Differential geometry of density states. (English) Zbl 1138.81336
Summary: We consider a geometrization, i.e., we identify geometrical structures, for the space of density states of a quantum system. We also provide few comments on a possible application of this geometrization for composite systems.
MSC:
| 81P15 | Quantum measurement theory, state operations, state preparations |
| 58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
References:
| [1] | Dirac, P. A.M., The Principles of Quantum Mechanics (1958), Pergamon: Pergamon New York · Zbl 0080.22005 |
| [2] | Carinena, J. F.; Marmo, G.; Nasarre, J., The nonlinear superposition principle and the Weyl-Norman method, Int. J. Mod. Phys., A13, 3601-4362 (1998) · Zbl 0928.34025 |
| [3] | Schrödinger, E., Zum Heisenbergschen Unschärfeprinzip, Ber. Kgl. Akad. Wiss., 296, 296-303 (1930) · JFM 56.0754.05 |
| [4] | Inner composite law of pure spin states, (Hilborn, R. C.; Tino, G. M., Spin-Statistics Connection and Commutation Relations, Vol. 545 (2000), American Institute of Physics: American Institute of Physics Napoli), 92-97 · Zbl 1115.81311 |
| [5] | Cirelli, R.; Gatti, M.; Maniá, A., On the nonlinear extension of quantum superposition and uncertainty principles, J. Geometry and Physics, 29, 54-86 (1999) · Zbl 0939.58005 |
| [6] | Jordan, P.; von Neumann, J.; Wigner, E., On an algebraic generalization of the quantum mechanical formalism, Ann. Math., 35, 29-54 (1934) · JFM 60.0902.02 |
| [7] | Ashtekar, A.; Shilling, T. A., Geometrical formulation of quantum mechanics, gr-qc/9706059, (Harvey, A., On Einstein’s Path (1998), Springer: Springer Melville, N.Y.) |
| [8] | Chruściński, D.; Jamiołkowski, A., Geometric Phases in Classical and Quantum Mechanics (2004), Birkhäuser: Birkhäuser Berlin · Zbl 1075.81002 |
| [9] | Cirelli, R.; Maniá, A.; Pizzocchero, L., Quantum mechanics as an infinite-dimensional Hamiltonian system with uncertainty structure. II, J. Math. Phys., 31, 2898-2903 (1990) · Zbl 0850.70207 |
| [10] | Esposito, G.; Marmo, G.; Sudarshan, G., From Classical to Quantum Mechanics (2004), Cambridge University Press: Cambridge University Press Boston · Zbl 1086.81001 |
| [11] | Man’ko, V. I.; Marmo, G.; Sudarshan, E. C.G.; Zaccaria, F., On the relation between Schrödinger and von Neumann equation, J. Russ. Laser Res., 20, 5, 421-437 (1999) |
| [12] | Grabowski, J.; Landi, G.; Marmo, G.; Vilasi, G., Generalized reduction procedure, Fortschr. Phys., 42, 5, 393-427 (1994) |
| [13] | Fubini, G., Sulle metriche definite da una forma Hermitiana, Atti Istituto Veneto, 6, 501 (1903) |
| [14] | Aldaya, V.; Guerrero, J.; Marmo, G., Quantization on a Lie group: Higher-order polarizations, (Gruber, B.; Ramek, M., Symmetries in Science X (1998), Plenum Press: Plenum Press Cambridge), 1-35 |
| [15] | Ercolessi, E.; Marmo, G.; Morandi, G.; Mukunda, N., Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach, Int. J. Mod. Phys., A15, 5007-5032 (2001) · Zbl 1026.81019 |
| [16] | Alekseevsky, D.; Grabowski, J.; Marmo, G.; Michor, P. W., Poisson structures on double Lie groups, J. Geom. Phys., 26, 340-379 (1998) · Zbl 1020.17016 |
| [17] | Benvegnú, A.; Sansonetto, N.; Spera, M., Remarks on geometrical quantum mechanics, J. Geom. Phys., 51, 229-243 (2004) · Zbl 1078.81029 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
