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


This paper introduces the parastatistics induced by Z2×Z2-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 Z2 ×Z2-graded superalgebras and producing parafermions have been considered).
It is shown how to detect Z2×Z2-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 Z2- and Z2 × Z2- gradings produces 9 inequivalent multi-particle
Hilbert spaces of a 4 × 4 matrix oscillator. The Z2 × Z2-graded parabosonic Hilbert space is
one of them.