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It seems to make sense to ask the peculiar question "If a baseball-sized rock on the bottom of the Amazon river were displaced one meter, how soon would a word spoken by you in your present or future conversation be different?". One intuitively feels that the next words one is about to say are already determined by the classical forces of chemistry and physics. After all, we do know for sure that the behavior of computers are exquisitely determined by present conditions, and we generally suppose that the vast assemblages of molecules making up neurons "swamp out" any minute differences that arise from quantum indeterminacy. But the question is difficult to phrase precisely, especially when quantum indeterminacy is taken into account, and especially when two parallel worlds are taken into account---one where the rock is transported a foot and the other where it is not. Probably relevant facts also come from the amusements of the famous mathematician/physicist Emil Borel, who used to like to calculate the changes induced by tiny shifts of matter very far away. In one example, he considered a gram of material in the star Sirius moved one centimeter from its present location. (One may suppose, I guess, that the gram of hot gas is overlaid on a nearby cubic centimeter, which, then, for an instant has twice the normal density of star matter, and the vacuum left in its wake quickly filled.) He obtained that the "state" of a liter of gas here on Earth under STP is changed within a second. That is (I presume) that most of the collisions that would have taken place do not take place, and that the pressure is maintained by collisions between different molecules. (Here is a math exercise that can actually be done in one's head. Suppose that several one centimeter spheres are to bounce off one another exactly, and say that they are separated by about one hundred centimeters. Now suppose that the flight of the first sphere is deflected by one millionth of a degree from the straight path that it would have taken towards the second sphere. It will strike the second sphere in a different spot on the second sphere by approximated the tangent of one millionth of a degree, which is about 100*(1/1000000), (since they are separated by a factor of 100). But from the point of view of the second sphere, the strike occurs at about 1/10000 of a degree different from where it would have struck. (This is because the same distance makes about 100 times more difference in degrees from the point of view of the second sphere, since its radius is 1/100 the distance that the first sphere has flown.) The first sphere then rebounds from the second at 1/10000th of a degree and, by the same logic, strikes the third sphere 1/100th of a degree different from where it would have stuck it. Thus each time the error is magnified by 100. It is easy to see ("Il est facile de voir", as LaPlace used to say just before an impenetrable statement or equation) that since molecules have size 10^-10 meters and are separated by distances on the order of 10^-5 meters, a spherical analogy in their case would cause the "error" to increase by a fantastic 10^5 each time! It is therefore clear that Borel only needed the tiniest influences supplied by gravity from Sirius to get the molecules in a liter of gas to fail to meet their predestined classical collision schedules.) Magnification Chains What are the most likely "stories" that could follow the displacement of the stone? Clearly life is a sensitive magnifier, so we imagine how fish, birds, and people might soon be affected. It's important to consider only the likely cases, not the less probable examples like the rock happening to strike the toe of a famous politician visiting Brazil. The visual system of a passing fish might be the first macro- life affected. This in turn could be communicated to other river dwellers, and before too long to a human being. One can visualize a movie presentation of the two worlds---one with the rock moved and one without---overlaid on the same screen. The first seen shows a double image of the rock, but just a single image of the fish; presently, however, the image of the fish would blur slightly. Later, other river dwellers' locations would also blur, and finally a double image of some creature would be observed. Because of the way sensitive dependence on initial conditions grows (e.g., see Gleick "Chaos"), these influences very rarely re-coalesce, and the divergence magnifies over time. At some point these easily visualizable influences are carried to far away nations, and the words you are classically destined to speak become affected at some point. But whether this might be days, years, centuries, or practically never probably has to remain a matter of conjecture. Build Up of Effects from Gases Near You Now It seems reasonable to suppose that since the macroscopic parameters of temperature and pressure remain virtually the same, the Borel-type differences induced in the air you are breathing by the stone's displacement have almost no effect. Yet the primary lesson of non-linear dynamics is that if these have any effects whatsoever, they grow exponentially. We may therefore suppose that on some timescale the extremely slight displacements of the molecules in the air---if not in your neurons themselves---begin to produce tiny macroscopic effects. My own guess is that within days something would build up from these alone that would alter some word in the set of sentences and exclamations you are deterministically destined to utter. What would be your guess? Enter Quantum Mechanics In his book "The Fabric of Reality" David Deutsch uses over and over again the phrase "a group of identical universes". One is supposed to picture (rather than branching universes) a stream of identical universes that become distinguished over time. That is, the width of the stream or river does not change, but the individual stream lines become different. (When two become identical again, we say that it is because interference has taken place.) Visualize a runner in a marathon. Given that he has just completed a stride with one leg, what is the probability that he will successfully complete a stride with the other? It is very high. In only a small proportion of universes does he trip, and in an even smaller proportion of universes is he struck by lightning, and in only an infinitesimal number of universes does he become an antelope. Yet by the laws of quantum mechanics, there is an amplitude for each of these outcomes, and so it is said in the Many Worlds Interpretation that each does obtain. But we can expect that the runner completes the next stride normally in 99.999% of the stream lines, the universes. [More later] |