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Communications in Theoretical Physics

On GCT-Invariant Operator Ordering Rule

Dehai Bao, Zhongyuan Zhu

Published Mar 1, 1993• 0 citations
Publisher

Abstract

With an observation that the general coordinate transformation (GCT) invariant operator ordering rule proposed by DeWitt is valid only for some cases, a careful classification for classical observables that can be related through the well-defined GCT is presented and a generalized GCT-invariant ordering rule is derived.
DOI
10.1088/0253-6102/19/2/185

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Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value ihihih where hhh is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.
By employing coordinate transformations, a generalized WKB quantization condition is derived which includes a modified WKB quantization rule and the related higher-order integrals. It is observed that a necessary condition for the first-order integral to give an exact quantization rule is the vanishing of the higher-order integrals. When this condition is satisfied, the resulting quantization condition is of the form of all previously known exact-quantization rules. The higher-order integrals are shown to vanish for certain cases of interest. An error in Paper I [C. Rosenzweig and J. B. Krieger, J. Math. Phys. 9, 849 (1968)] is noted and an examination of the higher-order integrals shows that a proposed quantization rule given there is not exact.
Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product" technique, and the mapping is manifestly apparent in the derivation. New generalized coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.
Following Dirac, the rules of canonical quantization include classical and quantum contact transformations of classical and quantum phase space variables. While arbitrary classical canonical coordinate transformations exist that is not the case for some analogous quantum canonical coordinate transformations. This failure is due to the rigid connection of quantum variables arising by promoting the corresponding classical variable from a ccc-number to a qqq-number. A different relationship of ccc-numbers and qqq-numbers in the procedures of Enhanced Quantization shows the compatibility of all quantum operators with all classical canonical coordinate transformations.