Group Theory and Quasiprobability Integrals of Wigner Functions

Abstract

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 Hilbert space carrying the positive discrete series representations of the algebra su(1,1)or 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 dicrete variable Meixner polynomials.

11 pages, latex file

References in corpus (3)

• Realizations of $su(1,1)$ and $U_q(su(1,1))$ and generating functions for orthogonal polynomials
• Bounds on Integrals of the Wigner Function
• Non-positivity of the Wigner function and bounds on associated integrals

Cited by in corpus (6)

• Wigner-Weyl isomorphism for quantum mechanics on Lie groups
• Wigner distributions and quantum mechanics on Lie groups: the case of the regular representation
• The quantum state vector in phase space and Gabor's windowed Fourier transform
• Bounds on integrals of the Wigner function: the hyperbolic case
• Region Operators of Wigner Function: Transformations, Realizations and Bounds
• Non-positivity of the Wigner function and bounds on associated integrals