{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:40:35Z","timestamp":1759920035672,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T00:00:00Z","timestamp":1736467200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2023YFA1407100","12247106","12247101"],"award-info":[{"award-number":["2023YFA1407100","12247106","12247101"]}]},{"name":"National Natural Science Foundation of China","award":["2023YFA1407100","12247106","12247101"],"award-info":[{"award-number":["2023YFA1407100","12247106","12247101"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/e27010059","type":"journal-article","created":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T09:24:42Z","timestamp":1736501082000},"page":"59","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics"],"prefix":"10.3390","volume":"27","author":[{"given":"Yan","family":"Gu","sequence":"first","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9011-6300","authenticated-orcid":false,"given":"Jiao","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Physics and Fujian Provincial Key Laboratory of Low Dimensional Condensed Matter Physics, Xiamen University, Xiamen 361005, China"},{"name":"Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,10]]},"reference":[{"key":"ref_1","unstructured":"Laplace, P.S. (1812). Th\u00e9orie Analytique des Probabilit\u00e9s, Mme ve Courcier."},{"key":"ref_2","unstructured":"Gibbs, J.W. (1902). Elementary Principles in Statistical Mechanics, Yale University Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/BF01203155","article-title":"Grundlagen der Wahrscheinlichkeitrechnung","volume":"5","year":"1919","journal-title":"Math. Z."},{"key":"ref_4","unstructured":"Kolmogorov, A.N. (1956). Foundations of the Probability Theory, Chelsea."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Halmos, P.R. (1950). Measure Theory, Springer.","DOI":"10.1007\/978-1-4684-9440-2"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1007\/BF01397477","article-title":"Zur Quantenmechanik der Sto\u00dfvorg\u00e4nge","volume":"37","author":"Born","year":"1926","journal-title":"Z. Physik"},{"key":"ref_7","unstructured":"von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics, Princeton University Press."},{"key":"ref_8","unstructured":"Barut, A.O., and Raczka, R. (1980). Theory of Group Representations and Applications, Polish Scientific Publishers."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Accardi, L. (1984). Some Trends and Problems in Quantum Probability. Lecture Notes in Mathematics, Springer.","DOI":"10.1007\/BFb0071706"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1007\/s00032-010-0134-3","article-title":"Quantum Probability: New Perspectives for the Laws of Chance","volume":"78","author":"Accardi","year":"2010","journal-title":"Milan J. Math."},{"key":"ref_11","unstructured":"Schlosshauer, M. (2007). Decoherence and the Quantum-to-Classical Transition, Springer."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"930","DOI":"10.2307\/1969387","article-title":"Postulates for general quantum mechanics","volume":"48","author":"Segal","year":"1947","journal-title":"Ann. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bratteli, O., and Robibson, D.W. (1979). Operator Algebras and Quantum Statistical Mechanics, Springer.","DOI":"10.1007\/978-3-662-02313-6"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Haag, R. (1992). Local Quantum Physics, Springer.","DOI":"10.1007\/978-3-642-97306-2"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Meyer, P.A. (1995). Quantum Probability for Probabilists, Springer.","DOI":"10.1007\/BFb0084701"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1310","DOI":"10.1103\/PhysRevA.32.1310","article-title":"Group-theoretical formalism of quantum mechanics based on quantum generalization of characteristic functions","volume":"32","author":"Gu","year":"1985","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Moretti, V. (2019). Fundamental Mathematical Structures of Quantum Theory, Springer.","DOI":"10.1007\/978-3-030-18346-2"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Berg, C., Christensen, J.F.R., and Ressel, P. (1984). Harmonic Analysis on Semigroups, Springer.","DOI":"10.1007\/978-1-4612-1128-0"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Parthasarathy, K.R. (1992). An Introduction to Quantum Stochastic Calculus, Springer.","DOI":"10.1007\/978-3-0348-0566-7"},{"key":"ref_20","unstructured":"Pedersen, G.K. (1979). C\u2217-Algebras and Their Automorphism Groups, Academic Press."