New ladder operators for the monopole harmonics. (English) Zbl 1121.33015
Summary: Using the ladder operators shifting the index \(m\) of the associated Jacobi functions, for a given \(n\), the monopole harmonics and their corresponding angular momentum operators are, respectively, extracted as the irreducible representation space and generators of \(\text{su}(2)\) Lie algebra. The indices \(n\) and \(m\) play the role of principal and azimuthal quantum numbers. By introducing the ladder operators shifting the index \(n\) of the same associated Jacobi functions, we also get a new type of the raising and lowering relations which are realized by the operators shifting only the index \(n\) of the monopole harmonics. Moreover, other symmetries, including the transformation of the irreducible representation spaces into each other, are derived based on the operators that shift the indices \(n\) and \(m\) of the monopole harmonics simultaneously and agreeably as well as simultaneously and inversely. Our results are reduced to spherical harmonics by eliminating magnetic charge of the monopole.
MSC:
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
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