Inequivalent quantizations from gradings and ${\mathbb Z}_2\times {\mathbb Z}_2$ parabosons

Resumo

This paper introduces the parastatistics induced by {\it \zzg algebras}. It accommodates four kinds of particles: ordinary bosons and three types of parabosons which mutually anticommute when belonging to different type
(so far, in the literature, only parastatistics induced by {\it \zzg superalgebras} and producing parafermions have been considered).\par
It is shown how to detect \zzg parabosons in the multi-particle sector of a quantum model.
The difference with respect to a system composed by ordinary bosons is spotted by measuring some selected observables on certain given eigenstates. The construction
of the multi-particle states is made through the appropriate braided tensor product.\par
The application of ${\mathbb Z}_2$- and ${\mathbb Z}_2\times {\mathbb Z}_2$- gradings produces $9$ inequivalent multi-particle Hilbert spaces of a $4\times 4$ matrix oscillator.
The \zzg parabosonic Hilbert space is one of them.