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Journal of Xihua Teachers College

Requirements of the Canonical Quantization in Different Coordinates

Chen Yuan-li

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

This article illustrates that when we get the formula of _i_j from canonical commutation rules _i,_j=iδ_(ij),we must take into account at the same time the Hermiticity of _j.To get the right quantum Hamiltonian operator  in non-Cartesian coordinates,we always need to change  in Cartesian coordinates into non-Cartesian coordinates by using the coordinates transformation.

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This article illustrates that when we get the formula of _i_j from canonical commutation rules _i,j=iδ(ij),we must take into account at the same time the Hermiticity of _j.To get the right quantum Hamiltonian operator  in non-Cartesian coordinates,we always need to change  in Cartesian coordinates into non-Cartesian coordinates by using the coordinates transformation.
Canonical quantization relies on Cartesian, canonical, phase-space coordinates to promote to Hermitian operators, which also become the principal ingredients in the quantum Hamiltonian. While generally appropriate, this procedure can also fail, e.g., for covariant, quartic, scalar fields in five-and-more spacetime dimensions (and possibly four spacetime dimensions as well), which become trivial; such failures are normally blamed on the `problem' rather than on the `quantization procedure'. In Enhanced Quantization the association of ccc-numbers to qqq-numbers is chosen very differently such that: (i) there is no need to seek classical, Cartesian, phase-space coordinates; (ii) {\it every} classical, contact transformation is applicable and {\it no} change of the quantum operators arises; (iii) a new understanding of the importance of `Cartesian coordinates' is established; and (iv) although discussed elsewhere in detail, the procedures of enhanced quantization offer fully acceptable solutions yielding non-trivial results for quartic scalar fields in four-and-more spacetime dimensions. In early sections, this paper offers a wide-audience approach to the basic principles of Enhanced Quantization using simple examples; later, several significant examples are cited for a deeper understanding. An historical note concludes the paper.
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.
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.
The dynamical system of a point particle constrained on a torus is quantized à la Dirac with two kinds of coordinate systems respectively; the Cartesian and toric coordinate systems. In the Cartesian coordinate system, it is difficult to express momentum operators in coordinate representation owing to the complication in structure of the commutation relations between canonical variables. In the toric coordinate system, the commutation relations have a simple form and their solutions in coordinate representation are easily obtained with, furthermore, two quantum Hamiltonians turning up. A problem comes out when the coordinate system is transformed, after quantization, from the Cartesian to the toric coordinate system.
The definition and interpretation of canonical, phase space path integrals has evolved over many years to achieve a form that now admits a correct and rigorous formulation, which is also covariant under canonical coordinate transformations. Such formulations involve coherent state representations, which, in their modern version, were originally introduced as an alternative tool to construct phase space path integrals. Moreover, coherent state representations lead to physical interpretations that are more natural than those afforded by more traditional representations. Suitable continuous time regularization procedures lead to a covariant phase space path integral formulation that greatly clarifies the vague phrase that canonical quantization requires Cartesian coordinates.