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Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere

By M.R. Dennis


Any eigenfunction of the Laplacian on a sphere is given in terms of a unique set of directions: these are Maxwell's multipoles, their existence and uniqueness being known as Sylvester's theorem. Here, the theorem is proved by realizing the multipoles are pairs of opposite vectors in Majorana's sphere representation of quantum spins. The proof involves a physicist's standard tools of quantum angular momentum algebra, integral kernels and Gaussian integration. Various other proofs are compared, including an alternative using the calculus of spacetime spinors

Topics: QA, QC
Year: 2004
OAI identifier: oai:eprints.soton.ac.uk:29390
Provided by: e-Prints Soton

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  5. (1954). C 1891 A Treatise on Electricity and Magnetism vol 1, 3rd edn (Oxford: Clarendon Press; reprinted by NewYork:Dover,
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  14. (1984). Spinors and Space-time, vol 1: Two-spinor Calculus and Relativistic Fields (Cambridge:
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