--- title: "Inequivalent Z2n-graded brackets, n-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics • CBPF Index" description: "O CBPF INDEX é o sistema de acompanhamento da produção científica e tecnológica do CBPF. Nele é possível obter a lista de publicações de artigos, livros, relatórios técnicos, etc de toda a comunidade do CBPF." canonical_url: "https://index.cbpf.br/publicacoes/6ad2a767639f62b11adbd690" locale: pt meta: "og:description": "O CBPF INDEX é o sistema de acompanhamento da produção científica e tecnológica do CBPF. Nele é possível obter a lista de publicações de artigos, livros, relatórios técnicos, etc de toda a comunidade do CBPF." "og:image": "https://index.cbpf.br/_og/d/c_Item%2Ctitulo_~SW5lcXVpdmFsZW50IFoybi1ncmFkZWQgYnJhY2tldHMsIG4tYml0IHBhcmFzdGF0aXN0aWNzIGFuZCBzdGF0aXN0aWNhbCB0cmFuc211dGF0aW9ucyBvZiBzdXBlcnN5bW1ldHJpYyBxdWFudHVtIG1lY2hhbmljcw%2Ctipo_~Tm90YSBkZSBGw61zaWNh%2Ccor_~I2E4MmMwMA%2Cautores_~TS5NLiBCYWxiaW5vOyBJLlAuIGRlIEZyZWl0YXM7IFIuRy4gUmFuYTsgRi4gVG9wcGFu%2Cdetalhe_~MTkvMDkvMjAyMw%2Csecao_~UHVibGljYcOnw7Vlcw%2Cp_Ii9wdWJsaWNhY29lcy82YWQyYTc2NzYzOWY2MmIxMWFkYmQ2OTAi%2Cs_Z8VsK4Ogy9vzZ32w.png" "og:image:height": "600" "og:image:type": image/png "og:image:width": "1200" "og:title": "Inequivalent Z2n-graded brackets, n-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics • CBPF Index" "twitter:image": "https://index.cbpf.br/_og/d/c_Item%2Ctitulo_~SW5lcXVpdmFsZW50IFoybi1ncmFkZWQgYnJhY2tldHMsIG4tYml0IHBhcmFzdGF0aXN0aWNzIGFuZCBzdGF0aXN0aWNhbCB0cmFuc211dGF0aW9ucyBvZiBzdXBlcnN5bW1ldHJpYyBxdWFudHVtIG1lY2hhbmljcw%2Ctipo_~Tm90YSBkZSBGw61zaWNh%2Ccor_~I2E4MmMwMA%2Cautores_~TS5NLiBCYWxiaW5vOyBJLlAuIGRlIEZyZWl0YXM7IFIuRy4gUmFuYTsgRi4gVG9wcGFu%2Cdetalhe_~MTkvMDkvMjAyMw%2Csecao_~UHVibGljYcOnw7Vlcw%2Cp_Ii9wdWJsaWNhY29lcy82YWQyYTc2NzYzOWY2MmIxMWFkYmQ2OTAi%2Cs_Z8VsK4Ogy9vzZ32w.png" --- > Index of every page on this site: [llms.txt](https://index.cbpf.br/llms.txt) **Nota de Física****Publicado** # **Inequivalent **Z2n{\\mathbb Z}\_2^nZ2n​**-graded brackets, **nnn**-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics** **M.M. Balbino; I.P. de Freitas; R.G. Rana; F. Toppan** 19/09/2023 [**Baixar PDF **1,8 MB](https://index.cbpf.br/files/publicacao/6ad2a767639f62b11adbd690/pdf) [**Contatar responsável**](mailto:apub@cbpf.br?subject=Inequivalent%20Z2n-graded%20brackets%2C%20n-bit%20parastatistics%20and%20statistical%20transmutations%20of%20supersymmetric%20quantum%20mechanics) ## **Dados da publicação**
Autores
M.M. Balbino; I.P. de Freitas; R.G. Rana; F. Toppan
Tipo
Nota de Física
Status
Publicado
Data de publicação
19/09/2023
Descrição
Given an associative ring of ${\mathbb Z}_2^n$-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 ${\mathbb Z}_2^n\times {\mathbb Z}_2^n\rightarrow {\mathbb Z}_2$ mappings, are defined by consistent sets of commutators\slash 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 particles (bosons, parabosons, fermions, parafermions). As a consequence, the assignment of certain ``marked" operators to a given graded sector is a further mechanism to induce inequivalent graded Lie (super)algebras (the basic examples of quaternions, split-quaternions and biquaternions illustrate these features). \\ As a first application we construct ${\mathbb Z}_2^2$ and ${\mathbb Z}_2^3$-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). The extension to ${\mathbb Z}_2^n$-graded quantum Hamiltonians for $n>3$ is immediate.