Zitterbewegung and the internal geometry of the electron

Abstract

Schr\"odinger'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 $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}=mc\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}$ in the rest frame of the center of mass. His oscillatory "microscopic coordinate" vector is retained. In the rest frame, it takes the form $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}=\ensuremath{-}i(\frac{\ensuremath{\hbar}}{2mc})\ensuremath{\beta}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}$, and the Zitterbewegung is described in this frame in terms of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$, and the Hamiltonian $m{c}^{2}\ensuremath{\beta}$, as a finite three-dimensional harmonic oscillator with a compact phase space. The Lie algebra generated by $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}$ is that of SO(5), and in particular $[{Q}_{i},{P}_{j}]=\ensuremath{-}i\ensuremath{\hbar}{\ensuremath{\delta}}_{\mathrm{ij}}\ensuremath{\beta}$. 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).