$Z_2\times Z_2$-graded parastatistics in multiparticle quantum Hamiltonians

The recent surge of interest in $Z_2\times Z_2$-graded invariant mechanics poses the challenge of understanding the physical consequences of a $Z_2\times Z_2$-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 $Z_2\times Z_2$-graded parastatistics obeyed by the particles. The toy model of the $N=4$ supersymmetric/ $Z_2\times Z_2$-graded oscillator is used. In this set-up the one-particle energy levels and their degenerations are the same for both supersymmetric and $Z_2\times Z_2$-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 $Z_2\times Z_2$-graded particles. Therefore, $Z_2\times Z_2$-graded mechanics has experimentally testable consequences. Furthermore, the $Z_2\times Z_2$-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. (paper)