Abstract. 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.

PACS numbers: 03.65.Fd, 02.30.Px, 03.50.De

Print publication: Issue 40 (8 October 2004)
Received 26 July 2004
Published 22 September 2004