Z2×Z2-graded mechanics: The quantization

Abstract

In a previous paper we introduced the notion of Z2×Z2-graded classical mechanics and presented a general framework to construct, in the Lagrangian setting, the worldline sigma models invariant under a Z2×Z2-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 Z2×Z2-graded quantum Hamiltonian introduced by Bruce and Duplij. We prove that this is just the first example of a large class of Z2×Z2-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 Z2×Z2-graded symmetries possessed by the quantum Hamiltonians is given.