${\mathbb Z}_2\times {\mathbb Z}_2$-graded parastatistics in multiparticle quantum Hamiltonians

TL;DR

This paper demonstrates that ${ m Z}_2 imes { m Z}_2$-graded parastatistics in multiparticle quantum systems lead to observable differences from traditional bosons and fermions, with testable physical consequences in a supersymmetric oscillator model.

Contribution

It introduces the concept of ${ m Z}_2 imes { m Z}_2$-graded parastatistics in multiparticle quantum mechanics and shows how these can be experimentally distinguished from standard statistics.

Findings

Multiparticle measurements can discriminate between graded and ordinary particles.

${ m Z}_2 imes { m Z}_2$-grading imposes superselection rules on observables.

The multiparticle sector is modeled using a Hopf algebra with a braided tensor product.

Abstract

The recent surge of interest in ${\mathbb Z}_2\times {\mathbb Z}_2$-graded invariant mechanics poses the challenge of understanding the physical consequences of a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded symmetry. In this paper it is shown that non-trivial physics can be detected in the multiparticle sector of a theory, being induced by the ${\mathbb Z}_2\times{\mathbb Z}_2$-graded parastatistics obeyed by the particles. The toy model of the ${\cal N}=4$ supersymmetric/ ${\mathbb Z}_2\times {\mathbb Z}_2$-graded oscillator is used. In this set-up the one-particle energy levels and their degenerations are the same for both supersymmetric and ${\mathbb Z}_2\times{\mathbb Z}_2$-graded versions. Nevertheless, in the multiparticle sector, a measurement of an observable operator on suitable states can discriminate whether the system under consideration is composed by ordinary bosons/fermions or by ${\mathbb Z}_2\times {\mathbb Z}_2$-graded particles. Therefore, ${\mathbb Z}_2\times {\mathbb Z}_2$-graded mechanics has experimentally testable consequences. Furthermore, the ${\mathbb Z}_2\times {\mathbb Z}_2$-grading constrains the observables to obey a superselection rule. As a technical tool, the multiparticle sector is encoded in the coproduct of a Hopf algebra defined on a Universal Enveloping Algebra of a graded Lie superalgebra with a braided tensor product.