Deformation quantization of Frechet Poisson algebras - Convergence of the Moyal product -
by
Yoshiaki Maeda
Keio University, Japan
Coauthors: Hideki Omori (Science University of Tokyo), Akira Yoshioka (Science University of Tokyo), Naoya Miyazaki (Keio University)

Oral Communication

Beyond formal deformation quantization; deformation quantizations of Poisson algebras with the formal deformation parameter (cf.[BF]), one can ask whether the formal parameter converges. For this direction, Rieffel [R] presented a notion of strict deformation quantization; deformation quantization with the convergence product in the C^*-algebras. This suggests us opportunities of finding the various kind of notions of deformation quantizations with the convergence product for suitable categories of algebras.

The purpose of this talk is to give a notion of deformation quantization corresponding to the Rieffel's work on the strict deformation quantization in the Fréchet categories, and to show that several strange phenomena occur when we treat exponential functions of quadratic forms.

Denote by P(C²) the set of all polynomials of x and y, and denote by E(C²) the set of all entire functions on C². E(C²) is a complete topological vector space under the compact open topology.

For entire functions f=f(x, y) and g=g(x, y) on C², we set

\labeleq:poisson bracket{f, g} = \partialy f·\partialx g -\partialx f·\partialyg.

(\theequation)

This gives a Poisson bracket on C², which is canonical in the sense that the coordinates x, y turn to be a Darboux coordinates.

The Moyal product formula is given as follow (cf.[OMY]):

\labelmoy00 f*\h g = å k  (\frac\h i2)k \frac1k! f ( <-- \partial   y  · --> \partial   x  - <-- \partial   x  · --> \partial   y  )k g.

(\theequation)

In general, f*_\h g is not always defined for any f, g in E(C²), but the following properties hold:

•   *_\h defines an associative product on the space P(C²).
•   For every fixed p in P(C²), the left- and the right-multiplication p*_\h,   *_\hp extend to continuous linear mappings
  

  p*\h,   *\hp:  E(C2) --> E(C2).

We consider the following subspace of E(C² ): For every positive p > 0, set

Ep(C2 )={f in E(C2 ) |  ||f||p, s=  sup  |f| e(-s|\xi|p) < \infty, for alls > 0}

(\theequation)

where |\xi|=(|x|²+|y|²)^1/2. The family {||  ||_{p, s}}_{s > 0} induces a topology on E_p(C² ) and (E_p(C² ), ·) is an associative commutative Fréchet algebra, where the dott · is the ordinary multiplication for functions in E_p(C² ). It is easily seen that for 0 < p < p', we have a continuous embedding

Ep(C2 ) subset Ep'(C2 )

(\theequation)

as a commutative Fréchet algebra (cf.[GS]).

It is obvious that every polynomial is contained in E_p(C²) and P(C²) is dense in E_p(C²) for any p > 0. The Poisson bracket () is also well-defined on E_p(C²), and (P(C²), {, }, ·) is a dense Poisson subalgebra of (E_p(C²), {, }, ·). We remark that every exponential function e^{\alphax +\betay} is contained in E_p(C²) for any p > 1, but not in E₁(C²), and functions such as e^{ax2+by2 +2cxy} are contained in E_p(C²) for any p > 2, but not in E₂(C²).

Our main result in this talk is as follows : The Moyal product formula gives the following:

(i)

For 0 < p <= 2, the space (E_p(C²), *_\h ) is a deformation quantization of (E_p(C²), ·, {, }).

(ii)

For p > 2 and a fixed \h in R, the Moyal product formula gives a continuous bi-liner mapping of

Ep(C2)×Ep'(C2) --> Ep(C2), Ep'(C2)×Ep(C2) --> Ep(C2),

(\theequation)

for every p' such that \frac1p+\frac1p' >= 1.

Date received: April 10, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-27.