Algebraic derivation of some theorems concerning contact structures on the 3-sphere

Abstract

Two theorems involving curl eigenfields on the 3--sphere are obtained using angular momentum theory. Spinor hyperspherical harmonics are shown to form an explicit, convenient basis. In particular, a spin--one vector calculus is reviewed. An easy proof of the vanishing of `odd' eigenfields is given and related to the sign change of fermionic spinors under 2$\pi$ rotations. The theorem that curl eigenfields with constant norm have to be proportional to a fundamental eigenfield (Hopf field) is also rapidly obtained. Attention is drawn to the relevance of early work of Schrödinger on Maxwell theory in an expanding universe.

13 pages. Amendment to text and 1 page explanatory Added Note regarding choice of expansion bases