Contemporary Mathematics
Volume 315, 2002
Deformation quantization: pro and contra
Boris Fedosov
ABSTRACT. We list some advantages and drawbacks of deformation quantiza-
tion.
Advantages:
(1) Deformation quantization gives a flexible quantization procedure which
works for any classical phase manifolds: Kahler, symplectic or Poisson.
(2) The Hamiltonian reduction in the framework of deformation quantization
goes almost similarly to the classical case.
(3) Admits generalizations to non-trivial coefficient bundles and for some
infinite-dimensional manifolds.
(4) There is a trace functional giving rise to index theorems.
Drawbacks and questions:
(1) Difficulties with calculations.
(2) Is it possible to get predictions with the help of deformation quantiza-
tion?
(3) Is there a reasonable substitution for spectral theory? For the Schri:idinger
evolution?
Introduction
In the last 5 years deformation quantization became an intensively developing
field, especially after the seminal work of M. Kontsevich where he not only gave
an answer to the longstanding question of deformation quantization, namely, the
existence and classification of star-products for every Poisson manifold, but also
discovered new directions and generalizations connected with the formality theorem.
We all know that this work has found a wide resonance: there appeared a great
number of publications caused by Kontsevich's paper, from a simple re-exposition
of his results to completely new ideas leading to new proofs. There appeared also
new names, let me mention A. Kattaneo, G. Felder, D. Tamarkin. I vividly imagine
a picture: a powerful tractor plowing a field and a crowd of birds behind looking
for warms. The Fields Medal awarded to Kontsevich in 1998 promoted deformation
quantization to a fashionable field, so to say "science a la mode" . Within the stream
of information I have felt an increasing desire to analyze and valuate the deformation
quantization
as
a whole: what is its place among different quantum theories, what
are its merits and drawbacks, can we expect further intensive development and
2000 Mathematics Subject Classification. Primary 53D55.
Key words and phrases. Star-product, reduction, trace, index.
http://dx.doi.org/10.1090/conm/315/05470
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