Inequivalent quantizations from gradings and ${\mathbb Z}_2\times {\mathbb Z}_2$ parabosons

TL;DR

This paper explores new types of particles called ${ m Z}_2 imes { m Z}_2$-graded parabosons, their algebraic structure, and how to detect them in quantum models, expanding the understanding of parastatistics beyond traditional frameworks.

Contribution

It introduces ${ m Z}_2 imes { m Z}_2$-graded parabosons, constructs their multi-particle states, and demonstrates how to distinguish them from ordinary bosons in quantum systems.

Findings

Nine inequivalent multi-particle Hilbert spaces identified.

${ m Z}_2 imes { m Z}_2$-graded parabosonic space constructed.

Detection method for parabosons via observable measurements.

Abstract

This paper introduces the parastatistics induced by ${\mathbb Z}_2\times {\mathbb Z}_2$-graded algebras. It accommodates four kinds of particles: ordinary bosons and three types of parabosons which mutually anticommute when belonging to different type (so far, in the literature, only parastatistics induced by ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superalgebras and producing parafermions have been considered). It is shown how to detect ${\mathbb Z}_2\times {\mathbb Z}_2$-graded parabosons in the multi-particle sector of a quantum model. The difference with respect to a system composed by ordinary bosons is spotted by measuring some selected observables on certain given eigenstates. The construction of the multi-particle states is made through the appropriate braided tensor product. The application of ${\mathbb Z}_2$- and ${\mathbb Z}_2\times {\mathbb Z}_2$- gradings produces $9$ inequivalent multi-particle Hilbert spaces of a $4\times 4$ matrix oscillator. The ${\mathbb Z}_2\times {\mathbb Z}_2$-graded parabosonic Hilbert space is one of them.