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

TL;DR

This paper establishes a new theorem on the representation structure of the $SL(2, \\mathbb{C})$ group in the quantum Coulomb field, revealing inequivalence and unitarity conditions based on charge eigenvalues.

Contribution

It provides a novel classification of $SL(2, \\mathbb{C})$ representations in the quantum Coulomb field, highlighting the role of supplementary series components and charge eigenvalues.

Findings

Representations with charge eigenvalues $|n| \\leq \\sqrt{\\pi/e^2}$ are inequivalent if $|n_1| eq |n_2|$.

Representations with charge eigenvalues $|n| > \\sqrt{\\pi/e^2}$ are unitarily equivalent.

Representations with larger eigenvalues do not contain supplementary series components.

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.