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Thanks. I added journal name and jstor-link (here)
Added also:
and will try to finally add now the original articles by Kibble…
added the original reference:
Incidentally I think this topic should really be called “symplectic formulation…”.
It’s weird that all authors insist on the vague “geometric”. (Also the Heisenberg picture is “geometric” in the end, even if NC geometric.)
So on absolute grounds the entry might deserve renaming, but it would probably be unhelpful to the search engines, so I am hesitant.
Re #5: “Symplectic formulation of quantum mechanics” is potentially ambiguous, since it could be interpreted as referring to the geometric quantization picture of quantum mechanics.
That potential confusion with “geometric quantization” is certainly only made worse by saying “geometric quantum mechanics”.
On the other hand, the term “symplectic mechanics” is standard, as shorthand for “symplectic formulation of classical mechanics as about Hamiltonian flows”. With this in mind, “symplectic quantum mechanics” would exactly express what’s going in.
And then one could transparently ask:
“Why, conceptually, does geometric quantization of symplectic classical mechanics yield symplectic quantum mechanics?”
which is, I think, the most interesting question that a symplectic geometer wants to ask here, now viewining, with Kibble and followers, quantization as a process that takes one symplectic manifold to (not a deformation of symplectic geometry but) another symplectic manifold.
I would say it is unlikely that the nLab can unilaterally enact a change in terminology for this specific subject, which probably makes it prudent to stick to the existing terminology.
I see numerous papers using the word “geometric” and similar derived words, but none that use “symplectic quantum mechanics”. This is the case for the references in the 2015 paper by Heydari, for example
It is my perception that the title of the article is often taken to be (by the readers) as the most common choice of terminology, which in this case would inevitably lead to confusion when trying to search for papers.
Yes, that’s why I said in #5 I am hesitant.
But it’s dangerous to keep sacrificing what is right for what is mainstream.
added pointer to today’s:
I think that this way of thinking fits perfectly with coherent state in geometric quantization, which currently has very little written about it.
Thanks for contributing, Josh. Much appreciated.
I have added double square brackets “[[...]]” around a bunch of keywords appearing in the text you added, which makes them automatically hyperlinked to their corresponding entries.
Also touched the formatting of the quote. Note that:
“>” opens an indented quote-like environment
quotation marks must be like "...", since LaTeX-style quotation marks are read by the nLab parser as delimiting a \verb-like environment,
and for similar reasons square brackets ought to be coded by “\[...\]” (if they don’t contain a link itself coded by square brackets) or “[...]” (otherwise)
But allow me to ask: Is what you added really meant to refer to Kibble, Ashtekar & Schilling et al.’s perspective on quantization that this entry is about, or did you mean to add it under geometric quantization?
(Incidentally, in the discussion above I argued that indeed this entry here deserves a different title, not to be confusing.)
Well I think that Anatol Odzijewicz had this perspective to. What is written in the idea section of this entry is a special case of theorem 5 and the nearby text in his cited paper, when but I haven’t gotten to that yet. He stated a related equivalence of categories in that paper. I think this should be at least referenced on the geometric quantization page too, since it seems to be what quantization is about. If you think it’s out of place you can move it, but I’ll continue writing and maybe we can see where it fits.
Thanks for the formatting edits, I’m still learning how to properly format here.
Okay, thanks. I haven’t read Odzijewicz, maybe I’ll find the time.
But I was asking because the entry, to my mind, is about the observation that Schrödinger evolution is in fact Hamiltonian flow for a symplectic structure on complex projective space, and because this fact is not picked up on in the material that you added. It seems?
Oh I agree, so far I haven’t made the connection it but I’m planning too. I guess since they all fit together nicely I’m not sure how to split it.
Ah, okay. Great.
I’ve more–or–less said completed the discussion about the symplectic case (though it should be refined). I’m not sure if it all belongs on this page, but something about it should be mentioned on the geometric quantization page, coherent state page, etc.
Thanks again.
I have made some minor cosmetic touches, such as hyperlinking the cross-references to the three items.
There is a notational clash where the symbols “” first denote, I think, the projection operators on and then the projective space of .
By the way, the “geometrical formulation” of QM raises the following question which one might wonder about: Since it makes the output of geometric quantization be of the same type as its input, what is the meaning of iterating it?
I mean: If we start with a symplectic manifold, do a geometric quantization, regard the resulting as another symplectic manifold, consider its geometric quantization – what does this “double” (not to say “second”) quantization “mean” as per physics, if anything?
Oh well, I’m identifying points in the projective space of with rank–one orthogonal projections in so to me they mean they same thing. Maybe I should emphasize this more.
Well in a sense iterating doesn’t do anything, since the quantization of is essentially the same as the classical mechanics of when restricted to classical observables of the form However, this is just the canonical quantization of You can quantize it in other ways by using the Veronese embedding to embed it into higher dimensional projective spaces . From the perspective of geometric quantization this is taking the power of a line bundle. Physically, more classical observables on will be the form for so you get a better approximation to the complete classical mechanics of
By the way, John Baez has a discussion about this here: https://johncarlosbaez.wordpress.com/2018/12/27/geometric-quantization-part-3/ I don’t completely agree with what he says, since it seems important that the symplectic submanifolds of projective space have the overcompleteness property, but assuming they do I think what he is saying makes sense. He discusses the Veronese embedding in part 7, where he describes it as a cloning.
Ah, right, it’s just rank=1 projectors. Okay. Yeah, why not add a half-sentence saying it.
Regarding the “Well in a sense it doesn’t do anything…” I haven’t really thought about it, so I may be missing the obvious: Do we have a natural isomorphism from the base to the projectivization of a space of suitably polarized sections of a suitable pre-quantum line bundle over it?
(Sounds plausible, I am not doubting it. Just asking out of curiosity while being busy with other things.)
There is a natural isomorphism from to the space of holomorphic sections of the canonical bundle of I think I mentioned it: we can naturally identify vectors in the fibers of the canonical bundle with points in and using this identification the isomorphism is given by where ie. is a section of the canonical bundle. So the projectivization of the space of holomorphic sections is naturally identified with which is naturally identified with That this is true pretty much has to be the case due to the equivalence of categories.
11: there is also coherent state entry.
Do we have a natural isomorphism from the base to the projectivization of a space of suitably polarized sections of a suitable pre-quantum line bundle over it?
In the case of representations, whenever this is possible one talks about “coherent state representations”, see the survey by Lisiecki listed at coherent state,
Another term for the embedding into projective space which comes when the quantization line bundle is ample is the “coherent state embedding” (of the homogeneous space into the projective space).
I see. Interesting.
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