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arXiv:cond-mat/9705029

Coupled Harmonic Oscillators and Feynman's Rest of the Universe

D. Han, Y. S. Kim, Marilyn E. Noz

Published May 4, 1997• 4 citations
arXivHTMLPDF

Abstract

According to Feynman, the universe consists of two parts - the system in which we are interested and the rest of the universe which our measurement process does not reach. Feynman then formulates the density matrix in terms of the observable world and the rest of the universe. It is shown that coupled harmonic oscillators can serve as an illustrative example for Feynman's ``rest of the universe.'' It is pointed out that this simple example has far-reaching consequences in many branches of physics, including statistical mechanics, measurement theory, information theory, thermo-field dynamics, quantum optics, and relativistic quantum mechanics. It is shown that our ignorance of the rest of the universe increases the uncertainty and entropy in the system in which we are interested.

DOI
10.48550/arxiv.cond-mat/9705029

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According to Feynman, the universe consists of two parts - the system in which we are interested and the rest of the universe which our measurement process does not reach. Feynman then formulates the density matrix in terms of the observable world and the rest of the universe. It is shown that coupled harmonic oscillators can serve as an illustrative example for Feynman's ``rest of the universe.'' It is pointed out that this simple example has far-reaching consequences in many branches of physics, including statistical mechanics, measurement theory, information theory, thermo-field dynamics, quantum optics, and relativistic quantum mechanics. It is shown that our ignorance of the rest of the universe increases the uncertainty and entropy in the system in which we are interested.
Coupled harmonic oscillators occupy an important place in physics teaching. It is shown that they can be used for illustrating an increase in entropy caused by limitations in measurement. In the system of coupled oscillators, it is possible to make the measurement on one oscillator while averaging over the degrees of freedom of the other oscillator without measuring them. It is shown that such a calculation would yield an increased entropy in the observable oscillator. This example provides a clarification of Feynman’s rest of the universe.
If we reduce coherence in a given quantum system, the result is an increase in entropy. Does this necessarily convert this quantum system into a classical system? The answer to this question is No. The decrease of coherence means more uncertainty. This does not seem to make the system closer to classical system where there are no uncertainties. We examine the problem using two coupled harmonic oscillators where we make observations on one of them while the other oscillator is assumed to be unobservable or to be in Feynman's rest of the universe. Our ignorance about the rest of the universe causes an increase in entropy. However, does the system act like a classical system? The answer is again No. When and how does this system appear like a classical system? It is shown that this paradox can be resolved only if measurements are taken along the normal coordinates. It is also shown that Feynman's parton picture is one concrete physical example of this decoherence mechanism.
If we reduce coherence in a given quantum system, the result is an increase in entropy. Does this necessarily convert this quantum system into a classical system? The answer to this question is No. The decrease of coherence means more uncertainty. This does not seem to make the system closer to classical system where there are no uncertainties. We examine the problem using two coupled harmonic oscillators where we make observations on one of them while the other oscillator is assumed to be unobservable or to be in Feynman’s rest of the universe. Our ignorance about the rest of the universe causes an increase in entropy. However, does the system act like a classical system? The answer is again No. When and how does this system appear like a classical system? It is shown that this paradox can be resolved only if measurements are taken along the normal coordinates. It is also shown that Feynman’s parton picture is one concrete physical example of this decoherence mechanism.
Two illustrative examples are given for Feynman's rest of the universe. The first example is the two-mode squeezed state of light where no measurement is taken for one of the modes. The second example is the relativistic quark model where no measurement is possible for the time-like separation fo quarks confined in a hadron. It is possible to illustrate these examples using the covariant oscillator formalism. It is shown that the lack of symmetry between the position-momentum and time-energy uncertainty relations leads to an increase in entropy when the system is different Lorentz frames.
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement, decoherence, and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of by two-by-two matrices commonly called SL(2,c)SL(2,c)SL(2,c). Thus the coupled harmonic oscillators can therefore play the role of combining quantum mechanics with special relativity. Both Paul A. M. Dirac and Richard P. Feynman were fond of harmonic oscillators, while they used different approaches to physical problems. Both were also keenly interested in making quantum mechanics compatible with special relativity. It is shown that the coupled harmonic oscillators can bridge these two different approaches to physics.
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of two-by-two matrices commonly called SL(2,c)SL(2,c)SL(2,c). Thus the coupled harmonic oscillators can play the role of combining quantum mechanics with special relativity. It is therefore possible to relate the current issues of physics to the Lorentz-covariant formulation of quantum mechanics.