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by contemplating the scattering of photons off electrons. The photons are the means by which we ‘see’ the electron, just as we see everyday objects by scattering photons off them and collecting them in our eyes. Ordinarily, the light that bounces off an object disturbs the object imperceptibly, but that is not to deny our fundamental inability to absolutely isolate the act of measurement from the thing one is measuring. One might worry that it could be possible to beat the limitations of the Uncertainty Principle by devising a suitably ingenious experiment. We are about to show that this is not the case and the Uncertainty Principle is absolutely fundamental, because we are going to derive it using only our theory of clocks.

Deriving Heisenberg’s Uncertainty Principle from the Theory of Clocks

Rather than starting with a particle at a single point, let us instead think about a situation where we know roughly where the particle is, but we don’t know exactly where it is. If we know that a particle is somewhere in a small region of space then we should represent it by a cluster of clocks filling that region. At each point within the region there will be a clock, and that clock will represent the probability that the particle will be found at that point. If we square up the lengths of all the clock hands at every point and add them together, we will get 1, i.e. the probability to find the particle somewhere in the region is 100 per cent.

In a moment we are going to use our quantum rules to perform a serious calculation, but first we should come clean and say that we have failed to mention an important addendum to the clock-winding rule. We didn’t want to introduce it earlier because it is a technical detail, but we won’t get the correct answers when it comes to calculating actual probabilities if we ignore it. It relates to what we said at the end of the previous paragraph.

If we begin with a single clock, then the hand must be of length 1, because the particle must be found at the location of the clock with a probability of 100 per cent. Our quantum rule then says that, in order to describe the particle at some later time, we should transport this clock to all points in the Universe, corresponding to the particle leaping from its initial location. Clearly we cannot leave all of the clock hands with a length of 1, because then our probability interpretation falls down. Imagine, for example, that the particle is described by four clocks, corresponding to its being at four different locations. If each one has a size of 1 then the probability that the particle is located at any one of the four positions would be 400 per cent and this is obviously nonsense. To fix this problem we must shrink the clocks in addition to winding them anti-clockwise. This ‘shrink rule’ states that after all of the new clocks have been spawned, every clock should be shrunk by the square root of the total number of clocks.5 For four clocks, that would mean that each hand must be shrunk by √4, which means that each of the four final clocks will have a hand of length ½. There is then a (½)2 = 25 per cent chance that the particle will be found at the site of any one of the four clocks. In this simple way we can ensure that the probability that the particle is found somewhere will always total 100 per cent. Of course, there may be an infinite number of possible locations, in which case the clocks would have zero size, which may sound alarming, but the maths can handle it. For our purposes, we shall always imagine that there are a finite number of clocks, and in any case we will never actually need to know how much a clock shrinks.

Let’s get back to thinking about a Universe containing a single particle whose location is not precisely known. You can treat the next section as a little mathematical puzzle – it may be tricky to follow the first time through, and it may be worth rereading, but if you are able to follow what is going on then you’ll understand how the Uncertainty Principle emerges. For simplicity, we’ve assumed that the particle moves in one