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prompted by this physics.SE question I ended up adding to Idea-section of the entry quantization a new subsection titled Motivation from classical mechanics and Lie theory .
Interesting! So how does fit into the story? Is there a sense in which this corresponds to making the interesting Lie integral look more like the uninteresting translation group?
Does always have an extension by the ’boring’ ?
Wouldn’t the limit of hbar going to zero correspond to taking the quotient ?
When curvature gets larger gets smaller, as in the explanation for why large system appears classical at Planck’s constant. Can one say a circle bundle tends to a line bundle as curvature grows?
Just a quick reply for the moment, am still absorbed by weekend business.
concerning the second question in #2: in general the extension of is by , where is the group of periods of the symplectic form. See at geometric quantization of non-integral forms.
concerning the classical limit: I don’t see a good way of saying that this corresponds to turning the circle extension into a line extension, except the following: -principal connections, having underlying neccesarily trivial -principal bundles, have arbitrary square roots, hence “divisions”. You can always find another such bundle such that its th power is the given one, for any .
This is not the case for -principal connections. But of course as their class becomes very large, it becomes more and more true in that for more and more it does work.
Have to quit now. Weekend duties. :-)
I had another minute and have cross-linked Planck’s constant and geometric quantization of non-integral forms a bit more explicitly.
Incidentally, just the other day there was a question about this on MO here.
I find this very helpful - filling in a few of the many gaps of my knowledge of the religion known as ‘quantization’. How about up-ing it to a proper paper for wider dissemination?
The Lie-integration perspective on pre-quantization that is highlighted above is very much amplified in our article
See (just) the introduction for an exposition!
The remaining polarization/genuine quantization step is elucidated in
I like to think of both together as doing away with any “religion” or “mystery” in the concept of quantization. But judge for yourself.
By the way, higher formal deformation quantization works not by lifting through the map
as in higher geometric quantization (see at Poisson bracket Lie n-algebra)
but through
(see at Poisson n-algebra).
Clearly that’s analogous, but just as clearly it’s different. In particular one has to be careful: the “algebra of observables” in formal deformation quantization is really something different than any observables in geometric quantization. There is a “holographic” relation implicit here which has rarely been made explicit.
The story at Motivation from classical mechanics and Lie theory takes us naturally from classical systems to their quantized versions.
Witten (as discussed on MO here) is suggesting that we associate compact Lie group representations with classical systems whose quantization yields the representation:
A representation of a group should be seen as a quantum object. This representation should be obtained by quantizing a classical theory. The Borel-Weil-Bott theorem gives a canonical way to exhibit for every representation of a compact group a problem in classical physics, with symmetry, such that the quantization of this classical problem gives back as the quantum Hilbert space. One introduces the “flag manifold” , with being a maximal torus in , and for each representation one introduces a symplectic structure on , such that the quantization of the classical phase space , with the symplectic structure , gives back the representation . Many aspects of representation theory find natural explanations by thus regarding representations of groups as quantum objects that are obtained by quantization of classical physics.
Would only a certain class of classical system be included in this story?
What would be the quantomorphism group for ?
Hi David,
that remark by Witten on how the trace in the irrep that gives the Wilson loops in Chern-Simons theory is really itself to be thought of as being due to a 1-dimensional field theory on the Wilson loops we have expanded on a good bit in the above context: on the nLab this is at orbit method starting somewhere here. But if you are interested I’d advertize the dedicated discssion of this in section 3.4.5 of our A higher stacky perspective on Chern-Simons theory.
I was rooting about yesterday, after that comment, and found Beilinson-Bernstein localization to be content free.
If the orbit method and the Borel-Weil-Bott theorem find a natural setting in Chern-Simons theory,
and if
Beilinson-Bernstein’s uniform construction of all representations of Lie groups via the geometry of D-modules on flag varieties…has the Borel-Weil-Bott theorem and the Kazhdan-Lusztig multiplicity conjectures as immediate consequences,
is there then a Chern-Simons story to tell about Beilinson-Bernstein localization?
Sadly, although Ben-Zvi says that
the result is not difficult to state and prove,
I don’t feel able to write up the theorem precisely.
