Abstract
In 1862, George Boole derived an inequality for variables that represents a demarcation line between possible and impossible experience. This inequality forms an important milestone in the epistemology of probability theory and probability measures. In 1985 Leggett and Garg derived a physics related inequality, mathematically identical to Boole’s, that according to them represents a demarcation between macroscopic realism and quantum mechanics. We show that a wide gulf separates the “sense impressions” and corresponding data, as well as the postulates of macroscopic realism, from the mathematical abstractions that are used to derive the inequality of Leggett and Garg. If the gulf can be bridged, one may indeed derive the said inequality, which is then clearly a demarcation between possible and impossible experience: it cannot be violated and is not violated by quantum theory. This implies that the Leggett-Garg inequality does not mean that the SQUID flux is not there when nobody looks, as Leggett and Garg suggest, but instead that the probability measures may not be what Leggett and Garg have assumed them to be, when no data can be secured that directly relate to them. We show that similar considerations apply to other quantum interpretation-puzzles such as two-slit experiments.
Highlights
We show that a wide gulf separates the “sense impressions” and corresponding data, as well as the postulates of macroscopic realism, from the mathematical abstractions that are used to derive the inequality of Leggett and Garg
In 1985, Leggett and Garg [1] wrote “Despite sixty years of schooling in quantum mechanics, most physicists have a very non-quantum mechanical notion of reality at the macroscopic level, which implicitly makes two assumptions. (A1) Macroscopic realism: A macroscopic system with two or more macroscopically distinct states available to it will at all times be in one or the other of these states. (A2) Noninvasive measurability at the macroscopic level: It is possible, in principle, to determine the state of the system with arbitrary small perturbation on its subsequent dynamics.”
We only need to postulate that we have not arrived at the precise ultimate alternatives and their probability measures
Summary
In 1985, Leggett and Garg [1] wrote “Despite sixty years of schooling in quantum mechanics, most physicists have a very non-quantum mechanical notion of reality at the macroscopic level, which implicitly makes two assumptions. (A1) Macroscopic realism: A macroscopic system with two or more macroscopically distinct states available to it will at all times be in one or the other of these states. (A2) Noninvasive measurability at the macroscopic level: It is possible, in principle, to determine the state of the system with arbitrary small perturbation on its subsequent dynamics.” Leggett and Garg continue to state that “. . .experimental predictions of the conjunction of (A1) and (A2) are incompatible with those of quantum mechanics. . ..” We note here that Leggett and Garg later added conditions other than (A1) and (A2) which are related to counterfactual realism. There are still many physicists that do not feel at ease with the notion of quantum superposition at the macroscopic level It is the purpose of the present paper to show that (A1) and the data interpreted by using (A1) are still separated by a wide gulf from the mathematical abstractions and axioms used in the derivation of the Leggett-Garg inequality. This fact was already recognized by Boole in 1862 [3], who investigated the connection between data and mathematical abstractions representing them. A violation of the inequality suggested to him that he had to look for different “ultimate alternatives” and different probability measures, in order to do justice to the complexity of the data
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