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arXiv:0810.5622 · Progress of Theoretical Physics

Stationary Phase and the Theory of Measurement: 1/N Expansion

R. Fukuda

Published Dec 1, 2009• 1 citation
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Abstract

The measuring process is studied, where a macroscopic number N of particles in the detector interact with the object. When N → ∞, the fluctuation of the object between different eigenstates of the operator O to be measured is suppressed, frozen to one and the same state while the detector is on. During this period, the stationary phase accompanying the macrovariable is established to have a one to one correspondence with the eigenvalue of O. A model is studied which produces the ideal result when N → ∞ and the correction terms are calculated in powers of 1/N. It is identical to the expansion including the fluctuation of the object successively.

DOI
10.1143/ptp.122.1405

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The measuring process is studied, where a macroscopic number N of particles in the detector interact with the object. When N → ∞, the fluctuation of the object between different eigenstates of the operator O to be measured is suppressed, frozen to one and the same state while the detector is on. During this period, the stationary phase accompanying the macrovariable is established to have a one to one correspondence with the eigenvalue of O. A model is studied which produces the ideal result when N → ∞ and the correction terms are calculated in powers of 1/N. It is identical to the expansion including the fluctuation of the object successively.
Abstract Any microscopic object is assumed to be a point-like particle, fluctuating quite randomly in the time scale τ\tauτ, which is much smaller than the smallest time interval Δ\DeltaΔ we can discriminate. The trajectories in Δ\DeltaΔ scale are defined. They are exclusive of each other since they originate from one particle having a position which is an objectively existing value. The measurement of the object is performed by the wave function, which is defined by summing up trajectories in Δ\DeltaΔ scale together with the phase factor. The wave function is thus a linear combination of exclusive paths. The measurement is performed by the macrovariable denoted as X. It loses the fluctuation by the stationary phase mechanism. Using a simple model to measure the operator O of the object, it is shown that, when the detector is switched on, the stationary phase trajectory of X branches into many trajectories, each of which is mapped to the eigenstate of O. Although these trajectories are all summed up for the wave function, in the actual experiment X takes one stationary trajectory. Therefore, the object is frozen to the corresponding eigenstate and realizes the reduction. The process of the reduction is shown to be realized with the correct frequency when the experiment is repeated. A model of the position measurement is also studied and the estimation of the magnitude of the upper bound of Δ\DeltaΔ is given based on the present experimental data.
The measuring process of an object system and a detector are studied in the von Neumann model and one of its variants, in which the detector consists of a single particle. If the states of the particle after the interaction with the object system are described partly by coherent states, then the particle can play the role of a detector under certain conditions. The mass (or free Hamiltonian) of the detector does not affect the measuring process at all, while the mass of the object system plays an important role. In the von Neumann model, if the mass of the object system is quite large, the position measurement can be considered, and under certain conditions the statistical operator \ensuremath{\rho} tends to just its mixed-state part, i.e., quantum interference disappears. However, when the time after the interaction becomes very large, interference emerges again. On the other hand, if the mass of the object system is small, we can even consider the momentum measurement and show that \ensuremath{\rho} approaches its mixed-state part when the time becomes very large, i.e., \ensuremath{\rho} is equivalent to its mixed-state part. The momentum can also be measured almost completely in the variant model; the operator \ensuremath{\rho} becomes equivalent to its mixed-state part. The different quantum states distinguish the detectors from the object systems.
We present and discuss some models for the quantum measuring process. Our models indicate that the coherent state approach to classical limits in quantum theories becomes a powerful method for a unified description of various measuring processes. Each detector in the models has a parameter N , the number of its dynamical variables; the whole process of an interaction between an object system and the detector is calculable for any finite number N . It is shown that the detector states after the interaction are characterized by a macroscopic parameter, which is closely connected with the “reliability” of the detector. The reliability plays an essential role in eliminating the quantum interference between two states corresponding to two different positions of the pointer. Finally, the quantum mechanics of the detector reduces to classical mechanics on suitable “phase space” as N becomes large, as is required.
In this paper, we discuss the importance of measurement in quantum mechanics and the so-called measurement problem. Any quantum system can be described as a linear combination of eigenstates of an operator representing a physical quantity; this means that the system can be in a superposition of states that corresponds to different eigenvalues, i.e., different physical outcomes, each one incompatible with the others. The measurement process converts a state of superposition (not macroscopically defined) in a well-defined state. We show that, if we describe the measurement by the standard laws of quantum mechanics, the system would preserve its state of superposition even on a macroscopic scale. Since this is not the case, we assume that a measurement does not obey to standard quantum mechanics, but to a new set of laws that form a “quantum measurement theory”.
A quantum-mechanical measurement process is analyzed in terms of a model Hamiltonian describing the interaction between a quantum system (a ``particle'') and a macroscopic apparatus (a ``detector''), which is assumed to be made up of N two-level elementary constituents (``molecules''). The description of the molecule locations introduces an effective fluctuating coupling constant, and this provokes a loss of quantum-mechanical coherence in the limit of large N. It is argued that coherence is lost statistically, as a result of the interaction: The collapse of the wave function is indeed obtained when the same experiment is performed many times, as a result of the microscopic differences among macroscopically identical initial states of the detector. In this way, insight is obtained into the mechanism engendering the loss of coherence suffered by a quantum-mechanical system when interacting with a macroscopic apparatus, and the concept of ``wave-function collapse'' is replaced by that of a statistically defined dephasing process. No classical behavior of the detection system is postulated and the presence of no external observer is required.