Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics

Abstract

Given an associative ring of $Z_2^n$-graded operators, the number of inequivalent brackets of Lie-type which are compatible with the grading and satisfy graded Jacobi identities is $b_n=n+\lfloor n/2\rfloor +1$. This follows from the Rittenberg-Wyler and Scheunert analysis of "color" Lie (super)algebras which is revisited here in terms of Boolean logic gates. The inequivalent brackets, recovered from $Z_2^n\times Z_2^n\to Z_2$ mappings, are defined by consistent sets of commutators/anticommutators describing particles accommodated into an $n$-bit parastatistics (ordinary bosons/fermions correspond to $1$ bit). Depending on the given graded Lie (super)algebra, its graded sectors can fall into different classes of equivalence expressing different types of (para)bosons and/or (para)fermions. As a first application we construct $Z_2^2$ and $Z_2^3$-graded quantum Hamiltonians which respectively admit $b_2=4$ and $b_3=5$ inequivalent multiparticle quantizations (the inequivalent parastatistics are discriminated by measuring the eigenvalues of certain observables in some given states). As a main physical application we prove that the $N$-extended, $1D$ supersymmetric and superconformal quantum mechanics, for $N=1,2,4,8$, are respectively described by $s_N=2,6,10,14$ alternative formulations based on the inequivalent graded Lie (super)algebras. These numbers correspond to all possible "statistical transmutations" of a given set of supercharges which, for $N=1,2,4,8$, are accommodated into a $Z_2^n$-grading with $n=1,2,3,4$ (the identification is $N=2^{n-1}$). In the simplest $N=2$ setting (the $2$-particle sector of the de DFF deformed oscillator with $sl(2|1)$ spectrum-generating superalgebra), the $Z_2^2$-graded parastatistics imply a degeneration of the energy levels which cannot be reproduced by ordinary bosons/fermions statistics.

57 pages, 16 figures

References in corpus (12)

• Once more on parastatistics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie Symmetries of the Lévy-Leblond Equations
• Quantum simulation of driven para-Bose oscillators
• Is the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded sine-Gordon equation integrable?
• Paraboson quotients. A braided look at Green ansatz and a generalization
• On non-minimal N=4 supermultiplets in 1D and their associated sigma-models
• 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
• Symmetries of the Schrödinger Equation and Algebra/Superalgebra Duality
• The $Z_2\times Z_2$-graded Lie superalgebras $pso(2n+1|2n)$ and $pso(\infty |\infty )$, and parastatistics Fock spaces
• Riemannian structures on ${\mathbb{Z}}_2^n$-manifolds
• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras

Cited by in corpus (6)

• Integrable ${\mathbb{Z}}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories
• On the detectability of paraparticles beyond bosons and fermions
• Affine extensions of ${\mathbb{Z}}_2^2$-graded $osp(1|2)$ and Virasoro algebra
• A Superalgebra Within: representations of lightest standard model particles form a ${\mathbb{Z}}_2^5$-graded algebra
• Braided quantum mechanics and Majorana qubits at third root of unity: a color Heisenberg-Lie (super)algebra framework
• Bosonization in $R$-paraparticle Luttinger models