Covariant generalization of the Zitterbewegung of the electron and its SO(4,2) and SO(3,2) internal algebras

The internal geometry of the Dirac electron is studied in a proper-time formalism with ${\gamma }^0$-Hermitian operators. We solve the Heisenberg equations, separate external and internal coordinates, and identify the SO(3,2) internal algebra as the projection of an SO(3,3) geometry to the hyperplane (perpendicular to the center-of-mass momentum) where the Zitterbewegung takes place. We also give covariant intrinsic-spin and magnetic-moment operators. The system can be generalized to a larger system with the internal geometry SO(4,2) with the inclusion of dynamical variables ${\gamma }^5$ and i${\gamma }^5$ ${\gamma }^{\mu }$. The resultant internal algebras have higher-dimensional representations generalizing the Dirac electron to multifermion states.