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Deformation quantization of a certain type of open systems. (English) Zbl 1193.53181
Author’s abstract: We give an approach to open quantum systems based on formal deformation quantization. It is shown that classical open systems of a certain type can be systematically quantized into quantum open systems preserving the complete positivity of the open time evolution. The usual example of linearly coupled harmonic oscillators is discussed.

MSC:
53D55 Deformation quantization, star products
81S22 Open systems, reduced dynamics, master equations, decoherence
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[1] Basart H., Flato M., Lichnerowicz A., Sternheimer D.: Deformation theory applied to quantization and statistical mechanics. Lett. Math. Phys. 8, 483–494 (1984) · Zbl 0567.58011 · doi:10.1007/BF00400978
[2] Basart H., Lichnerowicz A.: Conformal Symplectic geometry, deformations, rigidity and geometrical (KMS) conditions. Lett. Math. Phys. 10, 167–177 (1985) · Zbl 0589.53037 · doi:10.1007/BF00398154
[3] Bayen F., Flato M., Frønsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. Ann. Phys. 111, 61–151 (1978) · Zbl 0377.53024 · doi:10.1016/0003-4916(78)90224-5
[4] Bordemann, M.: The deformation quantization of certain super-Poisson brackets and BRST cohomology. In: Dito, G., Sternheimer, D. (eds.) Conférence Moshé Flato 1999. Quantization, Deformations, and Symmetries, Mathematical Physics Studies, vol. 22, pp. 45–68. Kluwer, Dordrecht (2000) · Zbl 1004.53067
[5] Bordemann M., Römer H., Waldmann S.: A remark on formal KMS states in deformation quantization. Lett. Math. Phys. 45, 49–61 (1998) · Zbl 0951.53057 · doi:10.1023/A:1007481019610
[6] Breuer, H.P., Petruccione, F.: Concepts and methods in the theory of open quantum systems. In: Benatti, F., Floreanini, R. (eds.) Irreversible Quantum Dynamics. Lecture Notes in Physics, vol. 622, pp. 65–79. Springer, Berlin (2003) (quant-ph/0302047) · Zbl 1041.81518
[7] Brittin W.E.: A note on the quantization of dissipative systems. Phys. Rev. 77(3), 396–397 (1950) · Zbl 0036.14304 · doi:10.1103/PhysRev.77.396
[8] Bursztyn H., Waldmann S.: Algebraic rieffel induction, formal morita equivalence and applications to deformation quantization. J. Geom. Phys. 37, 307–364 (2001) · Zbl 1039.46052 · doi:10.1016/S0393-0440(00)00035-8
[9] Bursztyn H., Waldmann S.: Completely positive inner products and strong Morita equivalence. Pac. J. Math. 222, 201–236 (2005) · Zbl 1111.53071 · doi:10.2140/pjm.2005.222.201
[10] Bursztyn H., Waldmann S.: Hermitian star products are completely positive deformations. Lett. Math. Phys. 72, 143–152 (2005) · Zbl 1081.53078 · doi:10.1007/s11005-005-4844-3
[11] Dekker H.: On the quantization of dissipative systems in the Lagrange–Hamilton formalism. Zeitschrift für Physik B 21, 295–300 (1975) · doi:10.1007/BF01313310
[12] DeWilde M., Lecomte P.B.A.: Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7, 487–496 (1983) · Zbl 0526.58023 · doi:10.1007/BF00402248
[13] Dito G., Léandre R.: Stochastic Moyal product on the Wiener space. J. Math. Phys. 48, 023509 (2007) · Zbl 1121.81089 · doi:10.1063/1.2472184
[14] Dito G., Turrubiates F.J.: The damped harmonic oscillator in deformation quantization. Phys. Lett. A352, 309–316 (2006) · Zbl 1187.81174
[15] Eckel, R.: Quantisierung von Supermannigfaltigkeiten à la Fedosov. PhD thesis, Fakultät für Physik, Albert-Ludwigs-Universität, Freiburg, September 2000
[16] Fedosov B.V.: Quantization and the Index. Sov. Phys. Dokl. 31(11), 877–878 (1986) · Zbl 0635.58019
[17] Gitman D.M., Kupriyanov V.G.: Canonical quantization of so-called non-Lagrangian systems. Eur. Phys. J. C 50, 691–700 (2007) · Zbl 1191.81143 · doi:10.1140/epjc/s10052-007-0230-x
[18] Kaschek D., Neumaier N., Waldmann S.: Complete positivity of Rieffel’s deformation quantization. J. Noncommut. Geom. 3, 361–375 (2009) · Zbl 1172.53055 · doi:10.4171/JNCG/40
[19] Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003) · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf
[20] Kupriyanov V.G., Lyakhovich L.S., Sharapov A.A.: Deformation quantization of linear dissipative systems. J. Phys. A 38, 8039–8051 (2005) · Zbl 1097.81041 · doi:10.1088/0305-4470/38/37/008
[21] Omori H., Maeda Y., Yoshioka A.: Weyl manifolds and deformation quantization. Adv. Math. 85, 224–255 (1991) · Zbl 0734.58011 · doi:10.1016/0001-8708(91)90057-E
[22] Razavy M.: On the quantization of dissipative systems. Zeitschrift für Physik B 26, 201–206 (1977) · doi:10.1007/BF01325274
[23] Rudin W.: Real and Complex Analysis. 3rd edn. McGraw-Hill, New York (1987) · Zbl 0925.00005
[24] Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Operator Theory: Advances and Applications, vol. 37. Birkhäuser, Basel (1990) · Zbl 0697.47048
[25] Tarasov V.E.: Quantization of non-Hamiltonian and dissipative systems. Phys. Lett. A 288, 173–182 (2001) · Zbl 0970.81027 · doi:10.1016/S0375-9601(01)00548-5
[26] Waldmann S.: States and representation theory in deformation quantization. Rev. Math. Phys. 17, 15–75 (2005) · Zbl 1138.53316 · doi:10.1142/S0129055X05002297
[27] Waldmann S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung. Springer, Heidelberg (2007) · Zbl 1139.53001
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