Spectrum-Generating Superalgebra for Linear Harmonic Oscillators

TL;DR

This paper demonstrates that the Hilbert space of the linear harmonic oscillator is fully described by the osp(2,1;2) superalgebra, providing a minimal algebraic framework with broad implications across physics.

Contribution

It introduces the osp(2,1;2) superalgebra as the smallest spectrum-generating algebra for the harmonic oscillator's Hilbert space.

Findings

The Hilbert space forms a complete orbit of osp(2,1;2).

osp(2,1;2) is the minimal spectrum-generating superalgebra.

Implications for various physics domains due to oscillator ubiquity.

Abstract

We show that the Hilbert space of the standard linear harmonic oscillator is a complete orbit of the osp(2,1;2) spectrum-generating superalgebra, and that this is the smallest such algebraic structure. The ubiquitous appearance of the linear harmonic oscillator in virtually all domains of theoretical physics guarantees a corresponding ubiquity of appropriate generalizations of this spectrum-generating superalgebra.