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arXiv · 1612.07686

The Wigner distribution function for the $\mathfrak{su}(2)$ finite oscillator and Dyck paths

Roy Oste·Joris Van der Jeugt
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Abstract

Recently, a new definition for a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum, was developed. This distribution function is defined on discrete phase-space (a finite square grid), and can thus be referred to as the Wigner matrix. In the current paper, we compute this Wigner matrix (or rather, the pre-Wigner matrix, which is related to the Wigner matrix by a simple matrix multiplication) for the case of the $\mathfrak{su}(2)$ finite oscillator. The first expression for the matrix elements involves sums over squares of Krawtchouk polynomials, and follows from standard techniques. We also manage to present a second solution, where the matrix elements are evaluations of Dyck polynomials. These Dyck polynomials are defined in terms of the well known Dyck paths. This combinatorial expression of the pre-Wigner matrix elements turns out to be particularly simple.

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Roy Oste, Joris Van der Jeugt. 2016-12-22. The Wigner distribution function for the $\mathfrak{su}(2)$ finite oscillator and Dyck paths. https://doi.org/10.1088/1751-8113/47/28/285301

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