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Incompatibility of Standard Completeness and Quantum Mechanics

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Abstract

The completeness of quantum mechanics (QM) is generally interpreted to be or entail the following conditional statement (called standard completeness (SC)): If a QM system S is in a pure non-eigenstate of observable A, then S does not have value a k of A at t (where a k is any eigenvalue of A). QM itself can be assumed to contain two elements: (i) a formula generating probabilities; (ii) Hamiltonians that can be time-dependent due to a time-dependent external potential. It is shown that, given (i) and (ii), QM and SC are incompatible. Hence, SC is not the appropriate interpretation of the completeness of QM.

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Notes

  1. In particular, one might point out that we often seem to meet predictions without such references, e.g. when we say that a fair coin has probability 1/2 for landing heads up without being able or willing to specify when it will land heads up. Could not QM deliver such predictions? This objection is discussed in philsciarchive.pitt.edu/9080/, Sect. 3, where it is shown that a serious physical theory of the coin toss (as presented in, e.g., [16]) can deliver probabilities for time-indexed events—in contrast with QM & SC.

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  1. Philosophisches Seminar, Universität Erfurt, Postfach 900221, 99105, Erfurt, Germany

    Carsten Held

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Held, C. Incompatibility of Standard Completeness and Quantum Mechanics. Int J Theor Phys 51, 2974–2984 (2012). https://doi.org/10.1007/s10773-012-1179-6

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