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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 representationsacting in the eigenspaces of the total charge operator corresponding to theeigenvalues $n_1, n_2$ whose absolute values are less than or equal$\sqrt{\pi/e^2}$ are inequivalent if $|n_1| \neq |n_2|$ and contain thesupplementary series component acting as a discrete component. On the otherhand the representations acting in the eigenspaces corresponding to eigenvalueswhose absolute values are greater than $\sqrt{\pi/e^2}$ are all unitarilyequivalent and do not contain any supplementary series component.

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