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 TMF, 1976, Volume 29, Number 3, Pages 309–322 (Mi tmf3467)

Different quantizations and different classical limits of quantum theory

Yu. M. Shirokov

Abstract: For a canonical Hamiltonian system, a set of combined algebras $\mathscr B(U,W)$ analogous to the algebra $\mathscr B(1)$ of [1] is constructed. Each algebra gives a quantization prescription and a prescription for the transition from quantum theory to classical theory, and also a method for calculating the quantum corrections in powers of Ii to classical objects of different nature (observables, generators, equations of motion, etc). For each algebra, a set of quantities is found for which there exists a transition from the quantum to the classical theory (or vice versa). The set of quantizations reflects not only the different orderings of noncommuting operators but also different correspondences between the classical and quantum states for a given ordering. The set of transitions from the quantum to the classical theory reflects the ambiguity in the prescription “$\hbar\to 0$” associated with the fact that the constant $\hbar$ can be everywhere introduced (or eliminated) by a transformation of constants. The requirement adopted in the present paper of a passage to the limit of the quantum laws into classical laws is consistent. A methodological example is presented which shows that by an appropriate choice of the algebra $\mathscr B(U,W)$ one can obtain a good classical approximation to even the essentially quantum problem of the binding energy of the hydrogen atom.

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English version:
Theoretical and Mathematical Physics, 1976, 29:3, 1091–1100

Bibliographic databases:

Citation: Yu. M. Shirokov, “Different quantizations and different classical limits of quantum theory”, TMF, 29:3 (1976), 309–322; Theoret. and Math. Phys., 29:3 (1976), 1091–1100

Citation in format AMSBIB
\Bibitem{Shi76} \by Yu.~M.~Shirokov \paper Different quantizations and different classical limits of quantum theory \jour TMF \yr 1976 \vol 29 \issue 3 \pages 309--322 \mathnet{http://mi.mathnet.ru/tmf3467} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=468853} \transl \jour Theoret. and Math. Phys. \yr 1976 \vol 29 \issue 3 \pages 1091--1100 \crossref{https://doi.org/10.1007/BF01028231} 

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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, “Perturbation theory with respect to Planck's constant”, Theoret. and Math. Phys., 31:3 (1977), 488–492
2. Yu. M. Shirokov, “Unified formalism for quantum and classical scattering theories”, Theoret. and Math. Phys., 38:3 (1979), 206–211
3. M. A. Antonets, “Classical limit of Weyl quantization”, Theoret. and Math. Phys., 38:3 (1979), 219–228
4. G. K. Tolokonnikov, “Algebras of observables of nearly canonical physical theories. II”, Theoret. and Math. Phys., 61:2 (1984), 1072–1077
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