Z2 x Z2-graded mechanics: The quantization

In a previous paper we introduced the notion of Z(2) x Z(2)-graded classical mechanics and presented a general framework to construct, in the Lagrangian setting, the worldline sigma models invariant under a Z(2) x Z(2)-graded superalgebra. In this work we discuss at first the classical Hamiltonian formulation for a class of these models and later present their canonical quantization. As the simplest application of the construction we recover the Z(2) x Z(2)-graded quantum Hamiltonian introduced by Bruce and Duplij. We prove that this is just the first example of a large class of Z(2) x Z(2)-graded quantum models. We derive in particular interacting multiparticle quantum Hamiltonians given by Hermitian, matrix, differential operators. The interacting terms appear as non-diagonal entries in the matrices. The construction of the Noether charges, both classical and quantum, is presented. A comprehensive discussion of the different Z(2) x Z(2)-graded symmetries possessed by the quantum Hamiltonians is given. (C) 2021 The Author(s). Published by Elsevier B.V.