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The Inequality and Boole

Karl Hess

Published Oct 22, 2014• 0 citations
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Abstract

My first contact with Hans and Kristel was, as just mentioned, through e-mail exchanges. We commiserated about how easy it was to get something published that indicated a violation of Bell’s inequality and how complicated it was to publish anything opposed to the gospel of Bell. I also brought Tony Leggett’s work with Anupam Garg to the attention of Kristel and Hans and we started a detailed correspondence and very soon a collaboration that resulted in a visit of Hans and Kristel in Hawaii.

DOI
10.1201/b16809-17

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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.

•Jan 1, 2016•13 citations
My first contact with Hans and Kristel was, as just mentioned, through e-mail exchanges. We commiserated about how easy it was to get something published that indicated a violation of Bell’s inequality and how complicated it was to publish anything opposed to the gospel of Bell. I also brought Tony Leggett’s work with Anupam Garg to the attention of Kristel and Hans and we started a detailed correspondence and very soon a collaboration that resulted in a visit of Hans and Kristel in Hawaii.
There was a follow-up to the Leiden conference organized in Prague (Frontiers of Quantum and Mesoscopic Thermodynamics), and Vaclav Spicka sent me an e-mail with an invitation to give one of the plenary presentations. Zeilinger was also to give one in the same session, and Hans and Kristel told me that they would attend and give a presentation too. Looking forward to see also Andrei Khrennikov, Theo Nieuwenhuizen, and many other friends, I accepted immediately and started to prepare a presentation in collaborationwith Hans and Kristel. I alsowanted to go to Prague for two personal reasons. My grandmother was Czech and I had never been in Prague, because of the problems with the “iron curtain” that fortunately did not exist anymore. The second personal reason was that Robert Basler, my best friend in Illinois who had died the previous year, had visited Prague, where he had lived in his youth, as his last travel wish and had told me how great this city was. I also was looking forward to visit my mother in Vienna on my way back to Hawaii. Sylvia had collected mileage upgrades so that I could fly first class. Going any other way from Hawaii is painful, because of the enormous distances involved; one travels around half the world. I flew from Hawaii to San Francisco and stayed there for a night, and the next day to Frankfurt and Vienna, from where I took the bus. When I finally arrived at the hotel in Prague after a total of two travel days, I was pretty tired, in spite of the fact that I had my own bed on the 380 Airbus from San Francisco to Frankfurt.
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.