The Hamiltonian H=xp and classification of osp(1|2) representations

TL;DR

This paper explores the mathematical properties of the quantized Hamiltonian H=xp, relating it to osp(1|2) Lie superalgebra representations, with implications for understanding the Berry-Keating conjecture and the Riemann hypothesis.

Contribution

The paper reexamines the classification of unitary, irreducible star representations of osp(1|2) using elementary methods, clarifying their structure in the context of H=xp quantization.

Findings

Complete classification of unitary, irreducible star representations of osp(1|2)

Reexamination of Hughes' classification with elementary arguments

Insights into the mathematical structure underlying H=xp quantization

Abstract

The quantization of the simple one-dimensional Hamiltonian H=xp is of interest for its mathematical properties rather than for its physical relevance. In fact, the Berry-Keating conjecture speculates that a proper quantization of H=xp could yield a relation with the Riemann hypothesis. Motivated by this, we study the so-called Wigner quantization of H=xp, which relates the problem to representations of the Lie superalgebra osp(1|2). In order to know how the relevant operators act in representation spaces of osp(1|2), we study all unitary, irreducible star representations of this Lie superalgebra. Such a classification has already been made by J.W.B. Hughes, but we reexamine this classification using elementary arguments.