The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [23]
Many problems are far too difficult to solve in a single mental leap, and deep understanding rarely emerges in ‘eureka’ moments. The trick is to make sure that you understand each little step and after a sufficient number of steps the bigger picture should emerge. Either that or we realize we have been barking up the wrong tree and have to start over from scratch. The little steps we’ve outlined so far are not difficult in themselves, but the idea that we have decided to take a single clock and turn it into an infinity of clocks is certainly a tricky concept, especially if you try to imagine drawing them all. Eternity is a very long time, to paraphrase Woody Allen, especially near the end. Our advice is not to panic or give up and, in any case, the infinity bit is a detail. Our next task is to establish the rule that tells us what all those clocks should actually look like at some time after we laid down the particle.
The rule we are after is the essential rule of quantum theory, although we will need to add a second rule when we come to consider the possibility that the Universe contains more than just one particle. But first things first: for now, let’s focus on a single particle alone in the Universe – no one can accuse us of rushing into things. At one instant in time, we’ll suppose we know exactly where it is, and it’s therefore represented by a single, solitary clock. Our specific task is to identify the rule that will tell us what each and every one of the new clocks, scattered around the Universe, should look like at any time in the future.
We’ll first state the rule without any justification. We will come back to discuss just why the rule looks like it does in a few paragraphs, but for now we should treat it as one of the rules in a game. Here’s the rule: at a time t in the future, a clock a distance x from the original clock has its hand wound in an anti-clockwise direction by an amount proportional to x2; the amount of winding is also proportional to the mass of the particle m and inversely proportional to the time t. In symbols, this means we are to wind the clock hand anti-clockwise by an amount proportional to mx2/t. In words, it means that there is more winding for a more massive particle, more winding the further away the clock is from the original, and less winding for a bigger step forward in time. This is an algorithm – a recipe if you like – that tells us exactly what to do to work out what a given arrangement of clocks will look like at some point in the future. At every point in the universe, we draw a new clock with its hand wound around by an amount given by our rule. This accounts for our assertion that the particle can, and indeed does, hop from its initial position to each and every other point in the Universe, spawning new clocks in the process.
To simplify matters we have imagined just one initial clock, but of course at some instant in time there might already be many clocks, representing the fact that the particle is not at some definite location. How are we to figure out what to do with a whole cluster of clocks? The answer is that we are to do what we did for one clock, and repeat that for each and every one of the clocks in the cluster. Figure 4.2 illustrates this idea. The initial set of clocks are represented by the little circles, and the arrows indicate that the particle hops from the site of every initial