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arXiv:1610.06066

Pointer States and Decoherence in Quantum Unitarity: Energy Conservation for Weak Interaction Model

Kentaro Urasaki

Published Oct 19, 2016• 0 citations
arXivHTMLPDF

Abstract

The purpose of the present paper is to derive the pointer states of a macro-object using a simple perturbation method. We study the model Hamiltonian involving the weak interaction between the center of mass and its environment. The main conclusion is that the pointer states emerge as the small part of the total state vector when the interaction Hamiltonian is not dominant: the degrees of freedom being at the center of interaction can be localized. In the weak interaction case, if the disappearance of the energy contribution from the off-diagonal terms is caused by the decoherence, it threatens the conservation law. In this study, however, the energy conservation reasonably holds, since the non-classical (i.e., normal quantum) states give main contribution to it. The emergence of the pointer states also implies the special initial state in which the localized state makes the deep well of the potential. In the result, the unitary time evolution of a pointer state is very different from that of the majority of the original state vector of the total system.

DOI
10.48550/arxiv.1610.06066

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The purpose of the present paper is to derive the pointer states of a macro-object using a simple perturbation method. We study the model Hamiltonian involving the weak interaction between the center of mass and its environment. The main conclusion is that the pointer states emerge as the small part of the total state vector when the interaction Hamiltonian is not dominant: the degrees of freedom being at the center of interaction can be localized. In the weak interaction case, if the disappearance of the energy contribution from the off-diagonal terms is caused by the decoherence, it threatens the conservation law. In this study, however, the energy conservation reasonably holds, since the non-classical (i.e., normal quantum) states give main contribution to it. The emergence of the pointer states also implies the special initial state in which the localized state makes the deep well of the potential. In the result, the unitary time evolution of a pointer state is very different from that of the majority of the original state vector of the total system.
The purpose of the present study is to derive the pointer states of a macroscopic system interacting with its environment, under the general assumptions, i.e., without assuming any form of the interaction Hamiltonian. The lowest order perturbation leads that the interaction energy shifts the phase factors of the state vectors. For a macroscopic system, these factors are the macroscopic quantities even for the very weak interaction. When we group the state vector of the total system by the view point of environmental side, the destructive interference occurs and the stationary phase approximation can be adopted. Only the pointer states then survive and the decoherence also occurs. The present approach is within the standard quantum mechanics as same as the standard decoherence theory, but the meaning of the classical state is much clear.
We have studied the emergence of classical states in the perturbative interaction model. The states which interact with many other degrees of freedom, such as the center of mass of a macro-object, play important role. Although the random phase mechanism is effective as same as Zurek's strong correlation model, there are enormous states, each of which independently developes due to the orthogonality of the environmental states. In these privileged states, the subsystem picture with the separability is stable.
Abstract In quantum mechanics, pointer states are eigenstates of the observable of the measurement apparatus that represent the possible positions of the display pointer of the equipment. The origin of this concept lies in attempts to fill the blanks in Everett’s relative-state interpretation, and to make it a fully valid description of physical reality. To achieve this, it was necessary to consider not only the main system interacting with the measurement apparatus (like von Neumann and Everett did) but also the role of the environment in eliminating correlations between different possible measurements when interacting with the measurement apparatus. The interaction of the environment with the main system (and the measurement apparatus) is the core of the decoherence theory, which followed Everett’s thesis. In this article, we review the measurement process according to von Neumann, Everett’s relative state interpretation, the purpose of decoherence and some of its follow-up until Wojciech Zurek’s primordial paper that consolidated the concept of pointer states, previously presented by Heinz Dieter Zeh. Employing a simple physical model consisting of a pair of two-level systems—one representing the main system, the other the measurement apparatus—and a thermal bath—representing the environment—we show how pointer states emerge, explaining its contributions to the question of measurement in quantum mechanics, as well as its limitations. Finally, we briefly show some of its consequences. This paper is accessible to readers with elementary knowledge about quantum mechanics, on the level of graduate courses.
Abstract: In quantum mechanics, pointer states are eigenstates of the observable of the measurement apparatus that represent the possible positions of the display pointer of the equipment. The origin of this concept lies in attempts to fill the blanks in Everett's relative-state interpretation, and to make it a fully valid description of physical reality. To achieve this, it was necessary to consider not only the main system interacting with the measurement apparatus (like von Neumann and Everett did) but also the role of the environment in eliminating correlations between different possible measurements when interacting with the measurement apparatus. The interaction of the environment with the main system (and the measurement apparatus) is the core of the decoherence theory, which followed Everett's thesis. In this article, we review the measurement process according to von Neumann, Everett's relative state interpretation, the purpose of decoherence and some of its follow-up until Wojciech Zurek's primordial paper that consolidated the concept of pointer states, previously presented by Heinz Dieter Zeh. Employing a simple physical model consisting of a pair of two-level systems-one representing the main system, the other the measurement apparatus-and a thermal bath-representing the environment-we show how pointer states emerge, explaining its contributions to the question of measurement in quantum mechanics, as well as its limitations. Finally, we briefly show some of its consequences. This paper is accessible to readers with elementary knowledge about quantum mechanics, on the level of graduate courses
It is well-known that the pointer basis of a quantum system satisfies the condition to diagonalize the interaction Hamiltonian between the subsystems. We show that this condition can be translated into the form δΛ=0,δΛ=0,δΛ=0, where ΛΛΛ, so-called the action, is the time integrated interaction energy: it is found out naturally in the phase of state vectors due to diagonal interaction. The careful treatment of a two states system demonstrates that the states of the total system branch into the states with different values of the action. Mathematically the pointer states are selected out by the saddle point condition on the phase ΛΛΛ. This study helps us to understand the precise mechanism and the general dynamics of decoherence.