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Physical Review A

Causality and quantization of time-delay systems: A two-body model problem

R. A. Moore, T. C. Scott

Published Dec 1, 1995• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.52.4371

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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.
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 pair of events, one for each of two particles, is regarded as a point in 8-space (V8{V}_{8}V8​). The technique of Hamilton's optical method is applied to V8{V}_{8}V8​, giving rays and associated de Broglie waves. These waves are quantized by associating a phase-cycle with an increase of hhh in the characteristic function, and hence quantum rules appear as resonance conditions on the de Broglie waves, when the part of V8{V}_{8}V8​ occupied by rays is multiply connected. A Hamiltonian equation is written down, the interaction between the particles being taken to be of Coulomb form, but with ordinary distance replaced by Minkowskian separation. This leads without approximation to a simple expression for quantized proper energies of the system, and from this expression follow the correct approximate energy levels for the hydrogenic atom (with ${Z}_{\ensuremath{\alpha}}$), the mass-correction factor being included. There is no approximation based on smallness of relative velocity or ratio of masses, and mathematical complexities associated with retarded potentials are avoided by the use of the relativistic Coulomb interaction.
A classical many-body problem composed of an infinite number of mass points coupled together by springs is quantized. The masses and the spring constants in this system are chosen in such a way that the motion of each particle is exponentially damped. Because of the quadratic form of the Hamiltonian, the many-body wave function of the system can be written as a product of two terms: a time-dependent phase factor which contains correlations between the classical motions of the particles, and a stationary state solution of the Schrödinger equation. By assuming a Hartree type wave function for the many-particle Schrödinger equation, the contribution of the time-dependent part to the single particle wave function is determined, and it is shown that the time-dependent wave function of each mass point satisfies the nonlinear Schrödinger–Langevin equation. The characteristic decay time of any part of the subsystem, in this model, is related to the stiffness of the springs, and is the same for all particles.
Two different formulations of the problem of time-delay caused by scattering from an oscillating center of force are studied. In the first one, appropriate for local interactions, a semi-classical time-dependent solution of the Schrödinger equation is obtained by writing the wave function as a solution of two coupled equations one of which is the Hamilton–Jacobi equation. These equations are then solved using the narrow wave packet approximation. The classical motion of the center of this wave packet is found for two specific cases, a square well and a Morse potential. In the second formulation a time-independent many-channel solution of the Schrödinger equation is obtained for an exactly solvable quantum-mechanical problem with a potential consisting of a separable attractive part which oscillates and a fixed rigid wall. The results show that the time-delay is extremely sensitive to the values of the amplitude and the frequency of oscillations, and also to the energy of the incident particle.