M.M. Balbino, I.P. de Freitas, R.G. Rana, F. Toppan
{"title":"Inequivalent Z2n-graded brackets, n-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics","authors":"M.M. Balbino, I.P. de Freitas, R.G. Rana, F. Toppan","doi":"10.1016/j.nuclphysb.2024.116729","DOIUrl":null,"url":null,"abstract":"<div><div>Given an associative ring of <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-graded operators, the number of inequivalent brackets of Lie-type which are compatible with the grading and satisfy graded Jacobi identities is <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>+</mo><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span>. This follows from the Rittenberg-Wyler and Scheunert analysis of “color” Lie (super)algebras which is revisited here in terms of Boolean logic gates.</div><div>The inequivalent brackets, recovered from <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mo>×</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mo>→</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> mappings, are defined by consistent sets of commutators/anticommutators describing particles accommodated into an <em>n</em>-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).</div><div>As a first application we construct <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msubsup></math></span>-graded quantum Hamiltonians which respectively admit <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>4</mn></math></span> and <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mn>5</mn></math></span> inequivalent multiparticle quantizations (the inequivalent parastatistics are discriminated by measuring the eigenvalues of certain observables in some given states). The extension to <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-graded quantum Hamiltonians for <span><math><mi>n</mi><mo>></mo><mn>3</mn></math></span> is immediate.</div><div>As a main physical application we prove that the <span><math><mi>N</mi></math></span>-extended, one-dimensional supersymmetric and superconformal quantum mechanics, for <span><math><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>8</mn></math></span>, are respectively described by <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>14</mn></math></span> alternative formulations based on the inequivalent graded Lie (super)algebras. The <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> numbers correspond to all possible “statistical transmutations” of a given set of supercharges which, for <span><math><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>8</mn></math></span>, are accommodated into a <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-grading with <span><math><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span> (the identification is <span><math><mi>N</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>).</div><div>In the simplest <span><math><mi>N</mi><mo>=</mo><mn>2</mn></math></span> setting (the 2-particle sector of the de Alfaro-Fubini-Furlan deformed oscillator with <span><math><mi>s</mi><mi>l</mi><mo>(</mo><mn>2</mn><mo>|</mo><mn>1</mn><mo>)</mo></math></span> spectrum-generating superalgebra), the <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>-graded parastatistics imply a degeneration of the energy levels which cannot be reproduced by ordinary bosons/fermions statistics.</div></div>","PeriodicalId":54712,"journal":{"name":"Nuclear Physics B","volume":null,"pages":null},"PeriodicalIF":2.5000,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nuclear Physics B","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0550321324002955","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, PARTICLES & FIELDS","Score":null,"Total":0}
引用次数: 0
Abstract
Given an associative ring of -graded operators, the number of inequivalent brackets of Lie-type which are compatible with the grading and satisfy graded Jacobi identities is . 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 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 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 and -graded quantum Hamiltonians which respectively admit and inequivalent multiparticle quantizations (the inequivalent parastatistics are discriminated by measuring the eigenvalues of certain observables in some given states). The extension to -graded quantum Hamiltonians for is immediate.
As a main physical application we prove that the -extended, one-dimensional supersymmetric and superconformal quantum mechanics, for , are respectively described by alternative formulations based on the inequivalent graded Lie (super)algebras. The numbers correspond to all possible “statistical transmutations” of a given set of supercharges which, for , are accommodated into a -grading with (the identification is ).
In the simplest setting (the 2-particle sector of the de Alfaro-Fubini-Furlan deformed oscillator with spectrum-generating superalgebra), the -graded parastatistics imply a degeneration of the energy levels which cannot be reproduced by ordinary bosons/fermions statistics.
期刊介绍:
Nuclear Physics B focuses on the domain of high energy physics, quantum field theory, statistical systems, and mathematical physics, and includes four main sections: high energy physics - phenomenology, high energy physics - theory, high energy physics - experiment, and quantum field theory, statistical systems, and mathematical physics. The emphasis is on original research papers (Frontiers Articles or Full Length Articles), but Review Articles are also welcome.