On the Consistency between Lagrangian and Hamiltonian Formalisms in Quantum Mechanics. III

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Abstract
A consistent quantization procedure is proposed for the non-linear Lagrangian L = (1/8)×g-1/4 { gik, \dotqk } g1/2gij { gjm, \dotgm } g-1/4 - v(q). We extend the previous formalism, which was formulated by means of a q-number variation principle for cases with special gij, so that it is applicable to the general case with arbitrary gij. The commutation relations of q-number variations δqi and δ\dotqj with qi and \dotqj are taken as unknown functions in deriving the Euler-Lagrange equation. However, in the final expression of he Euler-Lagrange equation and the Hamiltonian, these unknown functions can be eliminated so that the equation of motion and the Hamiltonian are uniquely given by gij and their derivatives. In general the unknown functions are not well defined in the overall region of qi, contrary to the special cases given in previous papers. For illustration, a simple example of two-dimensional space is presented. The Hamiltonian as the time generator of the system is given by H = L + 2v - (\hbar2/4)R where R is the curvature scalar constructed from gij. With this H the canonical equation of motion is in accord with the Euler-Lagrange equation.

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