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,