-graded parastatistics in multiparticle quantum Hamiltonians

Abstract

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