We present a nonrelativistic wave equation for the electron in (3+1)-dimensions which includes negative-energy eigenstates. We solve this equation for three well-known instances, reobtaining the corresponding Pauli equation (but including negative-energy eigenstates) in each case.

The Schrödinger Equation and Negative Energies

Electron rest-frame internal canonical coordinates are reobtained by the free-particle Foldy-Wouthuysen transformation: Schrodinger “microscopic momentum”, Barut-Bracken “microscopic coordinate”, and the rest Hamiltonian, which describe Zitterbewegung in this frame.SO(4, 1) Snyder space-time invariant quantization is considered in order to construct a dynamical group for Zitterbewegung. The electron's internal structure appears associated with its secondorder self-energy process and governed by the 15-parameter dynamical groupSO(5, 1). This is a generalization of Barut-Bracken symmetry which describes Zitterbewegung as generated by an algebra of the rotation groupSO(5). This noncompact symmetrySO(5, 1) permits a natural interpretation for the operators of its algebra and introduces a generalization to higher-dimensional fermionic representations.

SO(5, 1) Dynamical Symmetry for Electron Zitterbewegung

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SO(5, 1) Dynamical Symmetry for Electron Zitterbewegung