Finite Self-Energies in Radiation Theory. Part I

Abstract

According to Dirac, electric particles display a finite radius ${r}_{0}=\frac{2{e}^{2}}{3m{c}^{2}}$ as the result of the damping term $(\frac{2{e}^{2}}{3m{c}^{3}})\frac{{d}^{3}x}{d{t}^{3}}$ in the equation of motion. If the finite radius is due to radiative damping, the same must necessarily be true for the finite self-energy that is inversely proportional to the radius. An infinitely large self-energy and an infinitely small radius (Coulomb's law $\frac{{e}^{2}}{r}$) results from Fermi's Fourier representation of classical electrodynamics. A certain change is necessary, but the change is to produce at once a finite self-energy and a finite radius ${r}_{0}$. Now, an electric particle vibrating in a field of frequency $\ensuremath{\nu}$ suffers a reduction ${R}_{\ensuremath{\nu}}$ of its vibrational energy due to radiative damping, the energy reduction factor being ${R}_{\ensuremath{\nu}}=\frac{1}{[1+{(\frac{\ensuremath{\nu}}{{\ensuremath{\nu}}_{0}})}^{2}]}$ where ${\ensuremath{\nu}}_{0}=\frac{3m{c}^{3}}{4\ensuremath{\pi}{e}^{2}}$. In view of the uncertainty of position due to damping we propose that the Fourier terms in the expression for the energy in Fermi's classical radiation theory be reduced by the same factor ${R}_{\ensuremath{\nu}}$ with Doppler effect for particles in motion. The result of this reduction is that Dirac's finite radius ${r}_{0}$ now occurs in a modified Coulomb energy $(\frac{{e}^{2}}{r})[1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}r}{{r}_{0}})]$, and the finite self-energy of a single particle becomes $\frac{{e}^{2}}{2{r}_{0}}=(\frac{3}{4})m{c}^{2}$. Whereas the force between charged particles of finite mass remains finite for $r=0$, the force on an ideal test charge of infinite mass becomes infinite for $r=0$. This is analogous to the difference between the field $E$ and the displacement $D$ in Born's unitary field theory. Of interest for nuclear reactions are the electrostatic forces between particles of different masses $m$ and $M$. The results are related to Sommerfeld's fine-structure constant and to the theory of mesons.