Z 2 × Z 2 -graded parastatistics in multiparticle quantum Hamiltonians

Source abstract

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