The Interpretation of Quantum Mechanics and the Measurement Process.
Douglas Kutach Department of Philosophy Rutgers University
Peter Mittelstaedt defines his project as spelling out the interpretational consequences of the claims (1) that quantum mechanics (QM) is universally true and (2) that measuring devices are physical systems, and hence fully describable by QM. This reasonable stance - which rules out QM breaking down at the classical level, conscious observers playing some special role in the interpretation, and a physically critical distinction between measurements and non-measurements - has been adopted by many participants in debates over the relative merits of various interpretations. Thus, much of the ground Mittelstaedt covers has been trodden before. In fact, this book is half a textbook on some formal results in quantum-mechanical measurement and half a rediscovery of the measurement problem. Nevertheless, what Mittelstaedt understands as the claims of quantum mechanics leads him to conclusions that are not at all widely accepted, providing the philosophical reader plenty of material for fruitful debate.
The text moves at a steady pace and is moderately technical, presupposing for most of the discussion that the reader has a basic understanding of the QM formalism. There are, in addition, several proofs demanding more advanced knowledge, but readers unfamiliar with the more difficult mathematics can grasp the essential points with the help of Mittelstaedtis lucid summaries of the technical results.
Mittelstaedt has several goals. First, he wants to specify some of the constraints placed on interpretations by QM. Specifically he tries to establish that probabilities in QM should be interpreted as relative frequencies among N distinct, identical systems in the limit as N approaches infinity. Second, he wants to clarify the measurement problem and the options it forces upon an interpreter. Third, he wants to determine the relevance, if any, of the self-referentiality of QM to liar paradoxes and Godel's incompleteness results. Due to the book's lack of emphasis on the third aim, I set aside its discussion.
In order to accomplish these tasks, Mittelstaedt concentrates on three interpretations which he thinks are sufficient to bring out the essential issues: the many-worlds interpretation, the minimal interpretation, and the realistic interpretation. The many-worlds interpretation is familiar enough to readers versed in QM interpretational issues, and anyone looking for new insights into it will be disappointed by the lack of any recognition (and hence discussion) of problems with this view. As Mittelstaedt sees it, the many-worlds interpretation makes no additional assumptions beyond those of QM itself, and 'can be read off from the formalism'. He thus concludes that there is no way 'to escape the strange consequence of many really existing worlds', if one leaves QM unmodified. Mittelstaedt defines one of his two new interpretations, the minimal interpretation, by adding to QM the postulate that after measurement processes, the states of measuring devices are well defined or objective. The other, the realistic interpretation, is defined by adding to the minimal interpretation the postulate that after a measurement of an observable A, the system assumes some objective value of A. The minimal and realistic interpretations are the primary objects of discussion throughout the book, in great part because they reveal the implications of the objectification of measurement.
Supposedly, an interpretation of a theory tells what the theory is about. Yet, by identifying the minimal and realistic interpretations only through the addition of statements to the effect that some or other element is objective, Mittelstaedt leaves undetermined their ontology. Is there something to being an objective measurement result above and beyond having a wave function of some appropriate kind? What happened to the infinite ensemble of worlds that existed when we didn't assume the objectivity of measurement outcomes? Fortunately, Mittelstaedt's analysis of the measurement problem allows one to piece together his view. Assumed throughout is that QM commits us to at least two claims: the wave function describing a system is a complete description of the physics of that system, and the wave function of a system always evolves according to the familiar Schrodinger equation. In order to connect up the QM formalism with reality as we know it, some additional assumptions must come into play. As a partial step in that direction, Mittelstaedt considers several ways of understanding how a physical system can have a property. The traditional way of understanding properties is as follows: a system's having a property amounts to that system's wave function being an eigenstate of the operator associated with that property. One weaker condition Mittelstaedt considers is that the system has a property just by having the appropriate eigenvalue of the operator. Mittelstaedt then demonstrates that under the realistic and minimal interpretations, these two conditions are equivalent.
With this condition on property possession, one has all the ingredients for a measurement problem. If one starts with measuring devices and measured things which are physical - hence completely describable by QM - and then considers what happens to them as they interact, one can see that typically the QM state for the measuring device plus measured system after an interaction is a superposition state not equal to any eigenstate corresponding to the measuring devices having, for instance, a definite pointer position. Since the measuring device has a definite pointer position if the system is in one of those eigenstates, it doesn't have any objective pointer position. This result, combined with the observation that measuring devices typically do have some objective pointer position (or at least it looks that way prima facie), just is the measurement problem.
Given the importance Mittelstaedt places on objectification and the depth of his demonstrations that objectification leads to the measurement problem, it would seem important to justify his conclusion that 'the objectification of measurement results leads to inconsistencies that cannot be resolved in an obvious way'. His discussion of the measurement problem, however, leaves out important possible solutions. The point of considering the minimal and realistic interpretations should have been that they imply that either there is some element of reality missed by the wave function description or that the wave function does not always undergo unitary evolution. Yet very little consideration is given to the possibility of additional variables that could provide an objective measurement outcome. Well-known modifications of QM like Bohm's theory are not mentioned at all, and various modal interpretations are set aside from the beginning in part because the objectivity they allow is contextual. Furthermore, consideration of modification of the QM dynamics is given only brief consideration. The Ghirardi, Rimini, and Weber proposal to allow wave-function collapse is considered but is rather quickly set aside without a strong case against it. Thus, the range of various major contending interpretations which purport to solve the measurement problem is not dealt with in depth.
Mittelstaedt's aim of establishing the proper interpretation of quantum mechanical probabilities suffers as well. He presents a number of proofs which demonstrate that the probability of a system's being in the [k.sup.th] eigenstate of an operator B is equal to the [k.sup.th] value of a relative frequency observable that pertains to an ensemble of N pointer values obtained by B-measurements on N equally prepared systems in the limit as N [approaches] [infinity]. Mittelstaedt claims that the proper referents of QM probability claims are these ensembles and not, for example, the likelihood that individual quantum systems, if B-measured will result in a particular value. The problem is that in real life, our evidence for the truth of quantum mechanics is composed of finite numbers of measurements of individual systems. What reason do we have for thinking that the frequencies we observe in individual systems will match the values of the relative frequency observable? It seems that any connection between the two will raise the issue again as to how best to interpret the probabilities used in QM.
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|Publication:||The British Journal for the Philosophy of Science|
|Article Type:||Book Review|
|Date:||Dec 1, 1998|
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