DEFORMATION QUANTIZATION: PRO AND CONTRA 7
besides,
Ttl= c~/1rh+n-l"
The the usual magic words follow: "Suppose that the reduced algebra, that is
A ( ClP'n),
corresponds to the algebra of operators in the eigenspace of the "operator"
H with the eigenvalue E. Then the possible spectrum of H is given by (6.1)
provided that Tt 1 is a positive integer". On the set (6.1) we have Tt 1 =
c:;:+k
which is greater than zero for k
~
0. Thus, we indeed have found the spectrum
with its multiplicities. Of course, one needs to give more precise meaning to the
word "corresponds".
7. What is the Schrodinger Evolution?
The original idea of F. Bayen et al. was to get an alternative approach to
quantum mechanics based on deformation quantization rather than on the operator
theory. Instead of the Schrodinger evolution operator exp(iHt/h) they introduced
the so-called star-exponential exp* ( iH
t
/h) which one can imagine intuitively as a
series
00
1
("t)k
L
k!
~
H
*
H
* ... *
H
k=O k times
or as a solution of the differential equation
aU
=
!:._
H
*
U U(O)
=
1.
at
h '
The idea was that the Fourier-Dirichlet series determines completely the spectrum
of Has the set offrequencies with spectral projectors as coefficients. Unfortunately,
this heuristic notion may be justified only in exceptional cases like the harmonic
oscillator. On the other hand, we badly need such a notion, for example to define
characters for group action. One of the possible ways to define it rigorously is to
interpret it as a distribution Tt
U(t)a
where
a
E
Cc(M)
with
t
E [0, T] fixed. It turns
out that under natural additional conditions the support of this distribution should
coincide with fixed-point set of the classical hamiltonian flow. The contribution
of the given fixed-point component also may be found up to a "constant" factor.
But examples show that we need to extend the "constants" by rapidly oscillating
exponents, understood as formal expressions. A problem arises of consistent choice
of rapidly oscillating factors for each fixed-point component. Besides, the fixed-
point set depends on
t
rather chaotically: the components may fuse and split.
Thus, even if we succeed in definition of the star-exponential, one could hardly
expect to extract from it an information about spectra. On the contrary, we rather
could expect that the geometry of fixed-point components and the oscillating factors
are governed by the quantization conditions. In any case, it would be interesting
to understand the connection between quantization conditions and the behavior of
fixed-point components.
In conclusion, summarizing the merits and drawbacks of deformation quanti-
zation, we see that the main difficulties originate from calculations, as a rule they
require some artificial tricks. On the other hand, as long as we deal with construc-
tions in the algebra of observables, deformation quantization is a flexible tool and
gives us the maximal freedom.
MAX-PLANCK-INSTITUT FlJR MATHEMATIK, P.O.Box: 7280, D-53072 BONN, GERMANY
E-mail address:
fedosov!Dmpim-bonn .mpg. de
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