},{"key":"ref_21","first-page":"200","article-title":"Group-theoretical formalism of quantum mechanics and classical-quantum correspondence","volume":"35","author":"Gu","year":"1992","journal-title":"Sci. China A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"022318","DOI":"10.1103\/PhysRevA.74.022318","article-title":"Group-theoretical approach to entanglement","volume":"74","author":"Korbiez","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0167-2789(83)90134-3","article-title":"Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids","volume":"7","author":"Marsden","year":"1983","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kirillov, A.A. (1976). Elements of The Theory of Representations, Springer.","DOI":"10.1007\/978-3-642-66243-0"},{"key":"ref_25","unstructured":"Dufour, J.P., and Nguyen, T.Z. (2005). Poisson Structures and Their Normal Forms, Birkhauser Verlag."},{"key":"ref_26","first-page":"189","article-title":"The Heisenberg-Weyl Ring in Quantum Mechanics","volume":"Volume III","author":"Loebl","year":"1975","journal-title":"Group Theory and Its Applications"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0003-4916(89)90259-5","article-title":"The quantized Baker\u2019s transformation","volume":"190","author":"Balaz","year":"1989","journal-title":"Ann. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"115205","DOI":"10.1088\/1402-4896\/ab2cc1","article-title":"Steady states of quantum Brownian motion and decomposition of quantum states into ensembles of Gaussian packets having a uniform position variance","volume":"94","author":"Gu","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Breuer, H.P., and Petruccione, F. (2002). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1007\/3-540-44874-8_4"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Weiss, D. (2000). Quantum Dissipative Systems, World Scientific.","DOI":"10.1142\/9789812817877"},{"key":"ref_31","unstructured":"Louisell, W.H. (1973). Quantum Statistical Properties of Radiation, John Wiley & Sons."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Haake, F. (1973). Statistical Treatment of Open Systems by Generalized Master Equations, Springer. Springer Tracts in Modern Physics.","DOI":"10.1007\/978-3-662-40468-3_2"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Walls, D.F., and Milburn, G.J. (2008). Quantum Optics, Springer.","DOI":"10.1007\/978-3-540-28574-8"},{"key":"ref_34","unstructured":"Miguel, O. (2007). Quantum Optics, Springer."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Mehra, J. (1973). Time, Irreversibility and Structure. The Physicist\u2019s Conception of Nature, Reidel.","DOI":"10.1007\/978-94-010-2602-4"},{"key":"ref_36","unstructured":"Gu, Y., and Wang, J. (2024). Relaxation of A Thermally Bathed Harmonic Oscillator: A Study Based on the Group-theoretical Formalism. arXiv."},{"key":"ref_37","unstructured":"Gradshteyn, L.S., and Ryzhik, L.M. (1980). Table of Integrals, Series and Products, Academic Press."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1016\/0167-2789(91)90012-X","article-title":"The Arnol\u2019d cat: Failure of the correspondence principle","volume":"50","author":"Ford","year":"1991","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_39","unstructured":"Arnold, V.I., and Avez, A. (1968). Ergodic Problems of Classical Mechanics, Benjamin."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/0375-9601(90)90532-S","article-title":"Evidences of classical and quantum chaos in the time evolution of nonequilibrium ensembles","volume":"149","author":"Gu","year":"1990","journal-title":"Phys. Lett. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/S0375-9601(97)00194-1","article-title":"Time evolution of coarse-grained entropy in classical and quantum motions of strongly chaotic systems","volume":"229","author":"Gu","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"056208","DOI":"10.1103\/PhysRevE.63.056208","article-title":"Ergodicity and scars of the quantum cat map in the semiclassical regime","volume":"63","author":"Wang","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_43","unstructured":"Naimark, M.A., and Stern, A.I. (1980). Theory of Group Representations, Springer."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/1\/59\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:26:37Z","timestamp":1759919197000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/1\/59"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,10]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["e27010059"],"URL":"https:\/\/doi.org\/10.3390\/e27010059","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,1,10]]}}}