Abstract
Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied. It is shown that the sets have a common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vice versa, there exist tomograms of quantum states which are not admissible in classical mechanics. The role of different transformations of reference frames in the phase space of classical and quantum systems (scaling and rotation) determining the admissibility of tomograms as well as the role of quantum uncertainty relations are elucidated. The union of all admissible tomograms of both quantum and classical states is discussed in the context of interaction of quantum and classical systems. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively. The role of the semigroup of scaling transforms of the Planck's constant is discussed.
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Olga Man'ko and V. I. Man'ko, J. Russ. Laser Res., 18, 407 (1997); 21, 411 (2000); "Classical mechanics is not the ∼ ! 0 limit of quantum mechanics" Los Alamos ArXiv quant-ph/0407183 (2004).
V. I. Man'ko and R. V. Mendes, Physica D, 145, 330 (2000).
J. G. Muga and R. F. Snider, Europhys. Lett., 19, 569 (1992).
V. I. Man'ko, in: V. V. Dodonov and V. I. Man'ko (eds.), Theory of Nonclassical States of Light, Taylor & Francis, London & New York (2003), p. 219.
Y. M. Shirokov, Teor. Mat. Fiz., 28, 308 (1976) [Theor. Math. Phys., 28, 806 (1976)].
S. Mancini, V. I. Man'ko, and P. Tombesi, Phys. Lett., 213 A, 1 (1996).
S. Mancini, V. I. Man'ko, and P. Tombesi, Quantum Semiclass. Opt., 7, 615 (1995).
V. V. Dodonov and V. I. Man'ko, Phys. Lett. A, 239, 335 (1997).
V. I. Man'ko and O. V. Man'ko, JETP, 85, 430 (1997).
O. V. Man'ko, V. I. Man'ko, and G. Marmo, Phys. Scr., 62, 446 (2000).
O. V. Man'ko, V. I. Man'ko, and G. Marmo, J. Phys. A: Math. Gen., 35, 699 (2002).
S. Mancini, O. V. Man'ko, V. I. Man'ko, and P. Tombesi, J. Phys. A: Math. Gen., 34, 3461 (2001).
A. B. Klimov, O. V. Man'ko, V. I. Man'ko, Yu. F. Smirnov, and V. N. Tolstoy, J. Phys. A: Math. Gen., 35, 6101 (2002).
V. A. Andreev, O. V. Man'ko, V. I. Man'ko, and S. S. Safonov, J. Russ. Laser Res., 19, 340 (1998).
O. V. Man'ko, V. I. Man'ko, and S. S. Safonov, Theor. Math. Phys., 115, 185 (1998).
V. I. Man'ko and S. S. Safonov, Yad. Fiz., 4, 658 (1998).
V. A. Andreev and V. I. Man'ko, JETP, 87, 239 (1998).
O. Castaños, R. L´opez-Peña, M. A. Man'ko and V. I. Man'ko, J. Phys. A: Math. Gen., 36, 4677 (2003); 37, 8529 (2004); J. Opt. B: Quantum Semiclass. Opt., 5, 227 (2003).
M. A. Man'ko, J. Russ. Laser Res., 22, 168 (2001); S. De Nicola, R. Fedele, M. A. Man'ko, and V. I. Man'ko, Eur. Phys. J. B, 36, 385 (2003); J. Russ. Laser Res., 25, 1 (2004).
V. V. Dodonov and V. I. Man'ko, "Generalization of uncertainty relation in quantum mechanics" in: M. A. Markov (ed.), Invariants and the Evolution of Nonstationary Quantum Systems, Proceedings of the Lebedev Physical Institute, Nauka, Moscow (1987), Vol. 183, p. 5 [translated by Nova Science, New York (1989), p. 3].
E. Schr¨odinger, "Zum Heisenbergschen Unsch¨arfeprinzip" Ber. Kgl. Akad. Wiss. Berlin, 24, 296 (1930).
H. P. Robertson, Phys. Rev., 35, 667 (1930).
V. V. Dodonov, E. V. Kurmushev, and V. I. Man'ko, Phys. Lett. A, 79, 150 (1980).
V. V. Dodonov and V. I. Man'ko, in: V. V. Dodonov and V. I. Man'ko (eds.), Theory of Nonclassical States of Light, Taylor & Francis, London & New York (2003), p. 1.
V. V. Dodonov, J. Opt. B, 4, S98 (2002).
E. Wigner, Phys. Rev, 40, 749 (1932).
V. I. Man'ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Russ. Laser Res., 24, 507 (2003); Phys. Lett. A, 327, 353 (2004).
O. V. Man'ko, V. I. Man'ko, and G. Marmo, Phys. Scr., 62, 446 (2000); J. Phys. A: Math.Gen., 35, 699 (2002).
A. Bohm, N. L. Harshman, and H. Walther, Phys. Rev. A, 66, 012107 (2002); A. Bohm and M. Gadella, in: I. Antoniou (ed.), Proceedings of the XXII Solvay Conference on Physics: The Physics of Communication, World Scientific, Singapore (2003), p. 117.
A. J. Bracken and J. G. Wood, "Nonpositivity of Groenewold operator" Los Alamos ArXiv quant-ph/ 0407052 (2004).
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Man'ko, O., Man'ko, V.I. Classical Mechanics Is not the ħ, → 0 Limit of Quantum Mechanics. Journal of Russian Laser Research 25, 477–492 (2004). https://doi.org/10.1023/B:JORR.0000043735.34372.8f
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DOI: https://doi.org/10.1023/B:JORR.0000043735.34372.8f