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## AN ENIGMA FACED BY DIRAC RESOLVED

Dirac solved the problem of duplexity of point-charge electrons by introducing the matrix square root of the Laplacian operator in 1928.  This was a resolution of experimental problem of observed spectra having twice the expected number of stationary states as the ‘new quantum theory’ following Goudsmit and Uhlenbeck’s introduction of spin angular momentum for the electron.  DiracElectron

Now the Dirac eigenfunctions on $\mathbf{R}^3$ are not compactly supported which makes it harder to interpret the Dirac wavefunctions.  Dirac eigenfunctions on a compact manifold have a better chance of being localized.  On the round three-sphere $S^3$ the eigenspinors can be computed exactly and the localization can be checked directly.  We can make a reasonable conjecture that for perturbations of the great sphere embedded in $S^4(1/h)$ mildly the same localization holds.