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By Root 760 0

which means that we have to wind it 10.12 times. This is very close to 102 complete rotations – again a whole number of winds. We need to add this clock to the others at X and, as before, this will make the hand at X even longer. Continuing, there is also a point midway between points 1 and 2 and the clock hopping from there will get 101 complete rotations, which will add to the size of the final hand again. But here is the important point. If we now go midway again between these two points, we get to a clock that will be wound 100.5 rotations when it reaches X. This corresponds to a clock with a hand pointing to 6 o’clock, and when we add this clock we will reduce the length of the clock hand at X. A little thought should convince you that, although the points labelled 1, 2 and 3 each produce clocks at X reading 12 o’clock, and although the points midway between 1, 2 and 3 also produce clocks that read 12 o’clock, the points that are ¼ and ¾ of the way between points 1 and 3 and points 2 and 3 each generate clocks that point to 6 o’clock. In total that is five clocks pointing up and four clocks pointing down. When we add all these clocks together, we’ll get a resultant clock at X that has a tiny hand because nearly all of the clocks will cancel each other out.

This ‘cancellation of clocks’ obviously extends to the realistic case where we consider every possible point lying in the region between points 1 and 3. For example, the point that lies ⅛ of the way along from point 1 contributes a clock reading 9 o’clock, whilst the point lying ⅜ of the way reads 3 o’clock – again the two cancel each other out. The net effect is that the clocks corresponding to all of the ways that the particle could have travelled from somewhere in the cluster to point X cancel each other out. This cancellation is illustrated on the far right of the figure. The arrows indicate the clock hands arriving at X from various points in the initial cluster. The net effect of adding all these arrows together is that they all cancel each other out. This is the crucial ‘take home’ message.

To reiterate, we have just shown that, provided the original cluster of clocks is large enough and that point X is far enough away, then for every clock that arrives at X pointing to 12 o’clock, there will be another that arrives pointing to 6 o’clock to cancel it out. For every clock that arrives pointing to 3 o’clock, there will be another that arrives pointing to 9 o’clock to cancel it out, and so on. This wholesale cancellation means that there is effectively no chance at all of finding the particle at X. This really is very encouraging and interesting, because it looks rather like a description of a particle that isn’t moving. Although we started out with the ridiculous-sounding proposal that a particle can go from being at a single point in space to anywhere else in the Universe a short time later, we have now discovered that this is not the case if we start out with a cluster of clocks. For a cluster, because of the way all the clocks interfere with each other, the particle has effectively no chance of being far away from its initial position. This conclusion has come about as a result of an ‘orgy of quantum interference’, in the words of Oxford professor James Binney.

For the orgy of quantum interference and corresponding cancellation of clocks to happen, point X needs to be far enough away from the initial cluster so that the clocks can rotate around many times. Why? Because if point X is too close then the clock hands won’t necessarily have the chance to go around at least once, which means they will not cancel each other out so effectively. Imagine, for example, that the distance from the clock at point 1 to point X is 0.3 instead of 10. Now the clock at the front of the cluster gets a smaller wind than before, corresponding to 0.32 = 0.09 of a turn, which means it is pointing just past 1 o’clock. Likewise, the clock from point 3, at the back of the cluster, now gets wound by 0.52 = 0.25 of a turn, which means it reads 3 o’clock. Consequently, all of the clocks