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Nothing's Impossible thesis Nothing’s impossible Proofs of what is impossible, as John Bell out in a paper that has made Riccioli’s once remarked, often demonstrate little writings available from the original more than their authors’ own lack of The only surprising Latin text (arxiv:10113778), only two of imagination. Bell said this in discussion the seventy-seven proofs are obviously of one of the most famous, and famously thing is how these things religious. Most make eminently good sense erroneous, impossibility proofs of all time, always surprise us. given the empirical knowledge of the day. one constructed in 1932 by Hungarian “If Earth had a diurnal rotation”, physicist John von Neumann. It purported one proof observes, “then heavy bodies to show that no hidden-variables falling near the equator would have a interpretation of quantum theory — one this condition might not be true. It took fundamentally different motion than attributing definite trajectories and other Bell’s mental dexterity and perseverance identical bodies falling near the poles under objective properties to quantum systems to clarify how totally inappropriate was identical conditions.” Similarly, “Heavy that might account deterministically von Neumann’s use of this condition. bodies launched perpendicularly upwards for experimental outcomes — could The assumption rests on the idea of so- fall back upon the location from which they possibly reproduce all the predictions of called ‘non-contextuality’ — that physical were launched. If the Earth had diurnal and quantum theory. quantities can be thought of as having values annual motions, these bodies would follow Of course, von Neumann certainly had completely independently of the devices curved trajectories.” Indeed, Riccioli was more than a little imagination, although used to measure them. right — it’s only that the weakness of the this was perhaps not the only time it Of course any classical theory would Coriolis force he clearly envisaged meant showed its limitations. (In the 1950s, he work this way. But contextuality is precisely that it hadn’t yet been measured in 1651. strongly advised US authorities on the one of the most interesting things about Another of his observations runs along wisdom of an immediate unilateral nuclear quantum theory, and one of the things that the same lines: “If Earth moves, then the attack on the Soviet Union, apparently any hidden-variable theory needs to capture. clouds and the birds in the air would be because he could see no conceivable way Any such theory worth its salt has to allow seen to fly west, as they were left behind that the two nations might escape eventual the action of experimental devices to help by the Earth.” Riccioli was only making an mutual annihilation.) In any event, determine the values of a system’s variables. implicit assumption about the magnitude Bell’s phrase does capture a truth — that This, Bell showed, makes von Neumann’s of this effect. Again, a hidden presumption, impossibility proofs really demonstrate assumptions completely out of place — later revealed. The only surprising thing is only what is inconceivable to one mind, “absurd”, as he put it. how these things always surprise us. or one set of minds, which may (for all This example illustrates the general In recent times, we’ve seen a string of their other wonderful capacities) be problem — the inherent weakness of such proofs establishing the alleged impossibility blinkered. Such traps, as history illustrates, proofs to hidden assumptions, because such of breaking the security of various afflict us far more frequently than assumptions are sometimes so difficult to see schemes for quantum cryptography. Yet we expect. amidst the conceptual debris surrounding each progressively more intricate scheme Von Neumann’s mistake was subtle them. Proofs have loopholes that are only seems to inspire counter efforts that enough — and on a topic on which most effectively invisible to their authors. It is as identify new loopholes. Perhaps the only physicists were already convinced — that true today as it was four centuries ago. impossibility proofs that really work are it stood as received wisdom, beyond In 1651, an Italian Jesuit constructive proofs that actually establish questioning, for over two decades. He Giovanni Battista Riccioli published a positive results. For example, David Bohm’s started from five assumptions that, he book entitled Almagestum Novum, in 1952 alternative formulation of quantum thought, any hidden-variables interpretation which he gave seventy-seven proofs of theory gave a proof by explicit construction of quantum theory would clearly have to the impossibility of the Earth’s motion, that von Neumann’s argument couldn’t satisfy. To most physicists, these assumptions in particular its rotation. Riccioli’s book possibly be correct; indeed, that no seem eminently plausible. was written against the Copernicans, and argument could establish the impossibility His fifth and final assumption asserted he was no fool. In one historian’s view, of a consistent deterministic interpretation that the expectation values of summed Riccioli produced “the lengthiest, most of the non-locality inherent in quantum variables, evaluated for example in penetrating, and authoritative analysis of the theory; after all, Bohm had produced one. experiments on a set of systems prepared so question of Earth’s mobility or immobility It couldn’t then be proved out of existence. that all the hidden variables are identical, made by any author of the sixteenth and I’m sure this is a lesson humanity will be should satisfy the algebraic relation, seventeenth centuries.” learning for centuries to come. “All historical <X + Y> = <X> + <Y>: the average of a sum Riccioli’s proofs were, I expect, just experience”, as Max Weber wrote, “confirms of two variables should equal the sum of the as convincing to him and many of his that men might not achieve the possible if averages of those variables. contemporaries as von Neumann’s they had not, time and time again, reached This does indeed seem plausible, at least proofs about quantum theory were out for the impossible.” ❐ naively. It seems hard to get your head to physicists of the latter twentieth moving towards any conceptual space where century. As Christopher Graney points MARK BUCHANAN NATURE PHYSICS | VOL 7 | JANUARY 2011 | www.nature.com/naturephysics 5 © 2011 Macmillan Publishers Limited. All rights reserved.
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