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
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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":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}
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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。
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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.
期刊最新文献
Schottky anomaly of the Kalb-Ramond-de Sitter spacetime Quotient quiver subtraction Inequivalent Z2n-graded brackets, n-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics Gravitational waves driven by holographic dark energy Statistical and observation comparison of Weyl-type f(Q,T) models with the ΛCDM paradigm
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