Classically Radiationless Motions and Possible Implications for Quantum Theory

A simple general criterion is developed, using the retarded potentials of classical electromagnetic theory, for absence of radiation from arbitrary time-periodic charge-current distributions. The criterion is applied to rigid finitely extended distributions of charge which may undergo orbital motion with period $T$. It is found that, for this type of distribution, the condition for no radiation is that the extent $b$ of the distribution be an integer multiple of $\mathrm{cT}$. Some of these distributions may spin while orbiting. There exists at least one asymmetric spinning distribution which doesn't radiate under this condition; for this distribution, the (constant) spin angular velocity must be proportional to an integer ≥0 times $\frac{c}{b}$. This leads to the result that that part of the total (electromagnetic) angular momentum which is associated with the spin angular velocity must be an integer ≥0 times $\frac{e^2}{c}$ times a numerical constant whose value depends on the details of the distribution. It is shown that, when such nonradiating distributions are considered as stable particles, there exists an intrinsic uncertainty relation of the same form and with almost the same meaning as that of quantum theory.