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Evaluation of the Feynman Propagator by Means of the Quantum Hamilton-Jacobi Equation

M. Fusco Girard

Published Apr 24, 2023• 1 citation
Publisher

Abstract

It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton–Jacobi equation, namely, it is the quantum Hamilton's principal function (or quantum action). Therefore, the Feynman propagator can be computed either by means of the path integration, or by the way of the Hamilton–Jacobi equation. This is analogous to what happens in classical mechanics, where the Hamilton's principal function can be computed either by integrating the Lagrangian along the extremal paths, or as a solution of partial differential equation, namely the classical Hamilton–Jacobi equation. If the path is decomposed in the classical one and quantum fluctuations, the contribution of these quantum fluctuations satisfies a non-linear partial differential equation, whose coefficients depend on the classical action. When the contribution of the quantum fluctuations depend only on the time, it can be computed by means of a simple integration. The final results for the propagators in this case are equal to the Van Vleck–Pauli–Morette expressions, even though the two derivations are quite different. Quanta 2023; 12: 22–26.

DOI
10.12743/quanta.v12i1.223

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It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton–Jacobi equation, namely, it is the quantum Hamilton's principal function (or quantum action). Therefore, the Feynman propagator can be computed either by means of the path integration, or by the way of the Hamilton–Jacobi equation. This is analogous to what happens in classical mechanics, where the Hamilton's principal function can be computed either by integrating the Lagrangian along the extremal paths, or as a solution of partial differential equation, namely the classical Hamilton–Jacobi equation. If the path is decomposed in the classical one and quantum fluctuations, the contribution of these quantum fluctuations satisfies a non-linear partial differential equation, whose coefficients depend on the classical action. When the contribution of the quantum fluctuations depend only on the time, it can be computed by means of a simple integration. The final results for the propagators in this case are equal to the Van Vleck–Pauli–Morette expressions, even though the two derivations are quite different. Quanta 2023; 12: 22–26.
We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton–Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton–Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure. Quanta 2022; 11: 42–52.
One shows that the Feynman’s Path Integral designed for quantum mechanics has an analogous in classical mechanics, the so-called (min, +) Path Integral. This former is build on (min, +)-algebra and (min, +)-analysis which permit to handle in a linear way non-linear problems occurring in mathematical physics. The Hamilton-Jacobi equations and their solutions within this mathematical framework, are introduced and yield to a new interpretation expressed in a duality between action field and particle.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
In (1), a method for finding the propagator of a generalized quantum oscillator directly from the time-dependent Schrodinger equation is presented. This method assumes the form Q(t)exp[i (B(t)xx + A(t)yy + C(t)xy) ] which is inserted into the Schrodinger equation. The equation is broken into purely imaginary and real parts. Time dependent coefficients of xx, xy and yy are found leading to four differential equations which may be solved to ultimately yield a full solution for the propagator. In (2), it is suggested that the propagator is given by Q(t) exp(i Integral (t=0 to Tf) Lc dt1)). In order to use this approach, a solution to the classical Lagrangian Lc is found and x written as: XA(Tf-t)+YB(t-0) where X and Y are the initial and final x positions (and are numbers). Thus, dx/dt may be computed, and the classical action integral (t=0 to Tf) Lc dt) found through integration. In this note, we consider the quantum oscillator and try to demonstrate the two approaches are the same. A key point is that -id/dt (partial) in the Schrodinger equation really represents -i d/dTf (partial).
Abstract This chapter focuses on the Hamilton-Jacobi theory, the apotheosis of Lagrangian and Hamiltonian mechanics. Action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria, and are the solutions of a non-linear, first-order partial differential equation called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. An entire class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This connection is discussed in detail in the chapter, together with the Bohm hidden variable model and Feynman path integral method that are derived from it.