Group Theory and Quasiprobability Integrals of Wigner Functions

The integral of the Wigner function of a quantum mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0, 1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric disks and circles of given radius a. Their respective eigenvalues are determined as functions of a, in terms of the Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in a Hilbert space carrying the positive discrete series representation of the algebra su(1, 1) ≈ so(2, 1). The explicit relation between the spectra of operators associated with disks and circles with proportional radii, is given in terms of the discrete variable Meixner polynomials. PACS Nos: 03.65.Fd 02.30.Gp 02.20.Sv 42.50.Xa [email protected]; †[email protected] ; ‡[email protected]