Mathematical Aspects of the Weyl Correspondence

TLDR: In this article, the Weyl correspondence between classical and quantum observables is rigorously formulated for a linear mechanical system with a finite number of degrees of freedom, where a multiplication of functions and a *-operation are introduced to make the Hilbert space of Lebesgue square-integrable complex-valued functions on phase space into a H*-algebra.

Abstract: The Weyl correspondence between classical and quantum observables is rigorously formulated for a linear mechanical system with a finite number of degrees of freedom. A multiplication of functions and a *‐operation are introduced to make the Hilbert space of Lebesgue square‐integrable complex‐valued functions on phase space into a H*‐algebra. The Weyl correspondence is realized as a *‐isomorphism f → W(f) of this H*‐algebra onto the H*‐algebra of Hilbert‐Schmidt operators on the Hilbert space of Lebesgue square‐integrable complex‐valued functions on configuration space. Moreover, the kernel of W(f) is exhibited in terms of a Fourier‐Plancherel transform of f. Elementary properties of the Wigner quasiprobability density function and its characteristic function are deduced and used to obtain these results.