Chebyshev polynomials and Fourier transform of SU(2) irreducible representation character as spin-tomographic star-product kernel

TL;DR

This paper derives explicit formulas for star-product kernels in spin tomography using Chebyshev polynomials and Fourier transforms of SU(2) characters, revealing recurrence relations and sum rules for coefficients.

Contribution

It introduces a novel connection between star-product kernels in spin tomography and Chebyshev polynomials, providing explicit formulas and recurrence relations.

Findings

Explicit star-product kernels are expressed via Chebyshev polynomials.

Recurrence relations between kernels for different spins are established.

Sum rules for Clebsch-Gordan and Racah coefficients are derived.

Abstract

Spin-tomographic symbols of qudit states and spin observables are studied. Spin observables are associated with the functions on a manifold whose points are labelled by spin projections and 2-sphere coordinates. The star-product kernel for such functions is obtained in explicit form and connected with Fourier transform of characters of SU(2) irreducible representation. The kernels are shown to be in close relation to the Chebyshev polynomials. Using specific properties of these polynomials, we establish the recurrence relation between kernels for different spins. Employing the explicit form of the star-product kernel, a sum rule for Clebsch-Gordan and Racah coefficients is derived. Explicit formulas are obtained for the dual tomographic star-product kernel as well as for intertwining kernels which relate spin-tomographic symbols and dual tomographic symbols.