Talk:Geometric quantization

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Dubious[edit]

The article states, "However, as a natural quantization scheme, Weyl's map is not satisfactory. For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term 3ħ2/2. This extra term is actually physically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom."

This seems to be an old view, superseded by J. P. Dahl and W. P. Schleich, "Concepts of radial and angular kinetic energies", Phys. Rev. A,65 (2002). doi:10.1103/PhysRevA.65.022109 arXiv link. Since according to the paper this angular-momentum-squared example isn't really a problem (more of a resolved paradox, really), the extent to which the Weyl map is "unsatisfactory" is of chiefly mathematical rather than physical concern. Or does anyone have a different counterexample with physical meaning? Teply (talk) 07:10, 7 June 2012 (UTC)

It is important to state someplace that this article is not about representation changes in describing QM, but, instead, "Quantization", the quest of a new (quantum) theory out of some classical one. You might be able to tweak the wording so the message gets across without misunderstanding. The point made, I think, is that Weyl really believed he had discovered a "quantization functor": THE machine that unambiguously produces the quantum object out of the classical one. In effect, by some magic, start with the ħ-independent term, and automatically produce all the ħ dependence in the series of all observables! That was the old view, as you put it. We now appreciate this is impossible, and pointless, since several different quantum theories may have the same classical limit, ie the quantum theory has more information, to be put in by hand. (It is a good guide for the efforts detailed in this article, though).
The angular momentum paradox nicely explained by D&S is not a problem only if one understands the Weyl map as just a representation change (the new view), ie only if one appreciates the Wigner image of the quantum L2 is not just the classical angular momentum squared. (Phase-space quantum mechanics, the consumer of the map, like D&S, is quite happy with this, and uses it to explain the Bohr paradox, which made the above assumption, implicitly—it is only for quantization that it is a problem!). But Weyl's original vision of his map was that it was not just a change of represenation with physics in it, it was supposed to derive another theory, the quantum theory, out of the classical one — a functor! The counterexample/paradox, emphasizes easily how the ħ-squared offset terminates that vision.
So, as often in this area, the same mathematical twist serves a dual purpose. This article is on the "old" problem of quantization, the "mathematical concern" you see, and bears little conceptual linkage to the Phase space representation article (a different representation of the received quantum theory, where the quantization was achieved by trial-and-error), even though, maddeningly, the math is identical! I appended the D&S ref; there might be lots of room for pedagogical improvement there, though. Every time I find myself split hairs on the dual use of the term "quantization", I realize the term is expressly designed for systematic abuse....Cuzkatzimhut (talk) 10:48, 7 June 2012 (UTC)
Yes, I see the confusion here. The map is satisfactory if you want to go between quantum mechanical representations. The appearance spurious ħ terms distinguishing these representations is of no physical concern. Here the "Weyl quantization" should just be understood as "Weyl map." The map is unsatisfactory if you want to go all the way from classical mechanics to quantum mechanics ("quantization" in the absolute sense), where there is no surefire rule. Teply (talk) 17:31, 7 June 2012 (UTC)
Indeed, many, including myself, misuse "Weyl quantization" to mean "Weyl-map based representation"; in this article, they mean it, and they talk about a quantization recipe. You might enjoy Todorov's preprint. Also, as a pedantic aside, in the Husimi prescription/picture, the Weyl/Husimi map of the quantum SHO hamiltonian is also not the classical one, either, as it has an extra constant ħ/2 offset (which happens to be equal to the actual 0 pt energy). In the Weyl prescription, that term is missing, and emerges out of the *genvalue solution. Cuzkatzimhut (talk) 19:05, 7 June 2012 (UTC)
I was bold and branched off a Wigner–Weyl transform stub with the idea that most of the Weyl quantization material gets moved there. Only the part relating specifically to quantization would stay in the quantization article. Hopefully that will prevent this kind of confusion in the future. Teply (talk) 22:21, 7 June 2012 (UTC)