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Foundations of Physics

Can the statistical postulate of quantum theory be derived??A critique of the many-universes interpretation

L. E. Ballentine

Published Jun 1, 1973• 62 citations
Publisher
DOI
10.1007/bf00708440

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Critique of a recent claim of a formal difference between the Copenhagen and statistical interpretations of quantum theory

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Part III of this series [A. Landé, Amer. J. Phys. 33, 123 (1965), and 34, 160 (1966), quoted as Parts I and II] continues the critique of the customary interpretation of quantum mechanics. It is directed against the idea that electrons, in opposition to the Born statistical particle interpretation of the wave function, are “wavicles.” Serious conceptual and elementary mathematical defects in Bohr's famous “Discussion with Einstein” are pointed out. The doctrine that position q and momentum p of a particle are blurred over an uncertainty range because both allegedly cannot be measured simultaneously, that q and p do not exist simultaneously, is due to a confusion of analogy with identity of qualities. In its second part, the paper offers a nonquantal derivation of the probability interference law as a necessity under the postulate that the general interdependence between probabilities is to become the ordinary probability addition law in the average. The wave picture of matter violates the postulate of relativity.
It is shown that Schrödinger's equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability with a probability current which takes the form of a gradient. The third is a principle of maximal disorder as realized by the requirement of minimal Fisher information. The rule for calculating expectation values is obtained from a fourth postulate, the requirement of energy conservation in the mean. The fact that all these basic relations of quantum theory may be derived from premises which are statistical in character is interpreted as a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of quantum theory and classical statistical theories are compared, and some fundamental differences are identified.
In this paper, we do not try to solve the measurement problem, but rather to properly formulate it. One of the reasons why it still lacks a precise, agreed definition is that the problem may take very different aspects depending on the interpretation of quantum mechanics embraced. Inspired by the methodology of theory-independent results like Bell's theorem, we propose to identify the common root of the puzzle in an interpretation-independent way i.e. as a property of the empirical statistics only, before deriving its philosophical consequences. The key observation is that quantum matter can not be described by a Kolmogorovian probabilistic theory. Arguing that the Kolmogorov axioms of probability theory are the postulates of epistemic uncertainty leads us to reformulate the measurement problem as the impossibility to build an ontology independent of epistemology for quantum matter. Said differently, there exists no God's-eye view on quantum systems. Although it is meaningless to solve the measurement problem as defined in this way, insofar as it is a feature of the universe's statistics, such a formulation may on the other hand bring side benefits. In particular, we argue that it allows to: (i) shed a new light on the variety of interpretations; (ii) propose a fundamental reason why quantum mechanics and general relativity are so incompatible, not relying on purely mathematical or technical arguments; (iii) guide the quest for quantum gravity.