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Torus as phase space: Weyl quantization, dequantization, and Wigner formalism. (English) Zbl 1351.81066
Author’s abstract: The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.

81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
81S10 Geometry and quantization, symplectic methods
53D55 Deformation quantization, star products
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[1] Arnold, V. I.; Avez, A., Ergodic Problems in Classical Mechanics, (1968), Benjamin: Benjamin, New York · Zbl 0167.22901
[2] Hannay, J. H.; Berry, M. V., Quantization of linear maps on a torus - Fresnel diffraction by a periodic grating, Phys. D, 1, 267-290, (1980) · Zbl 1194.81107
[3] Keating, J. P., Asymptotic properties of the periodic orbits of the cat maps, Nonlinearity, 4, 277-307, (1991) · Zbl 0726.58036
[4] Keating, J. P., The cat maps: Quantum mechanics and classical motion, Nonlinearity, 4, 309-341, (1991) · Zbl 0726.58037
[5] Degli Esposti, M., Quantization of the orientation preserving automorphism of the torus, Ann. Inst. Henri Poincare, Sect. A, 3, 323-341, (1993) · Zbl 0777.58017
[6] Bouzouina, A.; De Biévre, S., Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun. Math. Phys., 178, 83-105, (1996) · Zbl 0876.58041
[7] Bianchi, A.; Cristadoro, G.; Lenci, M.; Ligabò, M., Random walks in a one-dimensional lévy random environment, J. Stat. Phys., 163, 22-40, (2016) · Zbl 1343.82039
[8] Degli Esposti, M.; Graffi, S.; Isola, S., Classical limit of the quantized hyperbolic toral automorphisms, Commun. Math. Phys., 167, 471-507, (1995) · Zbl 0822.58022
[9] Bonechi, F.; De Biévre, S., Exponential mixing and \(\left|\ln \mathit{ħ}\right|\) time scales in quantized hyperbolic maps on the torus, Commun. Math. Phys., 211, 659-686, (2000) · Zbl 1053.81032
[10] Bonechi, F.; De Biévre, S., Controlling strong scarring for quantized ergodic toral automorphisms, Duke Math. J., 117, 571-587, (2003) · Zbl 1049.81028
[11] Faure, F.; Nonnenmacher, S.; De Biévre, S., Scarred eigenstates for quantum cat maps of minimal periods, Commun. Math. Phys., 239, 449-492, (2003) · Zbl 1033.81024
[12] Faure, F.; Nonnenmacher, S., On the maximal scarring for quantum cat map eigenstates, Commun. Math. Phys., 245, 201-214, (2004) · Zbl 1071.81044
[13] Graffi, S.; Facchi, P.; Ligabò, The classical limit of the quantum Zeno effect, J. Phys. A: Math. Theor., 43, 032001, (2010) · Zbl 1183.81015
[14] Figalli, A.; Ligabò, M.; Paul, T., Semiclassical limit for mixed states with singular and rough potentials, Indiana Univ. Math. J., 61, 193-222, (2012) · Zbl 1264.81177
[15] Schwinger, J., Quantum Kinematics and Dynamics, (1970), Benjamin: Benjamin, New York · Zbl 1094.81500
[16] Cohen, L.; Scully, M., Joint Wigner distribution for spin-1/2 particles, Found. Phys., 16, 295-310, (1986)
[17] Wootters, W. K., A Wigner-function formulation of finite-state quantum mechanics, Ann. Phys., 176, 1-21, (1987)
[18] Cohendet, O.; Combe, P.; Sirugue, M.; Sirugue-Collin, M., A stochastic treatment of the dynamics of an integer spin, J. Phys. A: Math. Gen., 21, 2875-2884, (1988) · Zbl 0653.60106
[19] Galetti, D.; Toledo Piza, A. F. R., An extended Weyl-Wigner transformation for special finite spaces, Phys. A, 149, 267-282, (1988)
[20] Varadarajan, V. S., Variations on a theme of Schwinger and Weyl, Lett. Math. Phys., 34, 319-326, (1995) · Zbl 0830.22006
[21] Leonhardt, U., Discrete Wigner function and quantum-state tomography, Phys. Rev. A, 53, 2998, (1996)
[22] Miquel, C.; Paz, J. P.; Saraceno, M., Quantum computers in phase space, Phys. Rev. A, 65, 062309, (2002)
[23] Wootters, W. K., Picturing qubits in phase space, IBM J. Res. Dev., 48, 99-110, (2004)
[24] Man’ko, V. I.; Marmo, G.; Simoni, A.; Ventriglia, F., Semigroup of positive maps for qudit states and entanglement in tomographic probability representation, Phys. Lett. A, 372, 6490-6497, (2004) · Zbl 1225.81033
[25] Paz, J. P.; Roncaglia, A. J.; Saraceno, M., Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem, Phys. Rev. A, 72, 012309, (2005)
[26] Wootters, W. K., Quantum measurements and finite geometry, Found. Phys., 36, 112-126, (2006) · Zbl 1105.81023
[27] Chaturvedi, S.; Ercolessi, E.; Marmo, G.; Morandi, G.; Mukunda, N.; Simon, R., Wigner-Weyl correspondence in quantum mechanics for continuous and discrete systems-a Dirac-inspired view, J. Phys. A: Math. Gen., 39, 1405-1423, (2006) · Zbl 1088.81068
[28] Ibort, A.; Man’ko, V. I.; Marmo, G.; Simoni, A.; Ventriglia, F., An introduction to the tomographic picture of quantum mechanics, Phys. Scr., 79, 065013, (2009) · Zbl 1170.81007
[29] Facchi, P.; Ligabò, M., Classical and quantum aspects of tomography, AIP Conf. Proc., 1260, 3, (2010) · Zbl 1330.81002
[30] Facchi, P.; Ligabò, M.; Pascazio, S., On the inversion of the Radon transform: Standard versus M2 approach, J. Mod. Opt., 57, 239-243, (2010) · Zbl 1200.44003
[31] Zak, J., Doubling feature of the Wigner function: Finite phase space, J. Phys. A: Math. Theor., 44, 345305, (2011) · Zbl 1226.81113
[32] Facchi, P.; Ligabò, M.; Solimini, S., Tomography: Mathematical aspects and applications, Phys. Scr., 90, 074007, (2015)
[33] De Biévre, S., Chaos, quantization and the classical limit on the torus, (1998), Polish Scientific Publishers PWN
[34] Degli Esposti, M.; Graffi, S., The Mathematical Aspects of Quantum Maps, 618, 49-90, (2003), Springer · Zbl 1058.81542
[35] De Biévre, S., Recent Results on Quantum Map Eigenstates, 690, 367-381, (2006), Springer
[36] Zygmund, A., Trigonometric Series, (1988), Cambridge University Press · Zbl 0628.42001
[37] Folland, G., Harmonic Analysis in Phase Space, (1988), Princeton University Press
[38] Martinez, A., An Introduction to Semiclassical and Microlocal Analysis, (1987), Springer-Verlag: Springer-Verlag, New York
[39] Robert, D., Autour De L’Approximation Semi-Classique, (1987), Birkhauser: Birkhauser, Boston · Zbl 0621.35001
[40] Anderson, J.; Paschke, W., The rotation algebra, Houston J. Math., 15, 1-26, (1989) · Zbl 0703.22005
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