Wigner quantization of some one-dimensional Hamiltonians

TL;DR

This paper explores the Wigner quantization of two one-dimensional Hamiltonians, revealing their connection to the orthosymplectic Lie superalgebra osp(1|2) and introducing an extra representation parameter.

Contribution

It demonstrates the Wigner quantization of specific Hamiltonians related to osp(1|2), extending understanding of their mathematical and physical properties beyond canonical quantization.

Findings

Wigner quantization relates to osp(1|2) Lie superalgebra

Introduces an extra representation parameter in quantization

Canonical quantization is a special case of this framework

Abstract

Recently, several papers have been dedicated to the Wigner quantization of different Hamiltonians. In these examples, many interesting mathematical and physical properties have been shown. Among those we have the ubiquitous relation with Lie superalgebras and their representations. In this paper, we study two one-dimensional Hamiltonians for which the Wigner quantization is related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the Hamiltonian H = xp, is popular due to its connection with the Riemann zeros, discovered by Berry and Keating on the one hand and Connes on the other. The Hamiltonian of the free particle, H_f = p^2/2, is the second Hamiltonian we will examine. Wigner quantization introduces an extra representation parameter for both of these Hamiltonians. Canonical quantization is recovered by restricting to a specific representation of the Lie superalgebra osp(1|2).