DEFORMATION QUANTIZATION: PRO AND CONTRA

5

DEFINITION 4.1. A trace is a linear functional

having the property

Tr

(a*

b)

=

Tr (b

*a).

Here

Ac

E

A

is an ideal of compactly supported observables and at the right

stands the space of Laurent formal series in h with negative powers not exceeding

n

=

(1/2) dimM.

A trace functional gives rise to a very interesting generalisation of the Atiyah-

Singer index theorem for elliptic operators. In the simplest case when M is a

compact symplectic manifold and the coefficient bundle is Hom (E, E), the index

theorem consists in an explicit formula for the trace of the unit element:

(4.1)

Tr1

=

JM

chEexp

(2~h)

A(M).

Deformation quantization provides the most natural environment for the index

theorem and its various modifications. In the context of deformation quantization

they acquire quite different meaning than for elliptic operators (we will discuss this

subject in the next section).

The theory of traces in symplectic case is rather dull: there exists a unique

trace up to a normalization "constant". In contrast, the theory of traces for Poisson

manifolds is more rich. The traces are parametrized by volume forms dV such that

the Poisson tensor a is divergence-free. For each such volume form we have an index

formula similar to (4.1) with the replacement of exp(wj21rh) by i(exp(a/27rh))dV.

I hope enough was said about the merits of deformation quantization. Conclud-

ing the first part of my talk, I would like to stress once more the heuristic principle

that explains the merits of deformation quantization: any construction in the alge-

bra of classical observables can be lifted to the algebra of quantum observables in

the deformation quantization framework.

5. Difficulties with calculations

Now I am passing to the second part which can be titled "The drawbacks". To

begin with, let us note that in practice h is a small positive number, proportional

to the Planck constant with coefficients depending on physical parameters of the

system. Thus, for calculations we need to substitute this numerical value of h into

a formal power series. This operation is quite unclear except for the case when the

series terminates and we have a polynomial. Moreover, we know from traditional

quantum mechanics that physical parameters sometimes have discrete spectrum

and sometimes not. Is it possible to explain this in the deformation quantization

framework?

A reasonable way out is to treat formal power series in h as asymptotic ones, at

least as far as calculations are concerned. In fact, in many cases one can interpret

these series as asymptotic ones obtained by stationary phase method or the WKB

method for differential equations. But in general it is unclear if we can interpret

formal power series as asymptotic and in what sense. So, to my mind, most com-

putational algorithms should begin like this: "Suppose we have a representation

of our deformation business such that the formal power series in question are in