${\mathbb{Z}}_2^n$-Graded extensions of supersymmetric quantum mechanics via Clifford algebras

Abstract

It is shown that the $\mathcal{N}=1$ supersymmetric quantum mechanics (SQM) can be extended to a ${\mathbb{Z}}_2^n$-graded superalgebra. This is done by presenting quantum mechanical models which realize, with the aid of Clifford gamma matrices, the ${\mathbb{Z}}_2^n$-graded Poincaré algebra in one-dimensional spacetime. Reflecting the fact that the ${\mathbb{Z}}_2^n$-graded Poincaré algebra has a number of central elements, a sequence of models defining the ${\mathbb{Z}}_2^n$-graded version of SQM are provided for a given value of $n.$ In a model of the sequence, the central elements having the same ${\mathbb{Z}}_2^n$-degree are realized as dependent or independent operators. It is observed that as use the Clifford algebra of lager dimension, more central elements are realized as independent operators.

arXiv comment19 pages, no figure

References in corpus (12)

• Supersymmetry and Quantum Mechanics
• Once more on parastatistics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie Symmetries of the Lévy-Leblond Equations
• Double-Graded Supersymmetric Quantum Mechanics
• Quaternionic and Octonionic Spinors. A Classification
• On a ${\mathbb{Z}}_2^n$-Graded Version of Supersymmetry
• $\mathcal{N}$-Extension of duble-graded supersymmetric and superconformal quantum mechanics
• Special Reduced Multiplets and Minimal Representations for SO(p,q)
• The Z_2 x Z_2-graded Lie superalgebra pso(2m+1|2n) and new parastatistics representations
• Splitting theorem for ${\mathbb{Z}}_2^n$-supermanifolds
• The Graded Differential Geometry of Mixed Symmetry Tensors
• Z_n^3-graded colored supersymmetry

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• Inequivalent quantizations from gradings and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ parabosons
• 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
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• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
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• Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics
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• Symplectic ${\mathbb{Z}}_2^n$-manifolds
• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• $\mathcal{N}=2$ Double graded supersymmetric quantum mechanics via dimensional reduction