On the Quantization of Mass

Abstract

The Dirac equation may be written so as to give an eigenvalue problem whose eigenvalues would be values of the electron mass. The equation is solved in several cosmological spaces, none requiring any quantization of $m$. A space-time suggested by Eddington leads to wave equations that have solutions with quantized $m$, $m$ depending upon the radius of the universe, constants $c$ and $\ensuremath{\hbar}$, and a quantum number. If the radius of the universe is taken as ${10}^{28}$ cm, the lowest mass state in this group of solutions is of the order of ${10}^{\ensuremath{-}65}$ g. Conversely if the usual electron mass is considered as the lowest state, the radius of the universe is of the order of ${10}^{\ensuremath{-}10}$ cm. Since the only constants occurring in the theory are $\ensuremath{\hbar}$, $m$, and $c$, one would expect that the radius of the universe would come out in terms of $\frac{h}{\mathrm{mc}}$. Only an occurrence, as the result of quantization, of a large dimensionless number could lead to a reasonable result; but Dirac's equation evidently does not provide such a number, and is therefore unsuited to account for the electron mass in terms of the radius of the universe.