{"title":"从形变量子化的视角看非交换平面上的超对称量子力学","authors":"Md. Rafsanjany Jim, S. Hasibul Hassan Chowdhury","doi":"10.1016/j.aop.2024.169718","DOIUrl":null,"url":null,"abstract":"<div><p>A gauge invariant mathematical formalism based on deformation quantization is outlined to model an <span><math><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow></math></span> supersymmetric system of a spin <span><math><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span> charged particle placed in a nocommutative plane under the influence of a vertical uniform magnetic field. The noncommutative involutive algebra <span><math><mrow><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>[</mo><mrow><mo>[</mo><mi>ϑ</mi><mo>]</mo></mrow><mo>]</mo></mrow><mo>,</mo><msup><mrow><mo>∗</mo></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow></math></span> of formal power series in <span><math><mi>ϑ</mi></math></span> with coefficients in the commutative ring <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> was employed to construct the relevant observables, viz., SUSY Hamiltonian <span><math><mi>H</mi></math></span>, supercharge operator <span><math><mi>Q</mi></math></span> and its adjoint <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mi>†</mi></mrow></msup></math></span> all belonging to the 2 × 2 matrix algebra <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>[</mo><mrow><mo>[</mo><mi>ϑ</mi><mo>]</mo></mrow><mo>]</mo></mrow><mo>,</mo><msup><mrow><mo>∗</mo></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> with the help of a family of gauge-equivalent star products <span><math><msup><mrow><mo>∗</mo></mrow><mrow><mi>r</mi></mrow></msup></math></span>. The energy eigenvalues of the SUSY Hamiltonian all turned out to be independent of not only the gauge parameter <span><math><mi>r</mi></math></span> but also the noncommutativity parameter <span><math><mi>ϑ</mi></math></span>. The nontrivial Fermionic ground state was subsequently computed associated with the zero energy which indicates that supersymmetry remains unbroken in all orders of <span><math><mi>ϑ</mi></math></span>. The Witten index for the noncommutative SUSY Landau problem turns out to be <span><math><mrow><mo>−</mo><mn>1</mn></mrow></math></span> corroborating the fact that there is no broken supersymmetry for the model we are considering.</p></div>","PeriodicalId":8249,"journal":{"name":"Annals of Physics","volume":"467 ","pages":"Article 169718"},"PeriodicalIF":3.0000,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Supersymmetric Quantum Mechanics on a noncommutative plane through the lens of deformation quantization\",\"authors\":\"Md. Rafsanjany Jim, S. Hasibul Hassan Chowdhury\",\"doi\":\"10.1016/j.aop.2024.169718\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A gauge invariant mathematical formalism based on deformation quantization is outlined to model an <span><math><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow></math></span> supersymmetric system of a spin <span><math><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span> charged particle placed in a nocommutative plane under the influence of a vertical uniform magnetic field. The noncommutative involutive algebra <span><math><mrow><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>[</mo><mrow><mo>[</mo><mi>ϑ</mi><mo>]</mo></mrow><mo>]</mo></mrow><mo>,</mo><msup><mrow><mo>∗</mo></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow></math></span> of formal power series in <span><math><mi>ϑ</mi></math></span> with coefficients in the commutative ring <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> was employed to construct the relevant observables, viz., SUSY Hamiltonian <span><math><mi>H</mi></math></span>, supercharge operator <span><math><mi>Q</mi></math></span> and its adjoint <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mi>†</mi></mrow></msup></math></span> all belonging to the 2 × 2 matrix algebra <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>[</mo><mrow><mo>[</mo><mi>ϑ</mi><mo>]</mo></mrow><mo>]</mo></mrow><mo>,</mo><msup><mrow><mo>∗</mo></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> with the help of a family of gauge-equivalent star products <span><math><msup><mrow><mo>∗</mo></mrow><mrow><mi>r</mi></mrow></msup></math></span>. The energy eigenvalues of the SUSY Hamiltonian all turned out to be independent of not only the gauge parameter <span><math><mi>r</mi></math></span> but also the noncommutativity parameter <span><math><mi>ϑ</mi></math></span>. The nontrivial Fermionic ground state was subsequently computed associated with the zero energy which indicates that supersymmetry remains unbroken in all orders of <span><math><mi>ϑ</mi></math></span>. The Witten index for the noncommutative SUSY Landau problem turns out to be <span><math><mrow><mo>−</mo><mn>1</mn></mrow></math></span> corroborating the fact that there is no broken supersymmetry for the model we are considering.</p></div>\",\"PeriodicalId\":8249,\"journal\":{\"name\":\"Annals of Physics\",\"volume\":\"467 \",\"pages\":\"Article 169718\"},\"PeriodicalIF\":3.0000,\"publicationDate\":\"2024-06-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S000349162400126X\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Physics","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S000349162400126X","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Supersymmetric Quantum Mechanics on a noncommutative plane through the lens of deformation quantization
A gauge invariant mathematical formalism based on deformation quantization is outlined to model an supersymmetric system of a spin charged particle placed in a nocommutative plane under the influence of a vertical uniform magnetic field. The noncommutative involutive algebra of formal power series in with coefficients in the commutative ring was employed to construct the relevant observables, viz., SUSY Hamiltonian , supercharge operator and its adjoint all belonging to the 2 × 2 matrix algebra with the help of a family of gauge-equivalent star products . The energy eigenvalues of the SUSY Hamiltonian all turned out to be independent of not only the gauge parameter but also the noncommutativity parameter . The nontrivial Fermionic ground state was subsequently computed associated with the zero energy which indicates that supersymmetry remains unbroken in all orders of . The Witten index for the noncommutative SUSY Landau problem turns out to be corroborating the fact that there is no broken supersymmetry for the model we are considering.
期刊介绍:
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