W(infty)-Covariance of the Weyl-Wigner-Groenewold-Moyal Quantization

Abstract

The differential structure of operator bases used in various forms of the Weyl-Wigner-Groenewold-Moyal (WWGM) quantization is analyzed and a derivative-based approach, alternative to the conventional integral-based one is developed. Thus the fundamental quantum relations follow in a simpler and unified manner. An explicit formula for the ordered products of the Heisenberg-Weyl algebra is obtained. The W(infty) -covariance of the WWGM-quantization in its most general form is established. It is shown that the group action of W(infty) that is realized in the classical phase space induces on bases operators in the corresponding Hilbert space a similarity transformation generated by the corresponding quantum W(infty) which provides a projective representation of the former $W_{\infty}$. Explicit expressions for the algebra generators in the classical phase space and in the Hilbert space are given. It is made manifest that this W(infty)-covariance of the WWGM-quantization is a genuine property of the operator bases.