The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [81]
Let’s begin by exploring a system we have already met many times throughout the book: a world containing one single electron. The little circles in the ‘clock hopping’ figure on page 50 illustrate the various possible locations of the electron at some instant in time. To deduce the likelihood of finding it at some point X at a later time, our quantum rules say that we are to allow the electron to hop to X from every possible starting point. Each hop delivers a clock to X, we add up these clocks and then we are done.
We’re going to do something now that might look a little over-complicated at first, but of course there is a very good reason. It’s going to involve a few As, Bs and Ts – in other words we’re heading off into the land of tweed jackets and chalk dust again; it won’t last long.
When a particle goes from a point A at time zero to a point B at time T, we can calculate what the clock at B will look like by winding the clock at A backwards by an amount determined by the distance of B from A and the time interval, T. In shorthand, we can write that the clock at B is given by C(A,0)P(A,B,T) where C(A,0) represents the original clock at A at time zero and P(A,B,T) embodies the clock-winding and shrinking rule associated with the leap from A to B.1 We shall refer to P(A,B,T) as the ‘propagator’ from A to B. Once we know the rule of propagation from A to B, then we are all set and can figure out the probability to find the particle at X. For the example in Figure 4.2, we have lots of starting points so we’ll have to propagate from every one of them to X, and add all the resulting clocks up. In our seemingly overkill notation, the resultant clock C(X, T) = C(X1, 0)P(X1, X, T)+C(X2, 0)P(X2, X, T)+C(X3, 0)P(X3, X, T)+… where X1, X2, X3, etc. label all the positions of the particle at time zero (i.e. the positions of the little circles in Figure 4.2). Just to be crystal clear, C(X3,0)P(X3,X,T) simply means ‘take a clock from point X3 and propagate it to point X at time T’. Don’t be fooled into thinking there is something tricky going on. All we are doing is writing down in a fancy shorthand something we already knew: ‘take the clock at X3 and time zero and figure out by how much to turn and shrink it corresponding to the particle making the journey from X3 to X at some time T later and then repeat that for all of the other time-zero clocks and finally add all of the clocks together according to the clock-adding rule’. We’re sure you’ll agree that this is a bit of a mouthful, and the little bit of notation makes life easier.
We can certainly think of the propagator as the embodiment of the clock-winding and shrinking rule. We can also think of it as a clock. To clarify that bald statement, imagine if we know for certain that an electron is located at point A at time T = 0, and that it is described by a clock of size 1 reading 12 o’clock. We can picture the act of propagation using a second clock whose size is the amount that the original clock needs to be shrunk and whose time encodes the amount of winding we need. If a hop from A to B requires shrinking the initial clock by a factor of 5 and winding back by 2 hours, then the propagator P(A,B,T) could be represented by a clock whose size is ⅕ = 0.2 and which reads 10 o’clock (i.e. it is wound 2 hours back from 12 o’clock). The clock at B is simply obtained by ‘multiplying’ the original clock at A by the propagator clock.
As an aside for those who know about complex numbers, just as each of the C(X1,0) C(X2,0) can be represented by a complex number so can the P(X1,X,T), P(X2,X,T) and they are combined according to the mathematical rules for multiplying