Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics
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| Format: | Preprint |
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2023
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| _version_ | 1866913570911944704 |
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| author | Balbino, M. M. de Freitas, I. P. Rana, R. G. Toppan, F. |
| author_facet | Balbino, M. M. de Freitas, I. P. Rana, R. G. Toppan, F. |
| contents | 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 $Z_2^n\times Z_2^n\rightarrow Z_2$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_00965 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics Balbino, M. M. de Freitas, I. P. Rana, R. G. Toppan, F. High Energy Physics - Theory Statistical Mechanics Mathematical Physics Quantum Physics 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 $Z_2^n\times Z_2^n\rightarrow Z_2$ 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. |
| title | Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics |
| topic | High Energy Physics - Theory Statistical Mechanics Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2309.00965 |