Concerning the quantization of a mechanical system with second-order constraints

O. Yu. Shvedov

We study the semiclassical mechanics of systems with second-class constraints. We assume that the quantum mechanics of such a system is constructed by the Batalin–Fradkin–Tyutin method, where some additional coordinates and momenta are introduced and second-class constraints are converted into firstclass constraints. We also assume that the algebraic quantization method is used to quantize the extended system. To construct the semiclassical approximation, we use the Maslov complex germ theory. We study the semiclassical states of a system with second-class constraints, their scalar product, and the action of semiclassical observables in the first order of the semiclassical expansion. We consider the transformation of semiclassical states in the course of evolution.

D. Băleanu, Y. Güler

Jędrzej Śniatycki

Masayoshi Nakamura, Hiroaki Minowa, Nobuhiko Mishima

The problem of the quantization of an elementary model with the second class constraints, O(n) rigid rotator model, is investigated by using the Dirac method, the projection operator method (POM) and the nonlinear quantum mechanics (NLQM). It is proved that the quantum effects, which are equivalent to those of NLQM, appear in the equation of motion of POM. Furthermore, it is shown that POM as well as NLQM preserves the isometric symmetry, which the system possesses in the classical level, through the quantization process.

In this article we study the nature of time in Mechanics.The fundamental principle, according to which a mechanical system evolves governed by a second order differential equation, implies the existence of an absolute time-duration in the sense of Newton.There is a second notion of time for conservative systems which makes the Hamiltonian action evolve at a constant rate.In Quantum Mechanics the absolute time loses its sense as it does the notion of trajectory.Then, we propose two different ways to reach the time dependent Schrödinger equation.One way consists of considering a "time constraint" on a free system.The other way is based on the point of view of Hertz, by considering the system as a projection of a free system.In the later manner, the "time" appearing in the Schrödinger equation is a linear combination of the time-duration with the "time" quotient of the action by the energy on each solution of the Hamilton-Jacobi equation.Both of them are based on a rule of quantization that we explain in Section 4. Contents 1. Introduction 1 2. Time as duration and its absolute character 3 3. Classical Mechanics 6 4. Quantization of contravariant tensors.Dequantization of differential operators 10 5. Time dependent Schrödinger equation from a time constraint 14 6.Waves in Classical Mechanics 17 7.Time dependent Schrödinger equation from a Hertz constraint 21 8.