{mathbb Z}_2times {mathbb Z}₂-graded parastatistics in multiparticle quantum Hamiltonians

The recent surge of interest in {mathbb Z}_2times {mathbb Z}₂-graded invariant mechanics poses the challenge of understanding the physical consequences of a {mathbb Z}_2times{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}_2times{mathbb Z}₂-graded parastatistics obeyed by the particles. The toy model of the {cal N}=4 supersymmetric/ {mathbb Z}_2times {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}_2times{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}_2times {mathbb Z}₂-graded particles. Therefore, {mathbb Z}_2times {mathbb Z}₂-graded mechanics has experimentally testable consequences. Furthermore, the {mathbb Z}_2times {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.