A new theorem on the representation structure of the $SL(2,\mathbb{C})$ group acting in the Hilbert space of the quantum Coulomb field

Using the results obtained by Staruszkiewicz in Acta Phys. Pol. B 23, 591 (1992) and in Acta Phys. Pol. B 23, 927 (1992) we show that the representations acting in the eigenspaces of the total charge operator corresponding to the eigenvalues $n_1,n_2$ whose absolute values are less than or equal $\sqrt{\pi \mathrm{/}{e}^2}$ are inequivalent if $\mathrm{\mid }{n}_1\mathrm{\mid }\mathrm{\ne }\mathrm{\mid }{n}_2\mathrm{\mid }$ and contain the supplementary series component acting as a discrete component. On the other hand the representations acting in the eigenspaces corresponding to eigenvalues whose absolute values are greater than $\sqrt{\pi \mathrm{/}{e}^2}$ are all unitarily equivalent and do not contain any supplementary series component.