Introduction In a very elegant and rigorous paper, K. Hepp has discussed quantum measurement theory. He uses the C * algebra description of infinite quantum systems. Here an attempt is made to give a more popular account of some of his reasoning. Such an attempt seems worthwhile because many people not familiar with the C * algebra approach, and even somewhat intimidated by it, have been intrigued by the following statement in Hepp's abstract: In several explicitly soluble models, the measurement leads to macroscopically different ‘pointer positions’ and to a rigorous ‘reduction of the wave packet’ with respect to all local observables. This could look like a clean solution at last to the infamous measurement problem. But it is not so, nor thought by Hepp to be so. Here we will take one of his models and analyse it in elementary text-book terms. It will be insisted that the ‘rigorous reduction’ does not occur in physical time but only in an unattainable mathematical limit. It will be argued that the distinction is an important one. We will work at first in the Schrödinger picture, but later, with the extension to relativistic systems in mind, it will be argued that such considerations become particularly clear in the Heisenberg picture.
Introduction In a very elegant and rigorous paper, K. Hepp has discussed quantum measurement theory. He uses the C * algebra description of infinite quantum systems. Here an attempt is made to give a more popular account of some of his reasoning. Such an attempt seems worthwhile because many people not familiar with the C * algebra approach, and even somewhat intimidated by it, have been intrigued by the following statement in Hepp's abstract: In several explicitly soluble models, the measurement leads to macroscopically different ‘pointer positions’ and to a rigorous ‘reduction of the wave packet’ with respect to all local observables. This could look like a clean solution at last to the infamous measurement problem. But it is not so, nor thought by Hepp to be so. Here we will take one of his models and analyse it in elementary text-book terms. It will be insisted that the ‘rigorous reduction’ does not occur in physical time but only in an unattainable mathematical limit. It will be argued that the distinction is an important one. We will work at first in the Schrödinger picture, but later, with the extension to relativistic systems in mind, it will be argued that such considerations become particularly clear in the Heisenberg picture.
In this paper, a quantum dynamical model describing the quantum-measurement process is presented as an extensive generalization of the Coleman-Hepp model. In both the classical limit with very large quantum number and the macroscopic limit with very large particle number in the measuring instrument, this model generally realizes the wave-packet collapse in quantum measurement as a consequence of the Schr\"odinger time evolution in either the exactly solvable case or the non-exactly-solvable case. For the latter, the quasiadiabatic case is explicitly analyzed by making use of the high-order adiabatic-approximation method, which manifests the wave-packet collapse as well as in the exactly solvable case. By highlighting these analyses, it is finally found that an essence of the dynamical model of wave-packet collapse is the factorization of the Schr\"odinger evolution rather than the exact solvability. Therefore many dynamical models including the previous well-known ones, whether they are exactly solvable or not, can be shown to be only the concrete realizations of this factorizability.
We formulate the dynamics of the generic quantum system S_{c} comprising a microsystem S and a macroscopic measuring instrument I, whose pointer positions are represented by orthogonal subspaces of the Hilbert space of its pure states. These subspaces are simultaneous eigenspaces of a set of coarse grained intercommuting macroscopic observables and, most crucially, their dimensionalities are astronomically large, increasing exponentially with the number, N, of particles comprising I. We formulate conditions under which the conservative dynamics of S_{c} yields both a reduction of the wave packet describing the state of S and a one-to-one correspondence, following a measurement, between the pointer position of I and the resultant eigenstate of S; and we show that these conditions are fulfilled, up to utterly negligible corrections that decrease exponentially with N, by the finite version of the Coleman-Hepp model.
Abstract Precursing now popular ‘spontaneous reduction’ views, this chapter argues that the measurement problem in quantum mechanics is not a problem of measurement at all. Reductions happen in a variety of circumstances and are particularly important in the preparation of quantum states. But there is no reason to think there is any simple physical characterization of situations in which reduction of the wave packet occurs from when it does not. There is a difference in the mathematical characteristics of the evolution operator between the two (unitary operators versus non‐unitary), but this mathematical difference need mark no interesting physical difference distinguishing the situations to which operators apply.
By making use of Schwinger’s oscillator model of angular momentum, we put forward an interesting connection among three solvable Hamiltonians, widely used for discussions on the quantum measurement problem. This connection implies that a particular macroscopic limit has to be taken for these models to be physically sensible. PACS: 03.65.Fd; 03.80.+r; 42.52.+x; 03,65.BzThe quantum measurement problem [1] [2] has been a central issue in the foundations and interpretation of quantum mechanics, and is profoundly related to the investigation of the internal consistency of quantum theory. In order to find a resolution within the quantum mechanical framework, one usually describes a quantum measurement process in the following way: An elementary particle Q, on which the measurement is performed, interacts with a detection system D, which is considered to be made up of N elementary constituents. Generally, it is not easy to handle such a complicated system like Q + D, however, it is also true that several solvable models of measurement processes are now at hand. Among these are the Cini [3] and Coleman-Hepp [4] Hamiltonians and its modified version [5], which were all proposed
Once upon a time the well-known paradoxes of quantum mechanics (one might even say the “traditional” paradoxes of quantum mechanics!—I refer here to paradoxes such as the Schrödinger cat and the Einstein-Podolsky-Rosen Paradox) seemed especially difficult because of the rather bizarre behavior of the wave packet under the von Neumann theory of measurement, to which behavior these paradoxical situations seemed to draw especial attention. Quite recently quantum theoretical accounts of the measurement process have been appearing which sidestep direct reference to such things as “contraction” of the wave packet (see, for example, Krips and Jauch). With this development it has become increasingly popular to take a “cool” approach to these paradoxes, arguing that the new approach to measurement provides a consistent, nonbizarre account of the physical situations and that the paradoxicality was due largely to intuitive blunders—even plausible intuitive blunders—accentuated (but also disguised) by the unfortunate features of the von Neumann approach to measurement. Both Krips and Jauch, for example, adopt this approach.
Abstract We provide an affirmative answer to the question posed in the title. Our argument is based on a treatment of the Schroedinger dynamics of the composite, Sc, of a quantum microsystem, S, and a macroscopic measuring apparatus, I, consisting of N interacting particles. The pointer positions of this apparatus are represented by subspaces of its representative Hilbert space that are simultaneous eigenspaces of coarse-grained intercommuting macroscopic observables. By taking explicit account of their macroscopicality, we prove that, for suitably designed apparatus I, the evolution of the composite Sc leads both to the reduction of the wave-packet of S and to a one-to-one correspondence between the resultant state of this microsystem and the pointer position of I, up to utterly neglible corrections that decrease exponentially with N.