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Canadian Journal of Physics

Formal quantization of a chaotic model problem

R. A. Moore

Published Jan 1, 1999• 1 citation
Publisher

Abstract

A model problem, initially proposed by Feynman and Hibbs, is re-examined in this work. It is shown that the equation of motion is deterministic, in spite of its double time delay character, and has chaoticlike solutions. It is further shown that this model is amenable to the methods of higher than first-order Lagrangians, yielding a series of approximations to the initial model. Thus, for each approximation, a complete st of canonical variables can be defined, the Hamiltonian can be written down in canonical form and the conventional quantization rules applied, leading to exact quantizations. Extended sets of basis vectors follow along with the definition of physical state vectors, which are seen to contain all of the classical information. Finally, these results are used to give a possible explanation of the electron-position resonances seen in heavy-ion collision experiments that is consistent with the negative results found in low-energy electron scattering as well as other heavy-ion collision experiments.PACS Nos.: 03.20.+i, 03.65.-w, 05.45.+6

DOI
10.1139/cjp-77-3-221

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A model problem, initially proposed by Feynman and Hibbs, is re-examined in this work. It is shown that the equation of motion is deterministic, in spite of its double time delay character, and has chaoticlike solutions. It is further shown that this model is amenable to the methods of higher than first-order Lagrangians, yielding a series of approximations to the initial model. Thus, for each approximation, a complete st of canonical variables can be defined, the Hamiltonian can be written down in canonical form and the conventional quantization rules applied, leading to exact quantizations. Extended sets of basis vectors follow along with the definition of physical state vectors, which are seen to contain all of the classical information. Finally, these results are used to give a possible explanation of the electron-position resonances seen in heavy-ion collision experiments that is consistent with the negative results found in low-energy electron scattering as well as other heavy-ion collision experiments.PACS Nos.: 03.20.+i, 03.65.-w, 05.45.+6
A model problem consisting of two isolated particles with a mutual interaction depending upon retarded and advanced positions and constrained to one-dimensional motion is analyzed herein. This system has a number of features in common with a previously considered one-particle double-delay model problem. The equations of motion have closed-form solutions and the systems are both deterministic and causal. In both cases, standard low-order approximations to the exact problem, in general, have spurious solutions, which must be recognized and removed in order to extract the physically meaningful parts and to proceed reasonably with quantization. In addition, the two-particle system has specific characteristics. The center-of-momentum motion separates out. The dependence of the solutions on the mass ratio of the two particles can be examined. Further, not only does the relative motion have the ordinary solution, in which the two particles move out of phase, it can also have an extraordinary solution, in which they move in phase with each other. Finally, the total generalized linear momentum and Hamiltonian can be evaluated and seen to be constants of the motion.
In a time-symmetric form, classical relativistic formulations involve not only a reference time but also relatively advanced and retarded times. This work examines a simple model problem having this structure with the equations of motion having closed-form solutions. The complexity of more realistic models is avoided while the general features are retained. The determinism and casuality of the theory are clearly demonstrated. Some problems associated with the quantization of power series representations using conventional procedures are illustrated and discussed.
If higher order wave equations are reduced to first order by supplementary conditions, then quadratic Hamiltonians of the most general type will emerge. Even then, for several reasons elaborated below, canonical quantization can still not be applied consistently. The benefit of clinging to a Lagrangian description of non-localized phenomena is then questionable. Non canonical quantization procedures should be considered. Some initial attempts at Lie-admissible quantization are examined critically here, and it is concluded that further recourse to experiment seems to be necessary.
In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schrödinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical Hamiltonian in the NNN-dimensional metric configuration space defined by the classical kinetic energy of the system, with the classical conjugate momentum NNN-vector replaced by −iℏ-i\hbar−iℏ times the vector gradient operator in that space. We obtain these results by using conservation of probability, general tensor calculus, the Madelung transform, the Ehrenfest theorem and/or the Hamilton-Jacobi equation, and comparison with results for the charged harmonic oscillator in stochastic electrodynamics. We also provide two illustrative examples and a discussion of how coordinate trajectories could be compatible with wave properties such as interference, diffraction, and tunneling.
Nonperturbative analytical and numerical methods are presented for the solution of the coupled nonlinear Maxwell-Schr\"odinger equations. The theory has been derived within the Hamiltonian or canonical formalism. The canonical approach to dynamics, starting from the Maxwell and Schr\"odinger Lagrangians with a Lorenz gauge fixing term, yields a set of first order Hamilton equations. They form a well-defined initial value problem. The Maxwell-Schr\"odinger equations of motion are then represented in a spatial basis of Gaussian functions. In the limit of a complete basis this representation is exact. For any choice of finite basis it provides an approximate system of dynamical equations that can be integrated in time and made systematically more accurate by enriching the basis. The basis form of the theory has been implemented numerically and is used to investigate the dynamics of a single nonrelativistic spinless one electron atom interacting with the electromagnetic modes of a cavity.