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.
Categories: hep-th math-ph math.MP quant-ph