Inequivalent Z2n-graded brackets, n-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics

IF 2.5 3区 物理与天体物理 Q2 PHYSICS, PARTICLES & FIELDS Nuclear Physics B Pub Date : 2024-10-31 DOI:10.1016/j.nuclphysb.2024.116729
M.M. Balbino, I.P. de Freitas, R.G. Rana, F. Toppan
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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>&gt;</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":"1009 ","pages":"Article 116729"},"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}
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Abstract

Given an associative ring of Z2n-graded operators, the number of inequivalent brackets of Lie-type which are compatible with the grading and satisfy graded Jacobi identities is bn=n+n/2+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 Z2n×Z2nZ2 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 Z22 and Z23-graded quantum Hamiltonians which respectively admit b2=4 and b3=5 inequivalent multiparticle quantizations (the inequivalent parastatistics are discriminated by measuring the eigenvalues of certain observables in some given states). The extension to Z2n-graded quantum Hamiltonians for n>3 is immediate.
As a main physical application we prove that the N-extended, one-dimensional supersymmetric and superconformal quantum mechanics, for N=1,2,4,8, are respectively described by sN=2,6,10,14 alternative formulations based on the inequivalent graded Lie (super)algebras. The sN numbers correspond to all possible “statistical transmutations” of a given set of supercharges which, for N=1,2,4,8, are accommodated into a Z2n-grading with n=1,2,3,4 (the identification is N=2n1).
In the simplest N=2 setting (the 2-particle sector of the de Alfaro-Fubini-Furlan deformed oscillator with sl(2|1) spectrum-generating superalgebra), the Z22-graded parastatistics imply a degeneration of the energy levels which cannot be reproduced by ordinary bosons/fermions statistics.
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超对称量子力学的不等价 Z2n 级括号、n 位准统计量和统计嬗变
给定一个 Z2n 分级算子的关联环,与分级相容且满足分级雅可比等式的不等价括号的数量为 bn=n+⌊n/2⌋+1。从 Z2n×Z2n→Z2 映射中恢复的不等价括号是由描述容纳到 n 位准统计(普通玻色子/费米子对应 1 位)中的粒子的换向器/反换向器的一致集定义的。根据给定的分级李(超)代数,其分级扇区可以归入不同的等价类,表达不同类型的粒子(玻色子、旁玻色子、费米子、旁费米子)。因此,将某些 "标记 "算子分配给给定的分级扇区是诱导不等价分级李(超)代数的另一种机制(四元数、分裂四元数和双四元数的基本例子说明了这些特征)。作为第一个应用,我们构建了 Z22 和 Z23 梯度量子哈密顿,它们分别允许 b2=4 和 b3=5 不等价的多粒子量子化(通过测量给定状态下某些观测值的特征值来判别不等价的准量子化)。作为一个主要的物理应用,我们证明了 N=1,2,4,8 的 N 扩展一维超对称和超共形量子力学分别由 sN=2,6,10,14 基于不等价分级列(超)代数的替代公式描述。sN 数字对应于一组给定超电荷的所有可能的 "统计变换",对于 N=1,2,4,8,这些超电荷被容纳到一个 n=1,2,3,4(标识为 N=2n-1)的 Z2n 等级中。在最简单的 N=2 设置(具有 sl(2|1) 谱生成超代数的 de Alfaro-Fubini-Furlan 变形振荡器的 2 粒子扇区)中,Z22-分级副统计量意味着能级的退化,而普通的玻色子/费米子统计量无法再现这种退化。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Nuclear Physics B
Nuclear Physics B 物理-物理:粒子与场物理
CiteScore
5.50
自引率
7.10%
发文量
302
审稿时长
1 months
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.
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