Hyperbolic Angular Momentum

Author

• Jackson Lewis, Southern Illinois University CarbondaleFollow

Date of Award

5-1-2026

Degree Name

Master of Science

Department

Mathematics

First Advisor

Shajesh, Kuloth V.

Abstract

Schwinger’s oscillator construction provides an elegant algebraic realization of angular momentum operators in terms of pairs of independent harmonic oscillators. In addition to reproducing the standard $su(2)$ angular momentum algebra, this construction naturally gives rise to an alternative set of operators satisfying the $su(1,1)$ algebra, which we refer to as hyperbolic angular momentum.   In this thesis, we present a systematic development of Schwinger’s oscillator representation and analyze the algebraic structure of both the standard and hyperbolic angular momentum operators. We show that the hyperbolic operators provide a realization of the Lorentz algebra and establish their connection to the unitary irreducible representations of the Lorentz group classified by Bargmann. In particular, we demonstrate that the relationship between standard angular momentum and hyperbolic angular momentum arises from restricting Lorentz group representations to Bargmann’s discrete class $D_k^+$, the discrete classification with minimal state.   To clarify the structure of these representations, we provide a graphical framework for visualizing the Schwinger oscillator basis and the action of both the $su(2)$ and $su(1,1)$ ladder operators. This visualization provides an intuitive interpretation of angular momentum states and illustrates how hyperbolic operators connect states of different total spin.   As an application of the oscillator framework, we revisit the addition of angular momentum. Using the Schwinger representation, we provide a non-counting proof of the allowed coupled spin values, deriving the bounds on the total spin purely from the algebraic properties of the ladder operators rather than from dimension counting arguments.