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

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

2004-08-25

arxiv.org/abs/math-ph/0408046v1

J. Phys. A 37 (2004) 9487-9500

Physics

Mathematical Physics

16 pages, no figs

Scientific paper

10.1088/0305-4470/37/40/011

Any eigenfunction of the laplacian on the 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 realising the multipoles are pairs of opposite vectors in Majorana's sphere representation of quantum spins. The proof involves the 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.

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