${\mathbb Z}_2\times {\mathbb Z}_2$-graded mechanics: the quantization

TL;DR

This paper develops the Hamiltonian formulation and canonical quantization of ${ m Z}_2 imes { m Z}_2$-graded classical mechanics models, introducing new quantum Hamiltonians with rich symmetry structures and demonstrating their properties.

Contribution

It presents the first systematic canonical quantization of ${ m Z}_2 imes { m Z}_2$-graded classical models, expanding the class of such quantum systems with explicit interacting Hamiltonians.

Findings

Recovered the ${ m Z}_2 imes { m Z}_2$-graded quantum Hamiltonian from previous work.

Constructed interacting multiparticle quantum Hamiltonians with matrix differential operators.

Analyzed the symmetry properties of the quantum Hamiltonians.

Abstract

In the previous paper arXiv:2003.06470 we introduced the notion of ${\mathbb Z}_2\times{\mathbb Z}_2$-graded classical mechanics and presented a general framework to construct, in the Lagrangian setting, the worldline sigma models invariant under a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded superalgebra. In this work we discuss at first the classical Hamiltonian formulation of some of these models and later present their canonical quantization. As the simplest application of the construction we recover the ${\mathbb Z}_2\times{\mathbb Z}_2$-graded quantum Hamiltonian introduced by Bruce and Duplij in arXiv:1904.06975. We prove that this is the first example of a large class of ${\mathbb Z}_2\times{\mathbb Z}_2$-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 ${\mathbb Z}_2\times{\mathbb Z}_2$-graded symmetries possessed by the quantum Hamiltonians is given.