The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics
We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quan...
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2025-01-01
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| author | Yan Gu Jiao Wang |
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| description | We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well. |
| format | Article |
| id | doaj-art-96ef4789cfa54ffdbf8279fac46e0ee6 |
| institution | Kabale University |
| issn | 1099-4300 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
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| series | Entropy |
| spelling | doaj-art-96ef4789cfa54ffdbf8279fac46e0ee62025-01-24T13:31:51ZengMDPI AGEntropy1099-43002025-01-012715910.3390/e27010059The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical MechanicsYan Gu0Jiao Wang1Department of Modern Physics, University of Science and Technology of China, Hefei 230026, ChinaDepartment of Physics and Fujian Provincial Key Laboratory of Low Dimensional Condensed Matter Physics, Xiamen University, Xiamen 361005, ChinaWe show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.https://www.mdpi.com/1099-4300/27/1/59quantum probability theoryquantum statistical mechanicsgroup-theoretical characteristic functionquantum-optical master equationcat map |
| spellingShingle | Yan Gu Jiao Wang The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics Entropy quantum probability theory quantum statistical mechanics group-theoretical characteristic function quantum-optical master equation cat map |
| title | The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics |
| title_full | The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics |
| title_fullStr | The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics |
| title_full_unstemmed | The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics |
| title_short | The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics |
| title_sort | group algebraic formalism of quantum probability and its applications in quantum statistical mechanics |
| topic | quantum probability theory quantum statistical mechanics group-theoretical characteristic function quantum-optical master equation cat map |
| url | https://www.mdpi.com/1099-4300/27/1/59 |
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