On Geometric Paradox in Quantization of Natural Dynamical Systems

TL;DR

This paper revisits the Schrödinger quantization of natural Hamiltonian systems, revealing a coordinate-dependent quantum potential and non-uniqueness issues, with implications for quantum covariance and localization observables.

Contribution

It demonstrates the unique operator ordering in Schrödinger quantization and analyzes the coordinate dependence of the quantum potential and propagator in natural systems.

Findings

Quantum potential depends on coordinate choice, vanishing only in Cartesian coordinates.

Schrödinger quantization selects a specific operator ordering among many.

Quantum propagator's non-uniqueness depends on the path in configuration space.

Abstract

The Schrodinger variational approach (1926) to quantization of the natural Hamilton mechanics in $2n$-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 $V_n$. The quantum Hamilonian thus obtained has a paradoxical potential term depending on choice of coordinates in $V_n$, 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 $V_n$ are considered in detail. It is shown that, in general, the quantum-mechanical potential does not vanish even in the Euclidean configuration space $V_n$ 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 $V_n$ 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 $V_n$. Keywords: Hamilton function; Quantization; Quantum-Mechanical Potential; Observables of Localization; Quantum Anomaly of (non-relativistic) General Covariance.