Strange behaviour of quadratic star-exponential functions under Moyal product
by
Hideki Omori
Science University of Tokyo

The Weyl algebra is understood as the space of all polynomials of two variables u, v equiped with the Moyal product. This product, denoted by *, extends naturally to yield f*g where either f or g is a polynomial. Thus, one can consider the differential equation

\fracddtft = i(uv)*ft,        f0(u, v) = 1

The solution uniquely exists in the real analytic category in t. Though the solution has singularities in t, we write this by e_*^ituv and call this a *-exponential function. This function behaves strangely.

By the concrete form of the real analytic solution, we see that both i\int₀^\inftye_*^ituvdt and -i\int_-infty⁰e_*^ituvdt exist, and as a result, uv has two different inverses. The difference of two inverses is written in the form of the Fourier transform of the constant function 1 as \int_-\infty^\inftye_*^ituvdt. Thus, this may be viewed as the *-delta function. However, this is not a distribution in our expression but a Bessel function. Such strange calculus, however, has a deep connection with Sato's hyper function theory.

Date received: October 13, 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 # caek-61.