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First of all, I know almost nothing about physics. I was reading Kontsevich´s paper on Deformation quantization of Poisson manifolds, however I could not figure out what´s the intuition for such operation.

Why there is the word "quantization" and the Planck constant in the star product? Where's the physics in such formal deformation of the algebra $\mathscr{A} = \mathscr{C}^{\infty}(M)$ ?

In fact, the unique thing that I can understand is the "deformation" part, since (if I'm not wrong) it looks like a deformation of $\mathscr{A}$ along the 2-cocycle $\{ \cdot, \cdot\}$ of the Hochschild cochain complex.

Another thing, that I never got: why is the deformation done only in the global sections and not in the whole sheaf $\mathscr{C}^{\infty}$?

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Maybe this paper of Catteneo and Felder might help. – suresh Feb 18 '14 at 1:45

1 Answer 1

Quantization usually means the association of a Hilbert space to the classical phase space (in our case a Poisson manifold). However, in deformation quantization, this task is achieved indirectly, first through the construction of an associative $C^*$ algebra, in this case the deformed algebra of functions equipped with a star product which serves as the associative product of the $C^*$ algebra. This algebra is dependent on a formal parameter $\hbar$. Once an associative $C^*$ algebra is constructed, a Hilbert space representation can be constructed in principle by $C^*$ algebraic techniques such as the GNS construction. Please see for example the following article by Stefan Waldmann.

The motivation of deformation quantization is that in many physical models, the Taylor series with respect to the Planck's constant $\hbar$ gives a viable deformation quantization. The prototype of an explicitly known star product whose Taylor series in $\hbar$ defines a deformation quantization is the Moyal product on $\mathbb{R}^{2n}$. Also, there is the Gutt star product on the duals of Lie algebras, please see the following article by Monvel. There are also, the Wick and Anti-Wick star products and their generalizations in Berezin quantization of Kähler manifolds. Please, see for example this article by Bordemann, Brischle, Emmrich, Waldmann. Another known construction of geometric origin is the Fedosov star product, please see the following Philip Tillman thesis by .

The Hoschild closure of the deformation chains is needed to ensure the associatively of the star product, please see the following article by: McCurdy and Zumino (although treating a special case, but the relation between closure and associatively is clarified).

In Kontsevich construction, the sections need to be global, because the construction was performed locally for $\mathbb{R}^d$, and there is a need to globalize to a general Poisson manifold.

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I +1'd your answer, but the citation to McCurdy and Zumino's paper is incomplete...did you mean or some other paper? – Alex Nelson Feb 17 '14 at 17:55
Thanks for the answer. However I could not understand your justification for the fact of considering just the global sections. For instance, when one considers deformation of complex manifolds the deformation is performed in each chart, hence the deformation is done in each open set (or I'm missing something). You cited too that some physical models deals with taylor expansion with respect to the planck constant, could please show an example? – user40276 Feb 18 '14 at 0:59
@Alex Nelson - Corrected. Thank you. – David Bar Moshe Feb 18 '14 at 8:37
@user40276 - 1)I have added a few examples of explicitly known star products. 2) Consider for example vector fields which can be realized as differential operators on each chart, but not every set of differential operators defined on each chart is a vector field. Only the cases in which these differential operators satisfy the correct transformation properties on the overlaps make them vector fields, i.e., when they are global sections of $TM$. – David Bar Moshe Feb 18 '14 at 8:40
Thanks for the information added and sorry for my insistence, but I´m not getting where is the physics in such models. Apparently everything is a kind of generalization of the Weyl quantization (correct me whether I´m wrong, please), so maybe I should ask another question about the Weyl quantization. – user40276 Feb 18 '14 at 11:35

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