Different quantizations and different classical limits of quantum theory

Yu. M. Shirokov

Teoretičeskaâ i matematičeskaâ fizika, Tome 29 (1976) no. 3, pp. 309-322

Résumé

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.