Wigner distribution function for finite systems

Abstract

We construct a Wigner distribution function for finite data sets. It is based on a finite optical system; a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry. The dynamical group for this model is SU(2) and the wave functions are sets of N=2l+1 data points. The Wigner distribution function assigns classical c-numbers to the operators of position, momentum, and wave guide mode.

Wigner–Weyl isomorphism for quantum mechanics on Lie groups

“…It is then easy to see that we have the relations 15) which is in the spirit of eqns. (2.2,18), except that f is not restricted to be linear in any coordinate variables.…”

Wigner distributions and quantum mechanics on Lie groups: The case of the regular representation

“…The product of two elements A(q ′ ), A(q) is written as 14) where the n functions f r (q ′ ; q) of 2n real arguments each express the composition law in G. Certain important auxiliary functions play an important role; their definitions and some properties are summarised here: …”

A Wigner distribution function for finite oscillator systems

“…Also in the review paper [11], the Wigner function corresponding to an operator is the discrete Fourier transform of the matrix elements of that operator. A group theoretical approach was given in [12]: here, the Wigner operator is defined by extending (1) to an integral over a Lie group (using the invariant measure). The finite systems then arise through finite-dimensional representations of the Lie group.…”