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Journal of Applied Mathematics and Physics

Path Integral Quantization of Nonconservative Systems

Ola A. Jarab’ah, Khaled I. Nawafleh

Published Jan 1, 2018• 1 citation
Publisher

Abstract

In this paper nonconservative systems are investigated within the framework of Euler Lagrange equations. The solutions of these equations are used to find the principal function S, this function is used to formulate the wave function and then to quantize these systems using path integral method. One example is considered to demonstrate the application of our formalism.

DOI
10.4236/jamp.2018.68139

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In this paper nonconservative systems are investigated within the framework of Euler Lagrange equations. The solutions of these equations are used to find the principal function S, this function is used to formulate the wave function and then to quantize these systems using path integral method. One example is considered to demonstrate the application of our formalism.
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The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman's approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.
In the present paper we propose a method of quantization of mechanical systems that is applicable, inter alia, to systems acted on by nonconservative forces, such as dissipative systems. First, the correspondence between classical and quantum mechanics is considered and we make some important observations that can be used in defining the rules of quantization. Next, the quantization rules are presented and we give a few examples of the application of these rules to mechanical systems that are difficult to quantize within any other systemetic approach. Finally, it is shown that for any quantum-mechanical system there are stationary solutions of the evolution equations, and that the existence of these solutions is directly related to the proposed quantization rules.