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I see Gabriel Catren has a paper out:
Urs and Mike are referred to.
Thanks for the alert. I don”t have time to read it at the moment. But I gather it is a development of his point that real polarizations in geometric quantization may be understood as Lie algebra actions, and that hence the condition which picks out the “wavefunctions” (the polarized sections of the prequantum line bundle among all the sections) is mathematically the same as that which picks out gauge invariant observables in the BRST complex for gauge theories.
It seems Gabriel has been expanding this observation into a considerably bigger philosophical picture. (If that’s what it is, I should read the article.)
I can’t see anything on polarizations. From the abstract:
We distinguish two orientations in Weyl’s analysis of the fundamental role played by the notion of symmetry in physics, namely an orientation inspired by Klein’s Erlangen program and a phenomenological-transcendental orientation. By privileging the former to the detriment of the latter, we sketch a group(oid)-theoretical program—that we call the Klein-Weyl program–for the interpretation of both gauge theories and quantum mechanics in a single conceptual framework. This program is based on Weyl’s notion of a “structure-endowed entity” equipped with a “group of automorphisms”. First, we analyze what Weyl calls the “problem of relativity” in the frameworks provided by special relativity, general relativity, and Yang-Mills theories. We argue that both general relativity and Yang-Mills theories can be understood in terms of a localization of Klein’s Erlangen program: while the latter describes the group-theoretical automorphisms of a single structure (such as homogenous geometries), local gauge symmetries and the corresponding gauge fields (Ehresmann connections) can be naturally understood in terms of the groupoid-theoretical isomorphisms in a family of identical structures. Second, we argue that quantum mechanics can be understood in terms of a linearization of Klein’s Erlangen program. This stance leads us to an interpretation of the fact that quantum numbers are “indices characterizing representations of groups” ((Weyl, 1931a), p.xxi) in terms of a correspondence between the ontological categories of identity and determinateness.
I saw the abstract. Did you have a chance to look inside the article?
From what I’ve read I don’t see anything like what you wrote in #2. It’s describing what he takes to be the “Klein-Weyl program”, which sees “gauge theories and quantum mechanics in terms of a localization and a linearization of Klein’s Erlangen program respectively”.
So the principal bundles mark “the transition from Klein’s Erlangen program to gauge theories” which is “analogous to the transition from groups (encoding the automorphisms of a single structure) to groupoids (encoding the multiple self- and hetero-identifications in a family of structures)”. Then connections add local identifications.
Then QM linearises Klein since “the non-trivial identity of a homogenous quantum state is not simply encoded by a group, but rather by a linear representation of a group”.
Thanks! I see. Hm.
@Urs #2 has he written anything up about that approach to polarisations?
I think this is in Catren 13. Myself, I heard about it personally from him and his group members, when I was being a guest of their department a few years back. I seem to remember that this idea was part of the proposal that made Gabriel win that big ERC grant which he then used to organize, among other things, New Spaces for Mathematics and Physics.
In my recollection (I should re-read Gabriel’s article), the observation was that in the case of geometric quantization for a real (and maybe integrable) polarization, the polarization condition on sections to be wave-functions just says that these are invariant along the orbits. One can write down a BRST-like complex to resolve the wavefunctions among all sections.
I would attribute this to the fact that geometric quantization is in fact that: geometric; hence to quantize it uses the same kinds of quotients and intersections that appear elsewhere in differential geometry.
One should just beware that real (and maybe integrable) polarizations are rare, so that this analogy carries only so far. It seems that all the applications of geometric quantization that serve to do real work for us (as opposed to toy examples) use Kähler polarizations (namely that’s the case for perturbative quantization of interacting field theories, and for non-pertrubative quantization of Chern-Simons-type theories). This in turn should be seen in view of Bott’s observation that geometric quantization for Kähler polalizations is equivalently push-forward in complex K-theory. That opens a perspective on geometric quantization that also exhibits striking analogies with structures seen elsewhere (D-brane charges…) but, while still being “geometric” of sorts, it is different from that picture of real polarization.
But beware that I may be distorting Gabriel’s point here. Better you check his article.
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