6 BORIS FEDOSOV
fact asymptotic ones in such and such sense. Then ... ", and a necessary condition
follows for such a representation to exist.
Let me illustrate the situation on the example of the index theorem. Suppose we
have a representation of the algebra of quantum observables by means of operators
in a Hilbert space such that the formal power series for
Tr
a
is an asymptotic
expansion of the operator trace at h
--+
0 over some set A
C
(0, 1]. Then all the
indices must be integer-valued on the admissible set A modulo O(h00
).
Note that no
computational difficulties arise for indices since they are given by finite series. One
can show that in order to have integer-valued indices it is necessary and sufficient
that
(5.1)
that is, we have recovered the quantization condition of Maslov's asymptotic quan-
tization.
A similar application of the index theorem for elliptic operators we can see
even for a classical case of the Dirac operator. In this case the index is given by
the Atiyah-Singer formula
indD
=
L
chEA(X).
Surprisingly, the right-hand side is not always integer and its integrality is a neces-
sary condition for the existence of the spin structure.
6. Is there a Spectral Theory?
We all know about the important role that the spectral theory for operators in
Hilbert space plays in traditional quantum mechanics. In fact, it is responsible for
the variety of spectra. But in deformation quantization there is nothing which may
be compared in power with the spectral theorem though some separate notions
one can easily define. For example,
if
E
E
JR is a non-critical level of the real
Hamiltonian
H
E
C
00
(M)
we can construct an eigenstate up to a normalizing
factor
as
a non-trivial functional on the algebra of observables vanishing on the
subspace
(H-E)*
u, u EA. However, still we have problems
(1) how to treat critical values
(2) how to pull the eigenstates (critical and non-critical) together to obtain a
holomorphic calculus of quantum observables.
To my mind, a spectral theory for deformation quantization should be developed,
but maybe in a quite different way than for operators. It should incorporate Morse
theory (or rather the Floer theory) in the spirit of Witten,
as
well
as
the index
theory.
To conclude this section, let us consider an example where the spectrum may be
completely determined by means of reduction and the index theorem. So, consider
the quantum harmonic oscillator
H =
lxl
2
/2
E
C
00
(JR2(n+l)). For any
E
0 the
level manifold
H
=
E
is a sphere
lxl
2 =
2E
and the orbit space is CIF with the
symplectic form
w
=
2Ewps
where
wps
is the standard Fubini-Studi form on ClPm.
The quantization condition (5.1) yields
(6.1)
(
n+
1)
E
=
1rh k
+ -
2- ,
k
E
Z,
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