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