Quantum Mechanics and Hidden Superconformal Symmetry

TL;DR

This paper explores the construction of quantum mechanical models with hidden osp(1|2) superconformal symmetry, revealing that the Plyushchay family of models uniquely realize all unitary irreducible representations of this symmetry in one dimension.

Contribution

It demonstrates that the Plyushchay models are the complete set of quantum models with hidden osp(1|2) symmetry in one dimension, linking harmonic oscillators to Dirac systems.

Findings

Plyushchay models realize all unitary irreducible osp(1|2) representations.

Harmonic oscillator viewed as a Dirac system reveals hidden superconformal symmetry.

Constructs all quantum models with hidden osp(1|2) symmetry on a given state space.

Abstract

Solvability of the ubiquitous quantum harmonic oscillator relies on a spectrum generating osp(1|2) superconformal symmetry. We study the problem of constructing all quantum mechanical models with a hidden osp(1|2) symmetry on a given space of states. This problem stems from interacting higher spin models coupled to gravity. In one dimension, we show that the solution to this problem is the Plyushchay family of quantum mechanical models with hidden superconformal symmetry obtained by viewing the harmonic oscillator as a one dimensional Dirac system, so that Grassmann parity equals wavefunction parity. These models--both oscillator and particle-like--realize all possible unitary irreducible representations of osp(1|2).