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
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