC)

Somebody removed it: good and thanks whoever it was! Deamon138 (talk) 16:20, 4 May 2008 (UTC)

I didn't notice this comment before I removed it; I was working from a list of nonfree images in math articles, which have to be cleaned up every few months. There's a policy, WP:NFCC, which says among other things we can't use nonfree images when a free image could serve the same purpose. In this case, we could definitely make a free image that illustrates a coordinate transformation. — Carl (CBM · talk) 16:43, 4 May 2008 (UTC)

[edit] Confusing nature?

Somebody came and removed the confusing tag. Is this correct? I myself find it confusing, but then no more than a lot of Physics/Mathematics topics and though they're two subjects I know most about. Deamon138 (talk) 19:40, 4 May 2008 (UTC)