Talk:Noether's theorem

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

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 .

"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 (http://math.ucr.edu/home/baez/boosts.html) -- 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)

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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)

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)

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)

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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.

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 (http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html) by Nina Byers
• Note 3: A pseudotensor changes its sign under inversion by some transformation matrix. See note (http://mathworld.wolfram.com/Pseudotensor.html).

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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 (http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html). 3) I will strike the Hilbert statement in the Variational principle article, based on your comment. Ancheta Wis 21:13, 28 May 2005 (UTC)