The quasi-nonassociative exceptional $F(4)$ deformed quantum oscillator

TL;DR

This paper introduces a unique deformed quantum oscillator with $F(4)$ superalgebra, exhibiting specific degeneracies, octonionic covariance, and a connection to superconformal quantum mechanics, expanding understanding of exceptional Lie superalgebras in quantum models.

Contribution

It constructs a novel $F(4)$-based deformed quantum oscillator with octonionic covariance and detailed spectral properties, linking superconformal symmetry to exceptional algebra structures.

Findings

Energy levels are at (2/3)+n, with specific degeneracies.

Model admits an octonionic-covariant formulation.

Superconformal quantum mechanics associated with the model is presented.

Abstract

We present the deformed (for the presence of Calogero potential terms) one-dimensional quantum oscillator with the exceptional Lie superalgebra $F(4)$ as spectrum-generating superconformal algebra. The Hilbert space is given by a $16$-ple of square-integrable functions. The energy levels are $\frac{2}{3}+n$, with $n=0,1,2,\ldots$. The ground state is $7$ times degenerate. The excited states are $8$ times degenerate. The $(7,8,8,8,\ldots )$ semi-infinite tower of states is recovered from the $(7;8;1)$ supermultiplet of the ${\cal N}=8$ worldline supersymmetry. The model is unique, up to similarity transformations, and admits an octonionic-covariant formulation which manifests itself as "quasi-nonassociativity". This means, in particular, that the Calogero coupling constants are expressed in terms of the octonionic structure constants. The associated $F(4)$ superconformal quantum mechanics is also presented.