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

TL;DR

This paper proves Sylvester's theorem by representing Maxwell's multipoles as pairs of opposite vectors on Majorana's sphere, linking spherical eigenfunctions to quantum spin representations.

Contribution

It provides a new proof of Sylvester's theorem using Majorana's sphere and quantum angular momentum algebra, connecting classical spherical functions with quantum spin geometry.

Findings

Sylvester's theorem is proved via Majorana's sphere representation.

Maxwell's multipoles correspond to pairs of opposite vectors on Majorana's sphere.

The proof employs quantum angular momentum algebra and Gaussian integration.

Abstract

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.