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Can Quantum Mechanics be shown to be Incomplete in Principle

Carsten Held

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

The paper presents an argument for the incompleteness in principle of quantum mechanics. I introduce four principles (P0–P3) concerning the interpretation of probability, in general and in quantum mechanics, and argue that the defender of completeness must reject either P0 or all of P1–P3, which options both seem unacceptable. The problem is shown to be more fundamental than the measurement problem and to have implications for our understanding of quantum-mechanical contextuality.

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The paper presents an argument for the incompleteness in principle of quantum mechanics. I introduce four principles (P0–P3) concerning the interpretation of probability, in general and in quantum mechanics, and argue that the defender of completeness must reject either P0 or all of P1–P3, which options both seem unacceptable. The problem is shown to be more fundamental than the measurement problem and to have implications for our understanding of quantum-mechanical contextuality.
Abstract It is sometimes claimed that quantum mechanics undermines realism—that it should be taken only as an instrumental recipe, not as a description of the world. If so, then quantum mechanics has no implications for metaphysics. However, this chapter makes the case for a realistic interpretation of quantum mechanics. The EPR argument that quantum mechanics is an incomplete description of the world is presented, together with Bohr’s response. Bell’s theorem is explained, which shows that, subject to some plausible assumptions, no completion of quantum mechanics is possible. However, those assumptions can be challenged, leaving open several avenues for defending a realistic interpretation of quantum mechanics.
This study posits that Bohr failed to defend the completeness of the quantum mechanical description of physical reality against Einstein–Podolsky–Rosen’s (EPR) paper. Although there are many papers in the literature that focus on Bohr’s argument in his reply to the EPR paper, the purpose of the current paper is not to clarify Bohr’s argument. Instead, I contend that regardless of which interpretation of Bohr’s argument is correct, his defense of the quantum mechanical description of physical reality remained incomplete. For example, a recent trend in studies of Bohr’s work is to suggest he considered the wave-function description to be epistemic. However, such an interpretation cannot be used to defend the completeness of the quantum mechanical description.
The standard axiomatization of quantum mechanics (QM) is not fully explicit about the role of the time-parameter. Especially, the time reference within the probability algorithm (the Born Rule, BR) is unclear. From a probability principle P1 and a second principle P2 affording a most natural way to make BR precise, a logical conflict with the standard expression for the completeness of QM can be derived. Rejecting P1 is implausible. Rejecting P2 leads to unphysical results and to a conflict with a generalization of P2, a principle P3. All three principles are shown to be without alternative. It is thus shown that the standard expression of QM completeness must be revised. An absolutely explicit form of the axioms is provided, including a precise form of the projection postulate. An appropriate expression for QM completeness, reflecting the restrictions of the Gleason and Kochen-Specker theorems is proposed.
It is shown that the Einstein-Podolsky-Rosen conclusion concerning the `incompleteness' of Quantum Mechanics is invalidated by two logical errors in their argument. If it were possible to perform the proposed gedanken experiment it would, in fact, show that Quantum Mechanics is `complete' for the observables discussed. Because, however, of the non square-integrable nature of the wave function, the proposed experiment gives vanishing probabilities for measurements performed in finite intervals of configuration or momentum space. Hence no conclusion as to the `completeness', or otherwise, of Quantum Mechanics can be drawn from the experiment.
The Einstein, Podolski and Rosen (EPR) argument aiming to prove the incompleteness of quantum mechanics (QM) was opposed by most EPR's contemporary physicists and is not accepted within the standard interpretation of QM, which maintains that QM is a complete theory. An analysis of the semantic implications of the opponent positions shows that they imply different notions of truth. The introduction of a nonclassical notion of truth within the standard interpretation is usually justified by referring to known theorems that should prove that QM is a contextual and nonlocal theory. However, these theorems are based on a doubtful implicit epistemological assumption. If one renounces it, one can provide an alternative interpretation of QM that it realistic in a semantic sense. Within this interpretation the EPR viewpoint is recovered and QM is considered a (semantically) incomplete, noncontextual and local theory. Furthermore, the new interpretation provides several suggestions for constructing a more general theory embedding QM and for connecting QM with classical physics and relativity.
Quantum mechanics is a fundamentally probabilistic theory (at least so far as the empirical predictions are concerned). It follows that, if one wants to properly understand quantum mechanics, it is essential to clearly understand the meaning of probability statements. The interpretation of probability has excited nearly as much philosophical controversy as the interpretation of quantum mechanics. 20th century physicists have mostly adopted a frequentist conception. In this paper it is argued that we ought, instead, to adopt a logical or Bayesian conception. The paper includes a comparison of the orthodox and Bayesian theories of statistical inference. It concludes with a few remarks concerning the implications for the concept of physical reality.