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second? We wouldn’t be able to see such a tiny distance with our eyes, but it is still quite large on the scale of atoms. You can do the calculation quite easily yourself by substituting the numbers into Feynman’s winding rule.7 The answer is something like a hundred million years’ worth of clock winding. Imagine how much interference that allows for. The upshot is that the sand grain stays where it is and there is almost no probability that it will jump a discernible distance, even though we really have to consider the possibility that it secretly hopped everywhere in the Universe in order to reach that conclusion.

This is a very important result. If you had put the numbers in for yourself then you’d already have a feel for why this is the case; it’s the smallness of Planck’s constant. Written out in full, it has a value 0.0000000000000000000000000000000066260695729 kg m2/s. Dividing pretty much any everyday number by that will result in a lot of clock winding and a lot of interference, with the result that the exotic journeys of our sand grain across the Universe all cancel each other out, and we perceive this voyager through infinite space as a boring little speck of dust sitting motionless on a beach.

Our particular interest of course is in those circumstances where clocks do not cancel each other out, and, as we have seen, this occurs if the clocks do not turn by more than a single wind. In that case, the orgy of interference will not happen. Let’s see what this means quantitatively.

Figure 4.4. The same as Figure 4.3 except that we are now not committing to a specific value of the size of the clock cluster or the distance to the point X.

We are going to return to the clock cluster, which we’ve redrawn in Figure 4.4, but we’ll be more abstract in our analysis this time instead of committing to definite numbers. We will suppose that the cluster has a size equal to Δx, and the distance of the closest point in the cluster to point X is x. In this case, the cluster size Δx refers to the uncertainty in our knowledge of the initial position of the particle; it started out somewhere in a region of size Δx. Starting with point 1, the point in the cluster closest to point X, we should wind the clock corresponding to a hop from this point to X by an amount

Now let’s go to the farthest point, point 3. When we transport the clock from this point to X, it will be wound around by a greater amount, i.e.

We can now be precise and state the condition for the clocks propagated from all points in the cluster not to cancel out at X: there should be less than one full wind of difference between the clocks from points 1 and 3, i.e.

Writing this out in full, we have

We’re now going to consider the specific case for which the cluster size, Δx, is much smaller than the distance x. This means we are asking for the prospects that our particle will make a leap far outside of its initial domain. In this case, the condition for no clock cancellation, derived directly from the previous equation, is

If you know a little maths, you’ll be able to get this by multiplying out the bracketed term and neglecting all the terms that involve (Δx)2. This is a valid thing to do because we’ve said that Δx is very small compared to x, and a small quantity squared is a very small quantity.

This equation is the condition for there to be no cancellation of the clocks at point X. We know that if the clocks don’t cancel out at a particular point, then there is a good chance that we will find the particle there. So we have discovered that if the particle is initially located within a cluster of size Δx, then at a time t later there is a good chance to find it a long distance x away from the cluster if the above equation is satisfied. Furthermore, this distance increases with time, because we are dividing by the time t in our formula. In other words, as more time passes, the chances of finding the particle further away from its initial position increases. This is beginning to look suspiciously like a particle that is moving. Notice also that the