Good question. I don’t know at the moment. I should look into this.
I copied out snippets from a paper by Ben-Zvi and Nadler, so at least there’s something at Beilinson-Bernstein localization. This also indicates why Borel-Weil is a consequence. Obviously, it should all be redone in conversational language.
Were one to want the nPOV on this, and perhaps then the physics interpretation, I should imagine following the trail of Ben-Zvi and Nadler would be good, e.g., The Character Theory of a Complex Group and Beilinson-Bernstein localization over the Harish-Chandra center
Thanks. I have added some hyperlinks to the text.
By the way, morally there is no mystery here: D-modules are just a way of speaking about holomorphic sections in an algebraic context and so up to technicalities of explicit implementation in possibly different geometric context (and maybe up to the allowed freedom of choice of subgroup , if you wish), the statement of the the BB-localization is pretty much verbatim that of the BWB theory/orbit method.
What’s the localization in BB-localization? Does it mean in the nlab sense of localization?
…the statement of the the BB-localization is pretty much verbatim that of the BWB theory/orbit method.
The former concerns all reps of a Lie group. Does the latter go beyond irreps?
Yes. The “inverse problem” to the orbit method for all reps was solved in the first section of part II of Loop Groups and Twisted K-Theory. Joost Nuiten gives a geometric interpretation of this in terms of defect operators in higher geometric quantization in section 5.2.2 (see “The universal orbit method”) of his thesis.
So where Ben-Zvi and Nadler have
The Beilinson-Bernstein localization theorem is perhaps the quintessential result in geometric representation theory…Despite its importance, which is hard to overestimate…
they might as well have substituted “orbit method”? Isn’t the issue of whether the Lie group is compact significant? BB-localization goes beyond the compact case, but then I see a claim by Ben Wieland
Compact groups are pretty much the same as complex reductive groups, but one must replace finite dimensional unitary representations with finite dimensional holomorphic representations.
Anyway, with Beilinson-Bernstein being taken in a number theoretic direction (e.g., here and here), a thoroughly physics-based interpretation of the original might shed some light on ideas from physics appearing in number theory.
Right, so compactness of the group, hence of the orbit/flag manifold is what makes the space of holomophic sections of a line bundle over it be finite dimensional, hence is what makes the resulting representation be finite dimensional.
This is a common theme whenever any piece of physics is made rigorous (such as here the quantization of Wilson loop particles via the Borel-Weil theorem): typically things work well for the “finite”/”compact” case, while just as typically phsical application wants the “infinite”/”noncompact” case.
But the thing is that the notion of “compact” changes when the coefficient -ring changes:
For example the phase space a something as simple as the harmonic oscillator is not compact, hence the harmonic oscillator cannot be quantized, directly, by cohomological methods, hence by index theory over .
But consider then the (super-)string. This is effectively a big (infinite) collection of harmonic oscillators. We cannot quantize it cohomologically over . But we can change the coefficient -ring to . Then the string does have a cohomological quantization, given by the Witten genus. Hence we did quantize a highly non-compact phase space. But only after changing the base -ring.
That insight has got to go somewhere on nLab. Don’t know where.
Is there a way to know which phases spaces are compact for a given ? Is this about compact object in an (infinity,1)-category?
This is sort of the topic at motivic quantization, though I could highlight this particular point more, I suppose.
What in ordinary quantum mechanics is compact+polarizable for the phase space, in the general cohomological/motivic quantization becomes: orientable in -cohomology for the extended phase space. Here “extended” means that we also “de-transgress” the formation of phase spaces for field theories in the first place.
For instance when quatizing the 1+1-dimensional string sigma-model field theory, the “extended phase space” is really just target space itself. If that is -orientable then with the B-field regarded as the prequantum 2-bundle we quantize by forming its -index. The result is the Witten genus, the quantization of th heterotic string on that spacetime, coupled to that B-field.
The thing here is that what you would traditionally call the phase space of the string never explicitly appears in this construction. It would be a variant of the loop space of target space and certainly not compact. But since in the higher quantization everything is de-transgressed down to top codimension, we don’t actually quantize the loop space, but instead the target space of the -model. And so it’s that which needs to admit intergration – with coefficients in .
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