${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded parastatistics in multiparticle quantum Hamiltonians

Abstract

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 $\mathcal{N}=4$ 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.

37 pages

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