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

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.

30 pages. Final version to appear in Nucl. Phys. B

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• Inequivalent quantizations from gradings and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ parabosons
• Is the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded sine-Gordon equation integrable?
• Classification of minimal ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie (super)algebras and some applications
• Minimal bosonization of double-graded supersymmetric quantum mechanics
• Comments of ${\mathbb{Z}}_2^2$-supersymmetry in superfield formalism
• Irreducible representations of ${\mathbb{Z}}_2^2$-graded $\mathcal{N}=2$ supersymmetry algebra and ${\mathbb{Z}}_2^2$-graded supermechanics
• New aspects of the Z${}_{\text{2}}$ $\times$ Z${}_{\text{2}}$-graded 1D superspace: induced strings and 2D relativistic models
• The $Z_2\times Z_2$-graded Lie superalgebras $pso(2n+1|2n)$ and $pso(\infty |\infty )$, and parastatistics Fock spaces
• On classical $Z_2\times Z_2$-graded Lie algebras
• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
• Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics
• ${\mathbb{Z}}_2^3$-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics
• Orthosymplectic $Z_2\times Z_2$-graded Lie superalgebras and parastatistics
• Integrable ${\mathbb{Z}}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories
• Graded colour Lie superalgebras for solving Lévy-Leblond equations
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• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• Symplectic ${\mathbb{Z}}_2^n$-manifolds
• $\mathcal{N}=2$ Double graded supersymmetric quantum mechanics via dimensional reduction
• On the detectability of paraparticles beyond bosons and fermions
• Transmuted spectrum-generating algebras and detectable parastatistics of the Superconformal Quantum Mechanics
• Braided quantum mechanics and Majorana qubits at third root of unity: a color Heisenberg-Lie (super)algebra framework