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

Abstract

This paper introduces the parastatistics induced by ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded 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 ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded superalgebras and producing parafermions have been considered). It is shown how to detect ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded 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. 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 ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded parabosonic Hilbert space is one of them.

21 pages

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