Schrödinger's work on the Zitterbewegung of the free electron is reexamined. His proposed "microscopic momentum" vector for the Zitterbewegung is rejected in favor of a "relative momentum" vector, with the value $\overrightarrow{\mathrm{P}}=mc\overrightarrow{\alpha }$ in the rest frame of the center of mass. His oscillatory "microscopic coordinate" vector is retained. In the rest frame, it takes the form $\overrightarrow{\mathrm{Q}}=-i\left(\frac{\hslash }{2mc}\right)\beta \overrightarrow{\alpha }$, and the Zitterbewegung is described in this frame in terms of $\overrightarrow{\mathrm{P}}$, $\overrightarrow{\mathrm{Q}}$, and the Hamiltonian $m{c}^2\beta$, as a finite three-dimensional harmonic oscillator with a compact phase space. The Lie algebra generated by $\overrightarrow{\mathrm{Q}}$ and $\overrightarrow{\mathrm{P}}$ is that of SO(5), and in particular $[Q_i,P_j]=-i\hslash {\delta }_{\mathrm{ij}}\beta$. It is argued that the simplest possible finite, three-dimensional, isotropic, quantum-mechanical system requires such an SO(5) structure, incorporates a fundamental length, and has harmonic-oscillator dynamics. Dirac's equation is derived as the wave equation appropriate to the description of such a finite quantum system in an arbitrary moving frame of reference, using a dynamical group SO(3,2) which can be extended to SO(4,2). Spin appears here as the orbital angular momentum associated with the internal system, and rest-mass energy appears as the internal energy in the rest frame. Possible generalizations of these ideas are indicated, in particular those involving higher-dimensional representations of SO(5).

• Received 1 April 1980

DOI:https://doi.org/10.1103/PhysRevD.23.2454