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brief note on deformation quantization of the 2-sphere
(to go along with the existing geometric quantization of the 2-sphere)
I have added a paragraph (here) further highlighting the existence of different notions of deformation quantization with a pointer to Section 2 of Hawkins (2007) (arXiv:0706.2946) which goes to some length of tabulating and comparing them. Then I added a (currently empty) section for “Formal deformation quantization” to make very clear that this deserves its own discussion (I wasn’t interested in it when I created this entry, but please feel invited to add material!).
Then I highlighted that the section “Strict deformation quantization”, which has been there all along, is about one particular version of the notion (as extensively discussed in Hawkins’ article).
It arguably makes good sense to speak of “deformation quantization” in all these cases, given that they all concern successions of the algebras of observables parameterized by certain values of $\hbar$ accumulating at zero. Therefore I think the title of the entry is quite fine, certainly when it highlights the ambiguities, as I have tried to do now.
Yet more reasonable would be to refer to all this (including formal deformation quantization) as “algebraic quantization”, as this would nicely dual-rhyme on “geometric quantization”. Alas, the chance for this nice naming convention was missed early on and now seems to long have passed.
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