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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
Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere
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.
0305-4470
9487-9500
Dennis, M.R.
ff55cf66-eb8b-4eb9-83eb-230c2f223d61
Dennis, M.R.
ff55cf66-eb8b-4eb9-83eb-230c2f223d61

Dennis, M.R. (2004) Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere. Journal of Physics A: Mathematical and General, 37 (40), 9487-9500. (doi:10.1088/0305-4470/37/40/011).

Record type: Article

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.

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Published date: 2004
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 29390
URI: http://eprints.soton.ac.uk/id/eprint/29390
ISSN: 0305-4470
PURE UUID: afe0c7dd-5cc2-42b1-9822-6cbda2d78641

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Date deposited: 11 May 2006
Last modified: 16 Dec 2019 19:21

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