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Journal of Mathematical Physics

Quantization of a General Dynamical System by Feynman's Path Integration Formulation

K. S. Cheng

Published Nov 1, 1972• 152 citations
Publisher

Abstract

The Schrödinger equation is obtained by Feynman's path integration method of quantization for a general dynamical system. The meaning of the results is discussed.
DOI
10.1063/1.1665897

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The Lagrangian formulation of quantum dynamics in terms of path-integrals due to Feynman describes systems for which the Hamiltonian is classical in form and quantization is carried out in terms of commutators rather than anticommutators. The difficulty with the Feynman method is the actual evaluation of the pathintegral itself. An explicit evaluation is given for classical wave motion in one dimension. This requires an extension of the Feynman method which was introduced by Tobocman and studied in detail by Davies. The work of Corson on the question of a unified formulation of dynamics is discussed. (Author)
The path integral formulation in quantum mechanics corresponds to the first quantization since it is just to rewrite the quantum mechanical amplitude into many dimensional integrations over discretized coordinates xnx_nxn​. However, the path integral expression cannot be connected to the dynamics of classical mechanics, even though, superficially, there is some similarity between them. Further, the field theory path integral in terms of many dimensional integrations over fields does not correspond to the field quantization. We clarify the essential difference between Feynman's original formulation of path integral in QED and the modern version of the path integral method prevailing in lattice field theory calculations, and show that the former can make a correct second quantization while the latter cannot quantize fields at all and its physical meaning is unknown.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.