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:
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