On the twisted convolution product and the Weyl transformation of tempered distributions

Abstract

It is well known that the Weyl transformation in a phase space R 21, transforms the elements of L ( R 21) in trace class operators and the elements of L 2( R 21) in the Hilbert-Schmidt operators of the Hilbert space L 2( R 1); this fact leads to a general method of quantization suggested by E. Wigner and J.E. Moyal and developed by M. Flato, A. Lichnerowicz, C. Fronsdal, D. Sternheimer and F. Bayen for an arbitrary symplectic manifold, known under the name of star-product method. In this context, it is important to study the Weyl transforms of the tempered distributions on the phase space and that of the star-exponentials which gave the spectrum in this process of quantization. We analyze here the relations between the star-product, the twisted convolution product and the Weyl transformation of tempered distributions. We introduce symplectic differential operators which permit us to study the structure of the space O 1 λ λ ≠ 0, (similar to the space O 1 C) of the left (twisted) convolution operators of L ( R 21) which permit to define the twisted convolution product in the space L( R 21), and the structures of the admissible symbols for the Weyl transformation (i.e. the domain of the Weyl transformation). We prove that the bounded operators of L 2( R 1) are exactly the Weyl transforms of the bounded (twisted) convolution operators of L 2( R 21). We give an expression of the integral formula of the star product in terms of twisted convolution products which is valid in the most general case. The unitary representations of the Heisenberg group play an important role here.