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Can mathematics (including statistics, dynamical systems,...) combined with classical electromagnetism (using only the constants appearing in chargefree maxwell equations) be used to derive the planck constant? Can it be proven that planck's constant is truely a new physical constant?

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Look, Dr. Zaslavsky is completely correct. But. The great mathematician Jean Leray once, after being asked to think about Maslov's work on asymptotic methods to approximate the solutions of partial differential equations which were generalisations of the WKB method, decided, in the 70's, to write an entire book titled Lagrangian Analysis and Quantum Mechanics, note he gives his own special meaning to « Lagrangian Analysis.», MIT Press, see the nice abstract entitled « The meaning of Maslov's asymptotic method: The need of Planck's constant in mathematics.»

This is not a derivation of the magnitude of Planck's constatnt from Maxwell's equations, but it is a profound motivation for why there should be some finite, small, constant such as Planck's from the standpoint that the caustics you get in geometrical optics cannot be physical, and yet geometric optics ought to be a useful approximation to wave optics. From this point of view, there ought to be some constant like Planck's constant, at least in pure mathematics.

It is, however, very advanced: inaccessible unless you already know about Fourier integral operators in Symplectic manifolds, such as in Duistermaat's book or Guillemin and Sternberg, Symplectic Techniques in Physics. Maslov's original book is, although non-rigorous, very insightful and more accessible.

For a physicist, though, perhaps just the basics of the Hamiltonian relationship between geometrical optics and wave optics, and the basics of the WKB method, would be more important.

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for now I only find a relatively short article by Jean Leray with the same title: projecteuclid.org/euclid.bams/1183548218 is this the entire book or should I search harder? thanks for the reference btw! – propaganda Jan 23 '12 at 5:24
That's merely an abstract. The book is rather advanced, but yes it is an entire 200 page book. One should also read Maslov's original book which, although not rigorous, is tremendously insightful. the book by Guillemin and Sternberg (Symplectic Techniques in Physics) is also to be recommended, sort of, it is still more mathematical than physical, of course. – joseph f. johnson Jan 23 '12 at 5:27
I can not find any reference to the book itself :( – propaganda Jan 23 '12 at 5:31
It's on the shelf next to my bed right now. Look, information is not free. Leray and the translator put a lot of work into that book and they have to be paid for it...or their heirs or assigns... sorry. But the book by Maslov is a better introduction, and after that the abstract probably suffices. The book by Leray is very advanced and a little bit inaccessible unless you already know Fourier integral operators on symplectic manifolds, on the one hand, and Maslov's original work, on the other. So start there anyway, and put off Leray until you have got that far. And, for a physicsist, the – joseph f. johnson Jan 23 '12 at 5:36
basics of the Hamiltonian relation between geometrical optics and wave optics, and also the basics of the WKB method, are more important anyway, and will give you a lot of insight. – joseph f. johnson Jan 23 '12 at 5:37
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If you're talking about deriving the value of Planck's constant, then no, that is not possible. The value is simply a consequence of our chosen unit system.

If you're talking about deriving the fact that something analogous to Planck's constant has to exist at all, then I believe the answer is still no. To some extent that is also a consequence of our unit system, since if you use fully natural units, Planck's constant has a value of 1 and so it never shows up in the equations in the first place. But besides that, the original context in which the context was proposed was the quantization of energy, namely that the energy of an EM wave is quantized in units of $hf$. This could be considered the foundational assumption of quantum mechanics. Planck's constant is part of this assumption, so you can't really call it a derived result.

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I realize that textbooks dont derive planck's constant from maxwell equations, but can it be proven to be impossible to derive from maxwells equations using only more mathematics? – propaganda Jan 23 '12 at 4:35
If you can express the proposition "Planck's constant is impossible to derive from Maxwell's equation" in proper mathematical language, then perhaps yes, it is possible. But that would be a question for the math site. The (summarized) physics answer is that Planck's constant cannot be derived from Maxwell's equations because (1) Planck's constant is not something that can be derived, and (2) they deal with different areas of physics. – David Zaslavsky♦ Jan 23 '12 at 4:42
I was thinking perhaps similar to how hidden variable theories can be in some sense ruled out by Bell's theorems? quantum mechanics does seem to have a lot in common with bayesian statistics: prior knowledge of one variable affects expected probabilities of another – propaganda Jan 23 '12 at 4:43
also: using natural units shoves the value into the fine structure constant no? now you have a dimensionless unexplained constant – propaganda Jan 23 '12 at 5:46
I don't see any connection between Bell's theorem (which is a precise statement about correlations of measurements) and any relationship that might have existed between Planck's constant and Maxwell's equations. Also, I'm really not sure what you mean about natural units and the fine structure constant... yes, $\alpha$ can be calculated using $\hbar$, but it's a unitless number and thus independent of the actual value of $\hbar$. If you would like to continue this, let's take it to Physics Chat. – David Zaslavsky♦ Jan 23 '12 at 5:52
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Maxwell assumes only that a particle has charge, not that an electron has a frequency that depends on its rest mass. So one can not deduce Planck's constant from Maxwell. de Broglie fixed that.

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