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 TMF, 1976, Volume 28, Number 3, Pages 308–319 (Mi tmf4263)

Combined algebra for quantum and classical mechanics

Yu. M. Shirokov

Abstract: For a canonical Hamiltonian system, an algebra is constructed in which all the observables are realized by ordinary functions $A(p,q)$ of the momenta and coordinates and are simultaneously classical and quantum observables. The classical and quantum states are realized by density matrices $\rho(p,q)$ that are either coincident for the quantum and the classical theory or exist only in one of the theories. The entire difference between the quantum and classical descriptions reduces to the difference between the quantum and classical operations of multiplication of observables, their Poisson brackets, and thus between the evolutions of the observables (or states) in time. A transition from the quantum to the classical theory is proposed and investigated in which the observables and states do not change and the operations of quantum multiplication and taking of the quantum Poisson brackets go over as $\hbar\to0$ into the corresponding classical operations in a perfectly definite sense. It is shown that the quantum operations are infinitely differentiable with respect to $\hbar$ at zero. The transition to classical mechanics is possible for all observables but not for all states. Pure quantum states become mixed in the classical case. The quantum corrections destroy the Hamiltonicity of the classical equations of motion. For the space of observables a topology which admits unbounded operators is used.

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English version:
Theoretical and Mathematical Physics, 1976, 28:3, 806–813

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Citation: Yu. M. Shirokov, “Combined algebra for quantum and classical mechanics”, TMF, 28:3 (1976), 308–319; Theoret. and Math. Phys., 28:3 (1976), 806–813

Citation in format AMSBIB
\Bibitem{Shi76} \by Yu.~M.~Shirokov \paper Combined algebra for quantum and classical mechanics \jour TMF \yr 1976 \vol 28 \issue 3 \pages 308--319 \mathnet{http://mi.mathnet.ru/tmf4263} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=522660} \zmath{https://zbmath.org/?q=an:0335.70024} \transl \jour Theoret. and Math. Phys. \yr 1976 \vol 28 \issue 3 \pages 806--813 \crossref{https://doi.org/10.1007/BF01029172} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. M. Shirokov, “Different quantizations and different classical limits of quantum theory”, Theoret. and Math. Phys., 29:3 (1976), 1091–1100
2. G. K. Tolokonnikov, “Associative Hamiltonian algebras”, Theoret. and Math. Phys., 31:2 (1977), 441–445
3. Yu. M. Shirokov, “Perturbation theory with respect to Planck's constant”, Theoret. and Math. Phys., 31:3 (1977), 488–492
4. S. N. Sokolov, “Is relativistic invariance preserved in the limit $\hbar\to 0$?”, Theoret. and Math. Phys., 32:3 (1977), 790–794
5. Yu. M. Shirokov, “On admissible forms of canonical mechanics”, Theoret. and Math. Phys., 30:1 (1977), 3–6
6. Yu. M. Shirokov, “Unified formalism for quantum and classical scattering theories”, Theoret. and Math. Phys., 38:3 (1979), 206–211
7. M. A. Antonets, “Classical limit of Weyl quantization”, Theoret. and Math. Phys., 38:3 (1979), 219–228
8. Yu. M. Shirokov, “Algebra of one-dimensional generalized functions”, Theoret. and Math. Phys., 39:3 (1979), 471–477
9. G. K. Tolokonnikov, “Algebras of observables of nearly canonical physical theories. II”, Theoret. and Math. Phys., 61:2 (1984), 1072–1077
10. V. G. Budanov, “Methods of Weyl representation of the phase space and canonical transformations. I”, Theoret. and Math. Phys., 61:3 (1984), 1183–1195
11. Man'ko, O, “Classical mechanics is not the h -> 0 limit of quantum mechanics”, Journal of Russian Laser Research, 25:5 (2004), 477
12. Man'ko, OV, “Probability representation of classical states”, Journal of Russian Laser Research, 26:6 (2005), 429
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