4 BORIS FEDOSOV
G-invariant functions modulo the ideal generated by
J.L
coincides with the algebra
C
00
(B) with a Poisson structure defined by the symplectic form
WB.
What happens in deformation quantization? First of all, there exists a canonical
construction of a deformation quantization on a symplectic manifold parametrized
by a closed two-form
i
f2
=
hW
+
Wo
+
hw1
+ ... ,
called the Weyl curvature. Clearly, it may be chosen G-invariant if the form
n
is
invariant. Let *M and *B be the canonical star-products on
M
and
B
with the
Weyl curvatures
.
.
z z
OM=
hWM, f2B
=
hWB
respectively. Then we have a reduction theorem in deformation quantization which
sounds almost identically with the classical one, namely:
THEOREM
2.1. The algebra
(A(B) :=
C
00
(B)[[h]], *B) is isomorphic to the
quantum reduced algebra, which is the quotient of the subalgebra of G-invariants
(A( M),
*
M)
0
modulo the *-ideal generated by the moment map
J.L·
This is a statement of the type "reduction commutes with quantization", known
in geometric quantization as the Guillemin-Sternberg conjecture. While in the geo-
metric quantization framework it is a serious theorem, for deformation quantization
it is much more simple. Besides, it gives a natural correspondence between original
and reduced observables.
3. Non-trivial coefficients
Deformation quantization admits a natural generalization to the case of non-
trivial coefficients at least in the case of a symplectic manifold. For example, the
coefficients may be matrices and they may form a non-trivial bundle over
M,
i.e., we
may consider a deformation quantization with a coefficient bundle K =Hom (E, E)
for some vector bundle E over M. Moreover, in practice we need even more compli-
cated bundles. For instance, for the reduction theorem we need a bundle over the
base
B
whose fibers are quantum observables on
T*G.
The changes are as follows.
The algebra of classical observables is now a non-commutative algebra of sections of
the coefficient bundle
c= (
M, K), and we replace the condition 3 of Definition 1.1
by
ih ..
C1(a,b)
= -
2
w•3
Oiaaib
where 8 now means a covariant derivative in the coefficient bundle.
A canonical construction of deformation quantization with non-trivial coeffi-
cients is quite general. It is meaningful even in infinite-dimensional setting of mod-
uli spaces of connections on a Riemann surface, leading to a quantization procedure
commuting with reduction.
4. Trace and index
Besides the algebra of observables, a mechanical system (classical or quantum)
is characterized by its states: linear functionals on the algebra of observables with
some further additional properties. Usually one imposes a kind of positivity. We
will not go deeper into the subject, instead we consider a trace functional, which
may be introduced without positivity.
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