Applications of Supersymmetric Polynomials in Statistical Quantum Physics
We propose a correspondence between the partition functions of ideal gases consisting of both bosons and fermions and the algebraic bases of supersymmetric polynomials on the Banach space of absolutely summable two-sided sequences <inline-formula><math xmlns="http://www.w3.org/1998/Mat...
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MDPI AG
2023-12-01
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| Series: | Quantum Reports |
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| Online Access: | https://www.mdpi.com/2624-960X/5/4/43 |
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| author | Iryna Chernega Mariia Martsinkiv Taras Vasylyshyn Andriy Zagorodnyuk |
| author_facet | Iryna Chernega Mariia Martsinkiv Taras Vasylyshyn Andriy Zagorodnyuk |
| author_sort | Iryna Chernega |
| collection | DOAJ |
| description | We propose a correspondence between the partition functions of ideal gases consisting of both bosons and fermions and the algebraic bases of supersymmetric polynomials on the Banach space of absolutely summable two-sided sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo mathvariant="script">ℓ</mo><mn>1</mn></msub><mrow><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Such an approach allows us to interpret some of the combinatorial identities for supersymmetric polynomials from a physical point of view. We consider a relation of equivalence for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo mathvariant="script">ℓ</mo><mn>1</mn></msub><mrow><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, induced by the supersymmetric polynomials, and the semi-ring algebraic structures on the quotient set with respect to this relation. The quotient set is a natural model for the set of energy levels of a quantum system. We introduce two different topological semi-ring structures into this set and discuss their possible physical interpretations. |
| first_indexed | 2024-03-08T20:25:37Z |
| format | Article |
| id | doaj.art-fe7021270d5643b199b0152764213251 |
| institution | Directory Open Access Journal |
| issn | 2624-960X |
| language | English |
| last_indexed | 2024-03-08T20:25:37Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Quantum Reports |
| spelling | doaj.art-fe7021270d5643b199b01527642132512023-12-22T14:38:07ZengMDPI AGQuantum Reports2624-960X2023-12-015468369710.3390/quantum5040043Applications of Supersymmetric Polynomials in Statistical Quantum PhysicsIryna Chernega0Mariia Martsinkiv1Taras Vasylyshyn2Andriy Zagorodnyuk3Institute for Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, 3 b, Naukova Str., 79060 Lviv, UkraineFaculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, UkraineFaculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, UkraineFaculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, UkraineWe propose a correspondence between the partition functions of ideal gases consisting of both bosons and fermions and the algebraic bases of supersymmetric polynomials on the Banach space of absolutely summable two-sided sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo mathvariant="script">ℓ</mo><mn>1</mn></msub><mrow><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Such an approach allows us to interpret some of the combinatorial identities for supersymmetric polynomials from a physical point of view. We consider a relation of equivalence for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo mathvariant="script">ℓ</mo><mn>1</mn></msub><mrow><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, induced by the supersymmetric polynomials, and the semi-ring algebraic structures on the quotient set with respect to this relation. The quotient set is a natural model for the set of energy levels of a quantum system. We introduce two different topological semi-ring structures into this set and discuss their possible physical interpretations.https://www.mdpi.com/2624-960X/5/4/43quantum ideal gasgrand partition functionsupersymmetric polynomials on Banach spacesalgebraic basistopological semi-ringtropical semi-ring |
| spellingShingle | Iryna Chernega Mariia Martsinkiv Taras Vasylyshyn Andriy Zagorodnyuk Applications of Supersymmetric Polynomials in Statistical Quantum Physics Quantum Reports quantum ideal gas grand partition function supersymmetric polynomials on Banach spaces algebraic basis topological semi-ring tropical semi-ring |
| title | Applications of Supersymmetric Polynomials in Statistical Quantum Physics |
| title_full | Applications of Supersymmetric Polynomials in Statistical Quantum Physics |
| title_fullStr | Applications of Supersymmetric Polynomials in Statistical Quantum Physics |
| title_full_unstemmed | Applications of Supersymmetric Polynomials in Statistical Quantum Physics |
| title_short | Applications of Supersymmetric Polynomials in Statistical Quantum Physics |
| title_sort | applications of supersymmetric polynomials in statistical quantum physics |
| topic | quantum ideal gas grand partition function supersymmetric polynomials on Banach spaces algebraic basis topological semi-ring tropical semi-ring |
| url | https://www.mdpi.com/2624-960X/5/4/43 |
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