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Journal of Physics A Mathematical and General

Ordering problem in geometrical quantization

Dehai Bao, Zhongyuan Zhu

Published Apr 21, 1992• 3 citations
Publisher

Abstract

The operator ordering problem is investigated in the framework of geometrical quantization in the Schrodinger coordinate representation. It is shown that, for quantization of polarization-preserving classical observables pf(q), correct Hermitian quantum operators are derived; for quantization of polarization-unpreserving classical observables p 2 f(q), the BKS kernel method actually defines an ordering rule which is not Weyl's rule, but is the GCT-invariant rule proposed by DeWitt (1957).

DOI
10.1088/0305-4470/25/8/044

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Communications in Theoretical Physics•Nov 1, 2008•1 citation
The operator ordering problem is investigated in the framework of geometrical quantization in the Schrodinger coordinate representation. It is shown that, for quantization of polarization-preserving classical observables pf(q), correct Hermitian quantum operators are derived; for quantization of polarization-unpreserving classical observables p 2 f(q), the BKS kernel method actually defines an ordering rule which is not Weyl's rule, but is the GCT-invariant rule proposed by DeWitt (1957).
By examining three quantization schemes of a ray function in classical phase space (a geometric ray is expressed by δ(x-λq-νp)), we find that the Weyl quantization scheme can reasonably demonstrate the correspondence between classical functions and quantum mechanical operators, since δ(x-λq-νp) really maps onto the operator δ(x-λQ-νP), where [Q, P] = iℏ, and δ(x-λQ-νP) represents a pure state (the coordinate-momentum intermediate representation), while 𝔓-ordered, 𝔔-ordered quantization schemes δ(x-λq-νp) to two different Fresnel integration kernels in Weyl-ordered form. Thus, Weyl quantization is more reasonable and preferable.
A method for quantizing a classical system, based upon the notion of localization on phase space, is developed. This method, to which is given the name prime quantization, is based upon the mathematical theory of positive-operator-valued measures and their relationship to reproducing kernel Hilbert spaces, and to a notion of generalized kinematical variables related to phase space. For a system with openRn{\mathrm{openR}}^{n}openRn as its configuration space, it has the advantage of being able to connect the well-known problem of ordering in quantum mechanics to the choice of a measuring apparatus in a joint measurement of position and momentum (within the limits of the uncertainty principle). Moreover, for such a system, a choice of ordering in the present context also turns out to be a choice of polarization in the method of geometric quantization.
The Schrodinger variational approach (1926) to quantization of the natural Hamilton mechanics in 2n2n2n-dimensional phase space is revised in the modern paradigm of quantum mechanics in application to the system the Hamilton function of which is a positive-definite quadratic form of the n momenta with the coefficients depending on coordinates in the generic configuration space VnV_nVn​. The quantum Hamilonian thus obtained has a paradoxical potential term depending on choice of coordinates in VnV_nVn​, which was discovered first by B. DeWitt in 1952 in the framework of canonical quantization of the system by a particular ordering of the operators of observables of momenta and coordinates. It is shown that the Schrodinger approach in the standard paradigm of quantum mechanics determines uniquely the ordering selected by DeWitt among a continuum of other possibilties to determine the quantum Hamiltonian. Two particular classes of observables of localization in VnV_nVn​ are considered in detail. It is shown that, in general, the quantum-mechanical potential does not vanish even in the Euclidean configuration space VnV_nVn​ except the case when Cartesian coordinates are taken as the observables of localization of the system. It is noted also that, in the quasi-classical approach to the quantization considered by DeWitt in 1957, the quantum Green function (propagator) is also non-unique and depends on the choice of a line in VnV_nVn​ connecting these points. All three formalisms have the same local asymtotics of the quantum Hamiltonian if the normal Riemannian cordinates are used, at least implicitly, to localize the system in VnV_nVn​. Keywords: Hamilton function; Quantization; Quantum-Mechanical Potential; Observables of Localization; Quantum Anomaly of (non-relativistic) General Covariance.