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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 4th 2013
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   Author: Urs
   Format: MarkdownItexbrief note on _[[deformation quantization of the 2-sphere]]_ (to go along with the existing _[[geometric quantization of the 2-sphere]]_)

   brief note on deformation quantization of the 2-sphere

   (to go along with the existing geometric quantization of the 2-sphere)

2. • CommentRowNumber2.
   • CommentAuthorGuest
   • CommentTimeMar 20th 2023
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   Author: Guest
   Format: TextI think this page needs to be edited so as to not be misleading: the page is titled "deformation quantization of the 2-sphere" and at the beginning it states "As such one may consider the deformation quantization of its Poisson algebra of functions." However, the proceeding discussion really has nothing to do with deformation quantization, it is about what the page calls "strict deformation quantization", and despite the name the two are only related in special cases, and the 2-sphere isn't one of them. Actually, what the page calls "strict deformation quantization" is much closer to what Rieffel calls "strict quantization", and he defines "strict deformation quantization" to be something stronger. Actually, the term "strict deformation quantization" seems to be used inconsistently across the nlab, where it is sometimes stated to be essentially a convergent deformation quantization. As far as I know, a non-formal deformation quantization of the 2-sphere is not known to exist. - Josh

   I think this page needs to be edited so as to not be misleading: the page is titled "deformation quantization of the 2-sphere" and at the beginning it states "As such one may consider the deformation quantization of its Poisson algebra of functions." However, the proceeding discussion really has nothing to do with deformation quantization, it is about what the page calls "strict deformation quantization", and despite the name the two are only related in special cases, and the 2-sphere isn't one of them. Actually, what the page calls "strict deformation quantization" is much closer to what Rieffel calls "strict quantization", and he defines "strict deformation quantization" to be something stronger. Actually, the term "strict deformation quantization" seems to be used inconsistently across the nlab, where it is sometimes stated to be essentially a convergent deformation quantization. As far as I know, a non-formal deformation quantization of the 2-sphere is not known to exist.
   
   - Josh
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 20th 2023
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   Author: Urs
   Format: MarkdownItexI have added a paragraph ([here](https://ncatlab.org/nlab/show/deformation+quantization+of+the+2-sphere#Details)) further highlighting the existence of different notions of deformation quantization with a pointer to Section 2 of Hawkins (2007) ([arXiv:0706.2946](https://arxiv.org/abs/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. <a href="https://ncatlab.org/nlab/revision/diff/deformation+quantization+of+the+2-sphere/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/deformation+quantization+of+the+2-sphere/3">v3</a>, <a href="https://ncatlab.org/nlab/show/deformation+quantization+of+the+2-sphere">current</a>

   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.

   diff, v3, current

4. • CommentRowNumber4.
   • CommentAuthorGuest
   • CommentTimeMar 24th 2023
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   Author: Guest
   Format: TextAh this makes more sense to me, thanks for the prompt reply. I agree "algebraic quantization" would be a fitting term, too bad. Maybe I will add some things.

   Ah this makes more sense to me, thanks for the prompt reply. I agree "algebraic quantization" would be a fitting term, too bad. Maybe I will add some things.