\\ As a main physical application we prove that the ${\cal N}$-extended, one-dimensional supersymmetric and superconformal quantum mechanics, for ${\cal N}=1,2,4,8$, are respectively described by $s_{\cal N}=2,6,10,14 $ alternative formulations based on the inequivalent graded Lie (super)algebras. The $s_{\cal N}$ numbers correspond to all possible ``statistical transmutations" of a given set of supercharges which, for ${\cal N}=1,2,4,8$, are accommodated into a ${\mathbb Z}_2^n$-grading with $n=1,2,3,4$ (the identification is ${\cal N}= 2^{n-1}$).\\ In the simplest ${\cal N}=2$ setting (the $2$-particle sector of the de Alfaro-Fubini-Furlan deformed oscillator with $sl(2|1)$ spectrum-generating superalgebra), the ${\mathbb Z}_2^2$-graded parastatistics imply a degeneration of the energy levels which cannot be reproduced by ordinary bosons/fermions statistics.
Ano
2023
Número
CBPF-NF-002/23
Vínculo com tese
Não
Vínculo com bolsa
Não
```json {"@context":"https://schema.org","@graph":[{"@id":"https://index.cbpf.br/#website","@type":"WebSite","description":"O CBPF INDEX é o sistema de acompanhamento da produção científica e tecnológica do CBPF. Nele é possível obter a lista de publicações de artigos, livros, relatórios técnicos, etc de toda a comunidade do CBPF.","inLanguage":"pt-BR","name":"CBPF Index","url":"https://index.cbpf.br/","publisher":{"@id":"https://index.cbpf.br/#identity"},"workTranslation":[{"@id":"https://index.cbpf.br/en#website"},{"@id":"https://index.cbpf.br/es#website"}]},{"@id":"https://index.cbpf.br/publicacoes/6ad2a767639f62b11adbd690#webpage","@type":"WebPage","description":"O CBPF INDEX é o sistema de acompanhamento da produção científica e tecnológica do CBPF. Nele é possível obter a lista de publicações de artigos, livros, relatórios técnicos, etc de toda a comunidade do CBPF.","name":"Inequivalent Z2n-graded brackets, n-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics","url":"https://index.cbpf.br/publicacoes/6ad2a767639f62b11adbd690","isPartOf":{"@id":"https://index.cbpf.br/#website"},"potentialAction":[{"@type":"ReadAction","target":["https://index.cbpf.br/publicacoes/6ad2a767639f62b11adbd690"]}],"primaryImageOfPage":{"@id":"https://index.cbpf.br/#logo"}},{"@id":"https://index.cbpf.br/#identity","@type":"Organization","alternateName":"CBPF","description":"Unidade de pesquisa do Ministério da Ciência, Tecnologia e Inovação dedicada à pesquisa básica em Física, suas aplicações e à formação científica.","email":"web@cbpf.br","legalName":"Centro Brasileiro de Pesquisas Físicas/CBPF","name":"Centro Brasileiro de Pesquisas Físicas","taxID":"04.044.443/0001-35","telephone":"+55-21-2141-7100","url":"https://cbpf.br","address":{"@type":"PostalAddress","streetAddress":"Rua Dr. Xavier Sigaud, 150","addressLocality":"Rio de Janeiro","addressRegion":"RJ","postalCode":"22290-180","addressCountry":"BR"},"logo":{"@id":"https://index.cbpf.br/#logo"},"sameAs":["https://twitter.com/cbpf_mcti"]},{"@id":"https://index.cbpf.br/#logo","@type":"ImageObject","caption":"Centro Brasileiro de Pesquisas Físicas","contentUrl":"https://index.cbpf.br/icon-512x512-light.png","inLanguage":"pt-BR","url":"https://index.cbpf.br/icon-512x512-light.png"}]} ``` ## Sitemap See the full [sitemap](https://index.cbpf.br/sitemap.md) for all pages.