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

J. S. Dowker

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arXiv · 2304.13628

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

J. S. Dowker

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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 Schr\"odinger on Maxwell theory in an expanding universe.

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J. S. Dowker. 2023-04-26. Algebraic derivation of some theorems concerning contact structures on the 3-sphere. https://arxiv.org/abs/2304.13628

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