Suppose we are given a rest mass and asked to solve the Dirac equation with representing the Coulomb potential. For this note we assume that scaling is done with . We will have for each the equation

where is the scalar curvature which is a cubic for which there can be at most three roots for . If none of the three roots match an eigenvalue of the Dirac operator (which for a 4-sphere is , the Dirac equation will not have nontrivial solutions. In particular on the entirety of there are severe mass restrictions for the Dirac equation with Coulomb potential. This type of mass restriction is a consequence of the discreteness of the Dirac spectrum and the specific formula possible for the projection of onto eigenspaces.

This means that given a particular mass , the possible eigenspace of Dirac operator on which the particle can live are at most three. The wave equation is then solved my multiplication by

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