A Unified Treatment of the Characters of SU(2) and SU(1,1)

TL;DR

This paper presents a unified method for calculating characters of SU(2) and SU(1,1) groups using a Hilbert space approach, simplifying the evaluation of invariant integrals for different representations.

Contribution

It introduces a unified Hilbert space framework that simplifies character calculations for both SU(2) and SU(1,1), including their discrete and principal series representations.

Findings

Unified treatment of SU(2) and SU(1,1) characters

Evaluation of invariant integrals via geometric transformations

Distinct procedures for discrete and principal series of SU(1,1)

Abstract

The character problems of SU(2) and SU(1,1) are reexamined from the standpoint of a physicist by employing the Hilbert space method which is shown to yield a completely unified treatment for SU(2) and the discrete series of representations of SU(1,1). For both the groups the problem is reduced to the evaluation of an integral which is invariant under rotation for SU(2) and Lorentz transformation for SU(1,1). The integrals are accordingly evaluated by applying a rotation to a unit position vector in SU(2) and a Lorentz transformation to a unit SO(2,1) vector which is time-like for the elliptic elements and space-like for the hyperbolic elements in SU(1,1). The details of the procedure for the principal series of representations of SU(1,1) differ substantially from those of the discrete series.