2

BORIS FEDOSOV

so on. I am deeply indebted to the organizers of this meeting who gave me the

opportunity to speak here on this subject..

One can observe the ideas of deformation quantization at the very beginning

of quantum mechanics. I mean a non-commutative algebra of quantum observables

satisfying the correspondence principle and in the limit giving the classical commu-

tative algebra of observables. The only thing which was unrealized for a rather long

time is how to understand the limiting procedure to the classical mechanics, and,

in particular, what is the precise meaning of the correspondence principle. And

comparatively recently, in the late 70-s F. Bayen, M. Flato, C. Fronsdal, A. Lich-

nerowicz, D. Sternheimer suggested to consider quantum mechanics as a deforma-

tion theory. This point of view gave the maximal freedom for mathematicians with

their rigor that often seems to tie hands. It may sound funny, but for physicists the

idea of formal power series in h at first seemed a little bit strange. I remember I

gave a lecture course on deformation quantization for graduate students of Moscow

Institute of Physics and Technology, and one of the first questions was: "Don't

you find that the formal power series look very abstract, it is difficult to operate

with this notion". It took me some time to explain to them that this notion should

be the most familiar for physicists because they simply compute the coefficients

of various expansions and don't care of convergence or asymptotic nature of these

expansions. In other words, they actually work with formal power series. This

reminds me of one of Moliere's characters who did not know that he spoke prose.

Unfortunately, untying our hands we loose important phenomena, since defor-

mation quantization is unable to handle rapidly oscillating or rapidly decreasing

exponents . Thus, the typical quantum effects, such as diffraction pictures or tun-

nelling, remain outside the scope of deformation quantization. This is the main

contradiction which we come across again and again trying to realize the place

of deformation quantization. On one hand, the maximal freedom in the algebra

of quantum observables, on the other hand, a certainly incomplete description of

quantum phenomena. In my talk I will try to explain this contradiction in more

detail, illustrating it on several examples.

1.

Basic Notions

Let me recall the basic notions of deformation quantization. We consider a

Poisson manifold with an antisymmetric Poisson tensor aij which makes the com-

mutative algebra of functions

C

00

(M)

into a Lie algebra with the bracket

{u,v}

=

ohaiuaiv

satisfying the Jacobi identity and compatible with the commutative product

{u, vw}

=

{u, v}w

+

v{u,

w}.

A particular case is a symplectic manifold, when the Poisson tensor is non-degenerate.

Denoting Wij the inverse matrix to aii, one can see that the Jacobi identity is equiv-

alent to the closedness of the form

1 . .

w

=

2'

Wijdx'

1\

dx1

.

DEFINITION 1.1. A

deformation quantization

is an associative algebra structure

on the space A(M)

:=

C

00

(M)[[h]]

of formal power series in a formal parameter h

with respect to a product

*

(called a

star-product)

with the properties: