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Physical Review

Quantization and the Classical Hamilton-Jacobi Equation

Lloyd Motz

Published Apr 1, 1962• 20 citations
Publisher

Abstract

In this paper, a formalism for quantization is developed which starts out from the Hamilton-Jacobi expression, $\frac{\ensuremath{\partial}S}{\ensuremath{\partial}t}+H(\frac{\ensuremath{\partial}S}{\ensuremath{\partial}q}, q)$, and which leads to its usual quantum-mechanical operator equivalent by means of straightforward algebra. The quantum-mechanical operator equivalents of $H$ and $p$ are then seen to be the consequence of assigning a number of equally probable classical paths to a dynamical system.

DOI
10.1103/physrev.126.378

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In this paper, a formalism for quantization is developed which starts out from the Hamilton-Jacobi expression, \ensuremath∂S\ensuremath∂t+H(\ensuremath∂S\ensuremath∂q,q)\frac{\ensuremath{\partial}S}{\ensuremath{\partial}t}+H(\frac{\ensuremath{\partial}S}{\ensuremath{\partial}q}, q)\ensuremath∂t\ensuremath∂S​+H(\ensuremath∂q\ensuremath∂S​,q), and which leads to its usual quantum-mechanical operator equivalent by means of straightforward algebra. The quantum-mechanical operator equivalents of HHH and ppp are then seen to be the consequence of assigning a number of equally probable classical paths to a dynamical system.
Abstract The quantization of the dynamics of point systems is closely connected with the Hamilton-Jacobi theory of the calculus of variations for simple integrals, the latter being a suitable mathematical formalism for describing the classical laws of point dynamics. This connexion is evident from the fact that, in order to quantize a dynamical system, one has first to know what the pairs of canonically conjugate variables are, and also from the fact that the quantum laws, if expressed in terms of commutation brackets, have exactly the same form as the classical laws, if expressed in terms of Poisson brackets. In dealing with the quantization of the dynamics of continuous media, e.g. the electromagnetic and other fields, which has been developed along different lines, following Heisenberg and Pauli (1929), it seems tempting to try to base the method of quantization on an extended Hamilton-Jacobi theory of the calculus of variations for multiple integrals, the latter being the appropriate formalism for describing most relevant systems of continuum physics. In a previous paper (Weiss 1936), referred to as I, such an attem pt has been made, by extending the notion of pairs of conjugate variables. That attem pt led to quantum relations on an arbitrary space-like section in space-time, provided that one postulated that, if the space-like section becomes especially a space section, the quantum relations should go over into those of Heisenberg and Pauli.
The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.
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.
In this paper, we show that the Klein-Gordon quantum-mechanical operator, operating on the function ${e}^{\frac{\mathrm{iS}}{\ensuremath{\hbar}}}$, where SSS is the relativistic action of a free particle or of a particle in a field defined by a four vector, is completely equivalent to the relativistic Hamilton-Jacobi equation, provided one takes into account the vanishing of the divergence of the energy-momentum four vector. From this we see that classical relativistic mechanics can be formulated in terms of operators which are identical with those used in quantum mechanics.
An ordinary unambiguous integral representation for the finite propagator of a quantum system is found by starting of a privileged skeletonization of the functional action in phase space, provided by the complete solution of the Hamilton-Jacobi equation. This representation allows to regard the propagator as the sum of the contributions coming from paths where the momenta generated by the complete solution of the Hamilton-Jacobi equation are conserved -as it does happen on the classical trajectory-, but are not restricted to having the classical values associated with the boundary conditions for the original coordinates.