Contact manifolds and dissipation, classical and quantum. (English) Zbl 1404.81146
Summary: Motivated by a geometric decomposition of the vector field associated with the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation for finite-level open quantum systems, we propose a generalization of the recently introduced contact Hamiltonian systems for the description of dissipative-like dynamical systems in the context of (non-necessarily exact) contact manifolds. In particular, we show how this class of dynamical systems naturally emerges in the context of Lagrangian Mechanics and in the case of nonlinear evolutions on the space of pure states of a finite-level quantum system.
MSC:
| 81S22 | Open systems, reduced dynamics, master equations, decoherence |
| 53D10 | Contact manifolds (general theory) |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37J55 | Contact systems |
| 53Z05 | Applications of differential geometry to physics |
Keywords:
contact manifold; dissipation; nonlinear Schrödinger equation; Lagrangian mechanics; Hamiltonian mechanics; general linear group; GKLS equationReferences:
| [1] | Gorini, V.; Kossakowski, A.; Sudarshan, E. C.G., J. Math. Phys., 17, 5, 821-825, (1976) · Zbl 1446.47009 |
| [2] | Lindblad, G., Comm. Math. Phys., 48, 119-130, (1976) · Zbl 0343.47031 |
| [3] | Cariñena, J. F.; Clemente-Gallardo, J.; Jover-Galtier, J. A.; Marmo, G., J. Phys. A, 50, 36, 365301-365330, (2017) · Zbl 1376.81008 |
| [4] | Ciaglia, F. M.; Di Cosmo, F.; Ibort, A.; Laudato, M.; Marmo, G., Open Syst. Inf. Dyn., 24, 3, 1740003-1740038, (2017) · Zbl 1377.81082 |
| [5] | Ciaglia, F. M.; Di Cosmo, F.; Laudato, M.; Marmo, G., Int. J. Geom. Methods Modern Phys., 14, 8, 1740003-1740039, (2017) · Zbl 1373.81254 |
| [6] | Cantrijn, F., J. Math. Phys., 23, 9, 1589-1595, (1982) · Zbl 0496.70032 |
| [7] | de Ritis, R.; Marmo, G.; Platania, G.; Scudellaro, P., Internat. J. Theoret. Phys., 22, 10, 931-946, (1983) · Zbl 0573.70014 |
| [8] | Bravetti, A.; Cruz, H.; Tapias, D., Ann. Physics, 376, 17-39, (2017) · Zbl 1364.37138 |
| [9] | Rajeev, S. G., Ann. Physics, 322, 7, 1541-1555, (2007) · Zbl 1115.70013 |
| [10] | Chruściński, D.; Ciaglia, F. M.; Ibort, A.; Marmo, G.; Ventriglia, F., Stratified Manifold of Quantum States, actions of the complex special linear group, Ann. Phys., (2018), submitted for publication |
| [11] | Grabowski, J.; Kuś, M.; Marmo, G., J. Phys. A: Math. Gen., 38, 47, 10217-10244, (2005) · Zbl 1085.81017 |
| [12] | Grabowski, J.; Kuś, M.; Marmo, G., Open Syst. Inf. Dyn., 13, 04, 343-362, (2006) · Zbl 1110.81044 |
| [13] | Alipour, S.; Chruściński, D.; Facchi, P.; Marmo, G.; Pascazio, S.; Rezakhani, A. T., J. Phys. A, 50, 6, 065301-65318, (2017) · Zbl 1357.81116 |
| [14] | Chruściński, D.; Facchi, P.; Marmo, G.; Pascazio, S., Open Syst. Inf. Dyn., 19, 1, 1250002-1250010, (2012) · Zbl 1238.81152 |
| [15] | Ashtekar, A.; Schilling, T. A., (Harvey, A., On Einstein’s Path: Essays in Honor of Engelbert Schucking, (1999), Springer-Verlag: Springer-Verlag New York), 23-65 · Zbl 0913.00041 |
| [16] | Cariñena, J. F.; Ibort, A.; Marmo, G.; Morandi, G., Geometry From Dynamics, Classical and Quantum, (2015), Springer: Springer Dordrecht · Zbl 1364.81001 |
| [17] | Cirelli, R.; Mania, A.; Pizzocchero, L., J. Math. Phys., 31, 12, 2891-2903, (1990) · Zbl 0850.70207 |
| [18] | Cirelli, R.; Pizzocchero, L., Nonlinearity, 3, 4, 1057-1080, (1990) · Zbl 0718.58028 |
| [19] | Ercolessi, E.; Marmo, G.; Morandi, G., Riv. Nuovo Cimento, 33, 401-590, (2010) |
| [20] | Kibble, T. W.B., Comm. Math. Phys., 65, 2, 189-201, (1979) · Zbl 0412.58006 |
| [21] | Ciaglia, F. M.; Ibort, A.; Marmo, G., Differential Geometry of Quantum States, Observables and Evolution, (2018), UMI LN Springer, in press · Zbl 1391.81016 |
| [22] | Cruz, H.; Schuch, D.; Castaños, O.; Rosas-Ortiz, O., Ann. Physics, 360, 44-60, (2015) · Zbl 1360.81139 |
| [23] | Cruz, H.; Schuch, D.; Castaños, O.; Rosas-Ortiz, O., Ann. Physics, 373, 609-630, (2016) · Zbl 1380.81171 |
| [24] | Schuch, D., Quantum Theory from a Nonlinear Perspective, (2018), Springer International Publishing · Zbl 1388.81004 |
| [25] | Morandi, G.; Ferrario, C.; Lo Vecchio, G.; Marmo, G.; Rubano, C., Phy. Rep., 188, 3 and 4, 147-284, (1990) · Zbl 1211.58008 |
| [26] | Balachandran, A. P.; Marmo, G.; Skagerstam, B. S.; Stern, A., Nuclear Phys. B, 164, 427-444, (1980) |
| [27] | Grmela, M.; Öttinger, H. C., Phys. Rev. E, 56, 6, 6620-6632, (1997) |
| [28] | Koopman, B. O., Proc. Natl. Acad. Sci., 17, 5, 315-318, (1931) · Zbl 0002.05701 |
| [29] | Asorey, M.; Ciaglia, F. M.; Di Cosmo, F.; Ibort, A.; Marmo, G., Modern Phys. Lett. A, 32, 23, (2017), 1750122-15 · Zbl 1400.81134 |
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.
