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arXiv:quant-ph/9710023

The Problem of Measurement and the Theory of Quantum State Reduction

Masanao Ozawa

Published Oct 7, 1997• 0 citations
arXivHTMLPDF

Abstract

A new approach to the problem of measurement in quantum mechanics is proposed. In this approach, the process of measurement is described in the Heisenberg picture and divided into two stages. The first stage is to transduce the measured observable to the probe observable. The second stage is to amplify the probe observable to the macroscopic meter observable. Quantum state reduction is derived, based on the quantum Bayes principle, from the object-apparatus interaction in the first stage. The dynamical process of the second stage is described as a quantum amplification with infinite gain based on nonstandard analysis.

DOI
10.48550/arxiv.quant-ph/9710023

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A new approach to the problem of measurement in quantum mechanics is proposed. In this approach, the process of measurement is described in the Heisenberg picture and divided into two stages. The first stage is to transduce the measured observable to the probe observable. The second stage is to amplify the probe observable to the macroscopic meter observable. Quantum state reduction is derived, based on the quantum Bayes principle, from the object-apparatus interaction in the first stage. The dynamical process of the second stage is described as a quantum amplification with infinite gain based on nonstandard analysis.
The central problem in the quantum theory of measurement, how to describe the process of state reduction in terms of the quantum mechanical formalism, is solved on the basis of the relativity of quantal states, which implies that once the apparatus is detected in a well-defined state, the object state must reduce to a corresponding one. This is a process termed by Schrodinger disentanglement. Here, it is essential to observe that Renninger's negative result does constitute an actual measurement process. From this point of view, Heisenberg's interpretation of his microscope experiment and the Einstein-Podolsky-Rosen arguments are reinvestigated. Satisfactory discussions are given to various experimental situations, such as the Stern-Gerlach-type experiment, successive measurements, macroscopic measurements, and Schrodinger's cat. Finally it is proposed to regard a state vector in quantum mechanics as an irreducible physical construct, in Margenau's sense, that is not further analyzable both mathematically and conceptually.
New theory of measurement is presented to show that quantum mechanics can describe the whole processes of quantum-mechanical measurement leading to the so-called reduction of wave packet if applied properly to the total system of object and apparatus. We introduce a simple model, where momenta of particles are measured using a perfect rigid mirror, to yield the reduction of wave packet in a clear-cut way. Microscopic uncertainties of macroscopic quantities and the dynamical scattering theory play the essential roles. Before entering into the main part, we briefly summarize the controversial points in the theory of measurement.
The question as to whether quantum mechanics can be applied to a macroscopic system, which has recently been raised through discussions of the problem of how to interpret the so-called “reduction of a wave packet”, is discussed. It is shown by a detailed analysis of the destruction of interference which brings about the reduction, that a supplementary assumption on the preciseness of measurements must at least be introduced in order to interpret the reduction process as an objective process, and it does not, however, necessarily imply inapplicability of quantum mechanics to a macroscopic system.
Abstract The measurement problem in quantum mechanics still appears to be an unresolved issue. Here we present a new quantum theory of measurement that overcomes many of the difficulties previously found. It is based on a consistent use of the linear superposition principle and distinguishes two aspects: recording and observation. A recording elicits the full interaction of the object quantum system with the quantum measuring apparatus. No wavefunction collapse is introduced. Statistics may appear at the observation of the recording only and depends on filtering processes. The theory presented here uses the existing mathematical structure of quantum mechanics but requires no ad hoc measurement postulates. Well‐known paradoxical aspects in standard quantum mechanics, for instance, wave–particle duality, Schrödinger's cat, and Zeno effects do not appear in the current formulation. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
I start from the observation that the usual treatment of the object and the apparatus as independent systems in discussions of the quantum measurement process is not obviously consistent with a field theoretic picture. In such connection I consider a non relativistic second quantization model in which the role of the object is played by a particle out of the sea of the other particles in a macroscopic equilibrium state of the global system and the role of the apparatus index by the number of ions produced in certain given regions. In the framework of a formalism developed in preceding papers and in connection with the problem of decoherence, I also consider a specific, even if rather artificial, prescription to introduce a formal continuous monitoring of the particle distribution functions which would give an objective character to the macroscopic variables.
The problem of measurement taken at face value shows clearly that there is an inconsistency inside the quantum formalism.The phenomenon of decoherence is often presented as a solution to it.A widely debated question is to decide between two different interpretations.The first one is to consider that the decoherence process has the effect to actually project a superposed state into one of its classically interpretable component, hence doing the same job as the reduction postulate.For the second one, decoherence is only a way to show why no macroscopic superposed state can be observed and so, to explain the classical appearance of the macroscopic world, while the quantum entanglement between the system, the apparatus and the environment never disappears.In this case, explaining why only one single definite outcome is observed remains to do.In this paper, we examine arguments for and against both interpretations and defend a position according to which the outcome that is observed is relative to the observer in close parallel to the Everett interpretation.