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Zeno and Uncertainty |
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Something improbable is bound to happen. |
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Do events of probability zero ever occur? Consider a dartboard of unit area, and stipulate that a given dart will land at some “random” location on the board, with a uniform probability density distribution. The integral of the distribution over the board is 1, and the probability of landing within some small region of the board is proportional to the area of the region. As the size of the region goes to zero, the probability of landing within that region goes to zero. In the limit, the probability of the (center of the) dart landing on any specific point is zero. Hence, if events with probability zero never occur, one might think that the dart could not land in any specific place, contradicting the fact that it lands someplace with probability 1. |
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This is somewhat reminiscent of Zeno's famous “paradoxes” of motion. For example, he argued that before an arrow can reach its target it must reach the half-way point...etc., and so motion is impossible. Similarly we could imagine Zeno arguing that the probability of an arrow landing on any particular point of the target is zero, so the arrow can't possibly hit the target. As with Zeno’s actual arguments, one could uncritically accept the idealized premises and criticize the reasoning on the level of mathematics, but the physical premises themselves are actually more relevant. |
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For our dart board question, Zeno himself might have rejected the concept of probability in physics. He might have argued that a dart would go precisely where it was thrown, in a deterministic way, so it has probability 1 of landing where it lands, and zero probability of landing anywhere else. This basically denies any meaning to probability of counter-factual events. Granted, we may not be able to characterize the throwing motion and initial trajectory with perfect precision, but this (he might argue) is just our ignorance, and does not interfere with the ability of the arrow to land where it lands. |
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On the other hand, if we accept the physical existence of genuine irreducible randomness with continuous distributions, then there is (at least arguably) a real conundrum, because the choice of a specific real-valued number from any continuum generally represents an infinite amount of information. Recall that there is no uniform distribution over the real number line R, let alone over the plane R2, because these are of infinite extent (even though the cardinality is the same as for finite extents). We commonly deal with uniform distributions over finite areas, but this assumes a finite scale factor, and yet we can expand and re-scale any continuous finite region indefinitely while still mapping points to points. The real-valued coordinates of each point are scale-invariant. In contrast, the atoms and sub-atomic particles that comprise a physical dart and board have a definite scale, and apparently can be characterized by a finite amount of information (the quantum numbers). Hence a finite board really does consist of a finite number of particles, and the positions and momenta of those particles are subject to the uncertainty relation, expressing the fundamental quantum character of nature. |
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As with several of Zeno’s actual paradoxes, the premise of an actual physical continuum leads to conceptual difficulties. Of course, quantum mechanics itself is expressed in terms of Hilbert spaces and continuous wave functions, but the actual observed phenomena are reduced to discrete actions in terms of some basis of measurement. We still translate everything back to descriptions in terms of space and time (assumed continuous), even though, as Einstein remarked |
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…the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale… perhaps the success of quantum mechanics points to the elimination of continuous functions from physics… At the present time, however, such a program looks like an attempt to breath in empty space… |
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