DEFORMATION QUANTIZATION: PRO AND CONTRA 3
(1)
00
a*b= LhkCk(a,b)
k=O
where
ck
are bidifferential operators (the so-called locality property),
(2) the star-product is a formal deformation of the classical commutative
product, that is
(3)
Co(a, b)
=
ab,
C1(a,b)- C1(b,a)
=
ih{a,b},
(the so-called correspondence principle),
( 4) (optional)
for any
a E
A
An
equivalence
of two star-products
*
and
*'
is an isomorphism between the
algebras
(A, *)
and
(A, *')
given by a formal differential operator
Ta =a+ hT1a
+
h2T2 a
+ ...
with an identity operator as a leading term.
We see that the algebra of observables in deformation quantization is extremely
simple: we do not need a Hilbert space as in the traditional operator approach, we
do not need to care about domains of unbounded operators. Even comparing to
the C* -algebra approach we see that the C* -algebra structure is much more del-
icate one. Moreover, we don't need to worry how to construct the star-product:
the Kontsevich theorem gives a universal construction starting with an arbitrary
classical system. In contrast, other approaches may require quite non-trivial con-
structions (group representations, the theory of constrained systems).
2. Reduction
The algebra of quantum observables in deformation quantization may be treated
as an extension of the algebra of classical observables which serve as leading terms.
So, any construction in the former automatically descends to the latter. One can
ask if this principle admits a reversion. In other words, can we lift any construction
in the algebra of classical observables to the quantum level in the framework of
deformation quantization. Of course, there is no rigorous theorem of such sort, this
assertion may be viewed as an heuristic principle which explains the flexibility of
deformation quantization. Let me illustrate this on the example of the symplectic
reduction.
Consider a symplectic manifold ( M,
w)
with a Hamiltonian action of a compact
connected and simply connected Lie group
G.
"Hamiltonian" here means that
there exists the moment map
J-L :
M
--+
g*.
We assume, as usual, that zero is a
non-critical value of the moment map and
G
act freely on the zero level manifold.
Then the classical reduction theorem asserts that the manifold
B
=
M/ /G
:=
J-L-
1
(0)/G
is a symplectic manifold with a symplectic form ws such that its pull-back to
J-L-
1
(0) coincides with the restriction of w to J-L-
1
(0). The reduced algebra
A0 j
J1
of
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