Zernike System Stems from Free Motion
on the 3-Sphere
Kurt Bernardo Wolf, Natig M. Atakishiyev, George S. Pogosyan,
and Alexander Yakhno
Abstract Systems that stem from projection of free motion on a manifold are the
best candidates to exhibit remarkable symmetry properties. This is the case of free
motion on the 3-sphere which, properly projected on the 2-dimensional manifold of
a disk, yields the Zernike system. This exhibits separability in a variety of coordinate
systems, polynomial solutions, and interbasis expansion coefficients that are special
Clebsch–Gordan coefficients and Hahn orthogonal polynomials.
Keywords Spherical geometry · Zernike system · Separation of variables ·
Clebsch–Gordan coefficients
1 Introduction: The so(4) Algebra
The Lie algebras of the orthogonal groups have a basis of generators K i,j = −K j,i ,
whose commutation relations are
[K i,j , K k,, ] = δ j,k K i,, + δ i,, K j,k + δ k,i K + δ K k,i ,
(1)
K. B. Wolf ()
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos,
Mexico
e-mail: bwolf@icf.unam.mx; http://www.fis.unam.mx/~bwolf/
N. M. Atakishiyev
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos,
Mexico
G. S. Pogosyan
Yerevan State University, Yerevan, Armenia
Joint Institute for Nuclear Research, Dubna, Russian Federation
A. Yakhno
Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e Ingenierías,
Universidad de Guadalajara, Guadalajara, Mexico
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_16
169
on the 3-Sphere
Kurt Bernardo Wolf, Natig M. Atakishiyev, George S. Pogosyan,
and Alexander Yakhno
Abstract Systems that stem from projection of free motion on a manifold are the
best candidates to exhibit remarkable symmetry properties. This is the case of free
motion on the 3-sphere which, properly projected on the 2-dimensional manifold of
a disk, yields the Zernike system. This exhibits separability in a variety of coordinate
systems, polynomial solutions, and interbasis expansion coefficients that are special
Clebsch–Gordan coefficients and Hahn orthogonal polynomials.
Keywords Spherical geometry · Zernike system · Separation of variables ·
Clebsch–Gordan coefficients
1 Introduction: The so(4) Algebra
The Lie algebras of the orthogonal groups have a basis of generators K i,j = −K j,i ,
whose commutation relations are
[K i,j , K k,, ] = δ j,k K i,, + δ i,, K j,k + δ k,i K + δ K k,i ,
(1)
K. B. Wolf ()
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos,
Mexico
e-mail: bwolf@icf.unam.mx; http://www.fis.unam.mx/~bwolf/
N. M. Atakishiyev
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos,
Mexico
G. S. Pogosyan
Yerevan State University, Yerevan, Armenia
Joint Institute for Nuclear Research, Dubna, Russian Federation
A. Yakhno
Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e Ingenierías,
Universidad de Guadalajara, Guadalajara, Mexico
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_16
169