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

  title={Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere},
  author={Mark R. Dennis},
  journal={Journal of Physics A},
  • M. Dennis
  • Published 25 August 2004
  • Mathematics, Physics
  • Journal of Physics A
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… 
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