The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [61]
To streamline the discussion, let’s refer to the clock corresponding to particle 1 hopping to A and particle 2 hopping to B as clock 1. This is the clock associated with the upper picture in Figure 7.3. Clock 2 corresponds to the other option, where particle 1 hops to B instead. Here is an important realization: if we give clock 1 a turn before adding it to clock 2, then the final probability we calculate must be the same as if we choose to give clock 2 the same turn before adding it to clock 1.
To see this, notice that swapping the labels A and B around in our diagrams clearly cannot change anything. It is just a different way of describing the same process. But swapping A and B around swaps the diagrams in Figure 7.3 around too. This means that if we decide to wind clock 1 (corresponding to the upper picture) before adding it to clock 2, then this must correspond precisely to winding the clock 2 before adding it to clock 1, after we’ve swapped labels. This piece of logic is crucial, so it’s worth hammering home. Because we have assumed that there is no way of telling the difference between the two particles, then we are allowed to swap the labels around. This implies that a turn on clock 1 must give the same answer as when we apply the same turn to clock 2, because there is no way of telling the clocks apart.
Figure 7.4. The upper part of the figure illustrates that adding clocks 1 and 2 together after winding clock 1 by 90 degrees is not the same as adding them together after winding clock 2 by 90 degrees. The lower part illustrates the interesting possibility that we could wind one of the clocks by 180 degrees before adding.
This is not a benign observation – it has a very important consequence, because there are only two possible ways of playing around with the winding and shrinking of clocks before adding them together that will deliver a final clock with the property that it does not depend upon which of the original clocks gets the treatment.
This is illustrated in Figure 7.4. The top half of the figure illustrates that, if we wind clock 1 by 90 degrees and add it to clock 2 then the resultant clock is not of the same size as the resultant we would get if we instead wound clock 2 by 90 degrees and add it to clock 1. We can see this because, if we first wind clock 1, the new hand, represented by the dotted arrow, points in the opposite direction to clock 2’s hand, and therefore partly cancels it out. Winding clock 2 instead leaves its hand pointing in the same direction as clock 1’s, and now the hands will add together to form a larger hand.
It should be clear that 90 degrees is not special, and that other angles will also give resultant clocks that depend upon which of clocks 1 and 2 we decided to wind.
The obvious exception is a clock wind of zero degrees, because winding clock 1 by zero degrees before adding to clock 2 is obviously exactly the same as winding clock 2 by zero degrees before adding to clock 1. This means that adding clocks together without any wind is a viable possibility. Similarly, winding both clocks by the same amount would work, but that really is just the same as the ‘no winding’ situation and corresponds simply to redefining what we call ‘12 o’clock’. This is tantamount to saying that we are always free to wind every clock around by some amount, as long as we do that to every clock. This will never impact on the probabilities we are trying to compute.
The lower part of Figure 7.4 illustrates that there is, perhaps surprisingly, one other way we can combine the clocks: we could turn one of them through 180 degrees before adding them together. This does not produce exactly the same clock in the two cases but it does produce the same size of clock, and that means it leads to the same probability to find one electron at A and a second at B.
A similar line of reasoning rules out the possibility of shrinking or expanding one