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Can the Quantum Measurement Problem be resolved within the framework of Schroedinger Dynamics and Quantum Probability?

Geoffrey L. Sewell

Published Nov 21, 2012• 2 citations
OpenAlex

Abstract

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.

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Can the Quantum Measurement Problem be Resolved within the Framework of Schroedinger Dynamics and Quantum Probability?

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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, J, 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 J, 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 J, up to utterly neglible corrections that decrease exponentially with N.

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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.
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, J, 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 J, 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 J, up to utterly neglible corrections that decrease exponentially with N.
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.
One of the fundamental conceptual difficulties in quantum mechanics arises from the fact that two of the axioms of the theory concerning the evolution of a microscopic system are contradictory. The first axiom assumes that the state of a quantum system evolving undisturbed by any observer obeys a linear deterministic law expressed by the SchrSdinger equation. The second one (wave-packet reduction principle) states that, when the quantum system interacts with an apparatus devised to measure some of its dynamical variables, its state changes abruptly and stochastically, jumping into a normalized eigenstate of the measured variable. This evolution is not deterministic nor linear and cannot be accounted for by the SchrSdinger equation. As a consequence the internal coherence of the theory becomes questionable. On one side the influence of the apparatus on the quantum system cannot be described as an interaction of the quantum constituents of the measuring macroscopic apparatus with the measured quantum system. On the other side, the impossibility of considering a macroscopic object as made by quantum constituents, i .e . the difficulty of quantum mechanics in describing the behaviour of macroscopic systems, is even more dramatic if we consider the evolution of the closed system consisting of the macroscopic apparatus and of the microscopic system. In such a case the appearance of superpositions of macr0scopically distinguishable states related to the apparatus cannot be avoided. These superpositions contradict both our classical experience and wave-packet reduction principle, which provides for statistical mixtures.
This is the first of two papers showing that the quantum problem of measurement remains unsolved even when the initial state of the apparatus is described by a statistical operator and when the results of measurement have a small probability of being erroneous. A realistic treatment of the measurement of observables of microscopic objects (e.g., the position or the spin of an electron) by means of observables of macroscopic apparatus (e.g., the position of a spot on a photographic plate) requires the consideration of errors. The first paper considers measurement procedures of the following type: An initial eigenstate of the object observable leads to a final statistical operator of the object plus apparatus which describes a mixture of “approximate” eigenstates of the apparatus observable. It is proved that each of a large class of initial states leads to a final statistical operator which does not describe any mixture containing even one “approximate” eigenstate of the apparatus observable. INTRODUCTION Several writers have tried to solve the quantum-mechanical problem of measurement through one or both of the following proposals: (a) describthe initial state of the measuring apparatus by a projection onto a subspace of the associated Hilbert space or by a statistical operator, thus taking into account the practical impossibility of knowing the exact quantum state of a macroscopic object; (b) recognizing that there may be some physical inaccuracy, such as a small error in the position of a pointer needle, in the final registration of the outcome of the measurement by the apparatus.
We study a class of quantum measurement models. A microscopic object is entangled with a macroscopic pointer such that each eigenvalue of the measured object observable is tied up with a specific pointer deflection. Different pointer positions mutually decohere under the influence of a bath. Object-pointer entanglement and decoherence of distinct pointer readouts proceed simultaneously. Mixtures of macroscopically distinct object-pointer states may then arise without intervening macroscopic superpositions. Initially, object and apparatus are statistically independent while the latter has pointer and bath correlated according to a metastable local thermal equilibrium. We obtain explicit results for the object-pointer dynamics with temporal coherence decay in general neither exponential nor Gaussian. The decoherence time does not depend on details of the pointer-bath coupling if it is smaller than the bath correlation time, whereas in the opposite Markov regime the decay depends strongly on whether that coupling is Ohmic or super-Ohmic.
Abstract The so-called quantum measurement problems are solved from a new perspective. One of the main observations is that the basic entities of our world are particles, elementary or composite. It follows that each elementary process, hence each measurement process at its core, is a spacetime, pointlike, event. Another key idea is that, when a microsystem ψ gets into contact with the experimental device, factorization of ψ rapidly fails and entangled mixed states appear. The wave functions for the microsystem-apparatus coupled system for different measurement outcomes then lack overlapping spacetime support. It means that the aftermath of each measurement is a single term in the sum: a “wave-function collapse”. Our discussion leading to a diagonal density matrix, ρ = diag(|c 1|2,…,|cn |2,…) shows how the information encoded in the wave function <?CDATA $|\psi \rangle =\displaystyle {\sum }_{n}{c}_{n}|n\rangle $?> | ψ 〉 = ∑ n c n | n 〉 gets transcribed, via entanglement with the experimental device and environment, into the relative frequencies Pn = |cn |2 for various experimental outcomes F = fn . Our discussion represents the first, significant steps towards filling in the logical gaps in the conventional interpretation based on Born’s rule, replacing it with a clearer understanding of quantum mechanics. Accepting objective reality of quantum fluctuations, independent of any experiments, and independently of human presence, one renounces the idea that in a fundamental, complete theory of Nature the result of each single experiment must necessarily be predictable.