Spherical harmonics, invariant theory and Maxwell's poles

Details Spherical harmonics, invariant theory and Maxwell's poles Spherical harmonics, invariant theory and Maxwell's poles

2008-05-13

arxiv.org/abs/0805.1904v2

Physics

Mathematical Physics

45 pages. JyTex; added analysis and references, minor corrections

Scientific paper

I discuss the relation between harmonic polynomials and invariant theory and show that homogeneous, harmonic polynomials correspond to ternary forms that are apolar to a base conic (the absolute). The calculation of Schlesinger that replaces such a form by a polarised binary form is reviewed. It is suggested that Sylvester's theorem on the uniqueness of Maxwell's pole expression for harmonics is renamed the Clebsch-Sylvester theorem. The relation between certain constructs in invariant theory and angular momentum theory is enlarged upon and I resurrect the Joos--Weinberg matrices. Hilbert's projection operators are considered and their generalisations by Story and Elliott are related to similar, more recent constructions in group theory and quantum mechanics, the ternary case being equivalent to SU(3).

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