Deformation quantization (survey, introduction, sources)

[marcus]

In case anyone is interested in DQ I came across this, which is moderately accessible in parts.
http://math.berkeley.edu/~alanw/242papers99/karaali.pdf

Evidently Alan Weinstein (Berkeley faculty) was teaching a graduate course Math 242 back in 1999 and put this online for his students.

If anyone likes this approach to quantization and has helpful source material please post it here.
Also Keraali has an interesting list of sources at the end of his survey.

I have some other links somewhere, and will go fetch them.
http://www.math.columbia.edu/~woit/wordpress/?p=7108&cpage=1#comment-214180

Hi, Peter. Have you ever taken a look at these papers:

Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. & Sternheimer, D. Deformation theory and quantization. I. Deformations of symplectic structures. Annals of Physics 111, 61-110 (1978),

Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. & Sternheimer, D. Deformation theory and quantization. II. Physical applications. Annals of Physics 111, 111-151 (1978),

which were seminal in launching the idea of quantization as deformation, with representation theory playing a secondary role?

The thing is that a classical Poisson algebra is more than just a Lie algebra, it also has a commutative product and a Leibniz rule for the bracket with respect to the product. Quantizing by simply representing the Lie algebra part of the structure can easily “break” the product structure and cause lots of problems, like Groenewold-type no-go results, the need to choose polarizations on the phase space, the need to choose the appropriate Lie sub-algebra to represent, deciding which of these choices lead to equivalent or inequivalent quantizations, deciding what constitutes a classical limit, etc.

All of these issues are dealt with head on in deformation quantization. Moreover, once a classical Poisson algebra is deformed to a quantum non-commutative algebra, the non-commutative product defines the commutator bracket, allowing Lie algebras of symmetries to be represented in it. GNS type constructions give you representations of the quantum algebra on Hilbert spaces. Thus, unitary representations of Lie algebras of symmetries still appear.

Of course, deformation quantization is not a construction or a prescription, rather it’s a definition. On the other hand, the same can be said about quantization as Lie algebra representation. However, once given a definition, one can go and find constructive methods that satisfy it. This has actually been done with great success for deformation quantization.

http://www.math.columbia.edu/~woit/wordpress/?p=7108&cpage=1#comment-214186
http://en.wikipedia.org/wiki/Wigner–Weyl_transform#Deformation_quantization

http://ncatlab.org/nlab/show/deformation+quantization

http://arxiv.org/abs/math/9809056
Deformation Quantization: Twenty Years After
Daniel Sternheimer
(Submitted on 10 Sep 1998)
We first review the historical developments, both in physics and in mathematics, that preceded (and in some sense provided the background of) deformation quantization. Then we describe...

 

[torsten]

Formal deformation quantization of Poisson manifolds

Dear Marcus,
interesting thread.
Deformation quantization is strongly connecetd with the work of Kontsevich, see
http://en.wikipedia.org/wiki/Kontsevich_quantization_formula (he won the Fields medal for it). It means that (at least from the fomal point of view) any Poisson manifold can be qunatized using deformation quantization.

I used a result of Tureav: the space of all holonomies on a surfaces (2D) can be quantized by deformation quantization, one obtains the skein space (3D) (space of singular knots) and the deformation parameter (Plancks constant) changes to a space dimension.

Torsten

 

[kith]

A while ago I downloaded the following pedagogical introduction but I haven't found the time to read it yet.

"Deformation quantization in the teaching of quantum mechanics"
Allen C. Hirshfeld and Peter Henselder
Am. J. Phys. 70 (5), May 2002
http://arXiv.org/abs/quant-ph/0208163v1