A New Theorem on the Representation Structure of the SL$(2, \mathbb {C})$ Group Acting in the Hilbert Space of the Quantum Coulomb Field

Abstract

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 $\sqrt{\pi/e^2}$ are inequivalent if $|n_1| \neq |n_2|$ 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 $\sqrt{\pi/e^2}$ are all unitarily equivalent and do not contain any supplementary series component.