General Solution and Canonical Quantization of the Conic Path Constrained Second-Class System

TL;DR

This paper provides a comprehensive analysis of a second-class constrained system describing motion along a conic path, offering a general solution, canonical quantization, and detailed constraint structure analysis.

Contribution

It introduces a novel method for solving the nonlinear equations and performs a consistent canonical quantization of the system using Dirac brackets.

Findings

Explicit general solution for the nonlinear differential equations

Complete Dirac brackets algebra in phase space derived

Physical realization of brackets in differential operator form

Abstract

We consider the problem of constrained motion along a conic path under a given external potential function. The model is described as a second-class system capturing the behavior of a certain class of specific quantum field theories. By exhibiting a suitable integration factor, we obtain the general solution for the associated non-linear differential equations. We perform the canonical quantization in a consistent way in terms of the corresponding Dirac brackets. We apply the Dirac-Bergmann algorithm to unravel and classify the whole internal constraints structure inherent to its dynamical Hamiltonian description, obtain the proper extended Hamiltonian function, determine the Lagrange multiplier and compute all relevant Poisson brackets among the constraints, Hamiltonian and Lagrange multiplier. The complete Dirac brackets algebra in phase space as well as its physical realization in terms of differential operators is explicitly obtained.