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The Uncertainty Principle is a relationship between measurements of pairs of attributes, position and momentum, as well as energy and time. Perfect precision of one attribute's measurement leads to a full lack of knowledge about the other attribute (most likely, because there is no solid quantity for the other attribute). This appears to be a fundamental attribute of the world at large. Bohmian mechanics posits hidden variables, and makes the claim that particles (ordinary objects) have attributes at all times.

Bertrand Russell developed an argument against Zeno's paradoxes which he called the "at-at theory of motion."

Bertrand Russell offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is a function of position with respect to time.

In order to measure anything's position, you would need to have a single precise instant of measurement. The same would apply to energy, which comes in discrete units. To measure momentum (mass times velocity) with the at-at theory, you need to take at least two measurements. To take a time measurement, such as comparing elapsed time as measured by two different timepieces, you would also necessarily need at least two measurements.

The Uncertainty Principle gives a precise mathematical formulation of an apparent fact of nature, which is unlikely to be something that a thought experiment could derive. However, the nature of the relationships appear to follow as natural consequences of Russell's response to Zeno. (And if space is quantized, it may be a fact of nature.)

Does the Uncertainty Principle actually follow from Bohmian mechanics and the at-at theory of motion? If not, what am I missing?

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My guess would be that no, the Uncertainty Principle does not follow from Bohmian mechanics and the at-at theory of motion. You can get the uncertainty principle pretty much simply from the definition of expectation values of operators and the Schwarz inequality. Calculus solves Zeno's paradox, and Bohmian mechanics has its problems (see, eg, physics.stackexchange.com/questions/7112). Also, remember that the uncertainty principle is not about measurements, it is a property of matter; the position and momentum of a particle are not defined with arbitrary precision. –  Goku Aug 21 '13 at 4:02
    
You may find this TRF article interesting. –  Dimensio1n0 Sep 4 '13 at 11:25
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