by Brian Cox
In the history of science, there are a handful of legendary gatherings of scientists – meetings that certainly appear to have changed the course of science. They probably didn’t, in the sense that the participants had usually been working on problems for years, but the Shelter Island Conference of June 1947, held at the tip of Long Island, New York, has a better claim than most for catalysing something special. The participant list alone is worth reciting, because it is short and yet a role-call of the greats of twentieth-century American physics. In alphabetical order: Hans Bethe, David Bohm, Gregory Breit, Karl Darrow, Herman Feshbach, Richard Feynman, Hendrik Kramers, Willis Lamb, Duncan MacInnes, Robert Marshak, John von Neumann, Arnold Nordsieck, J. Robert Oppenheimer, Abraham Pais, Linus Pauling, Isidor Rabi, Bruno Rossi, Julian Schwinger, Robert Serber, Edward Teller, George Uhlenbeck, John Hasbrouck van Vleck, Victor Weisskopf and John Archibald Wheeler. The reader has met several of these names in this book already, and any student of physics probably has heard of most of them. The American writer Dave Barry once wrote: ‘If you had to identify, in one word, the reason why the human race has not achieved, and never will achieve, its full potential, that word would be meetings.’ This is doubtless true, but Shelter Island was an exception. The meeting began with a presentation of what has become known as the Lamb shift. Willis Lamb, using high-precision microwave techniques developed during the Second World War, found that the hydrogen spectrum was not, in fact, perfectly described by old-fashioned quantum theory. There was a minute shift in the observed energy levels that could not be accounted for using the theory we have developed so far in this book. It is a tiny effect, but it was a wonderful challenge to the assembled theorists.
We shall leave Shelter Island there, poised after Lamb’s talk, and turn to the theory that emerged in the months and years that followed. In doing so we will uncover the origin of the Lamb shift, but, to whet your appetite, here is a cryptic statement of the answer: the proton and electron are not alone inside the hydrogen atom.
QED is the theory that explains how electrically charged particles, like electrons, interact with each other and with particles of light (photons). It is single-handedly capable of explaining all natural phenomena with the exception of gravity and nuclear phenomena. We’ll turn our attention to nuclear phenomena later on, and in doing so explain why the atomic nucleus can hold together even though it is a bunch of positively charged protons and zero charge neutrons which would fly apart in an electro-repulsive instant without some sub-nuclear goings-on. Pretty much everything else – certainly everything you see and feel around you – is explained at the deepest known level by QED. Matter, light, electricity and magnetism – it is all QED.
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 two complex numbers together. For those who do not know about complex numbers: it doesn’t matter because the description in terms of clocks is equally accurate. All we did was introduce a slightly different way of thinking about the clock-winding rule: we can wind and shrink a clock using another clock.
We are free to design our clock multiplication rule to make this all work: multiply the sizes of the two clocks (1 × 0.2 = 0.2) and combine the times on the two clocks such that we wind the first clock backwards by 12 o’clock minus 10 o’clock = 2 hours. This does sound a little bit like we are over-elaborating, and it is clearly not necessary when we only have one particle to think about. But physicists are lazy, and they wouldn’t go to all this trouble unless it saved time in the long run. This little bit of notation proves to be a very useful way of keeping track of all the winding and shrinking when we come to the more interesting case where there are multiple particles in the problem – the hydrogen atom, for example.
Regardless of the details, there are just two key elements in our method of figuring out the chances to find a lone particle somewhere in the Universe. First, we need to specify the array of initial clocks which codify the information about where the particle is likely to be found at time zero. Second, we need to know the propagator P(A,B,T), which is itself a clock encoding the rule for shrinking and turning as a particle leaps from A to B. Once we know what the propagator looks like for any pair of start and end points then we know everything there is to know, and we can confidently figure out the magnificently dull dynamics of a Universe containing a single particle. But we should not be so disparaging, because this simple state of affairs doesn’t get much more complicated when we add particle interactions into the game. So let’s do that now.
Figure 10.1 illustrates pictorially all of the key ideas we want to discuss. It is our first encounter with Feynman diagrams, the calculational tool of the professional particle physicist. The task
we are charged with is to work out the probability of finding a pair of electrons at the points X and Y at some time T. As our starting point we are told where the electrons are at time zero, i.e. we are told what their initial clock clusters look like. This is important because being able to answer this type of question is tantamount to being able to know ‘what happens in a Universe containing two electrons’. That may not sound like much progress, but once we have figured this out the world is our oyster, because we will know how the basic building blocks of Nature interact with each other.
To simplify the picture, we’ve drawn only one dimension in space, and time advances from left to right. This won’t affect our conclusions at all. Let’s start out by describing the first of the series of pictures in Figure 10.1. The little dots at T = 0 correspond to the possible locations of the two electrons at time zero. For the purposes of illustration, we’ve assumed that the upper electron can be in one of three locations, whilst the lower is in one of two locations (in the real world we must deal with electrons that can be located in an infinity of possible locations, but we’d run out of ink if we had to draw that). The upper electron hops to A at some later time whereupon it does something interesting: it emits a photon (represented by the wavy line). The photon then hops to B where it gets absorbed by the other electron. The upper electron then hops from A to X whilst the lower electron hops from B to Y. That is just one of an infinite number of ways that our original pair of electrons can make their way to points X and Y. We can associate a clock with this entire process – let’s call it ‘clock 1’ or C1 for short. The job of QED is to provide us with the rules of the game that will allow us to deduce this clock.
Figure 10.1. Some of the ways that a pair of electrons can scatter off each other. The electrons start out on the left and always end up at the same pair of points, X and Y, at time T. These graphs correspond to some of the different ways that the particles can reach X and Y.
Before getting into the details, let’s sketch how this is going to pan out. The uppermost picture represents one of the myriad ways that the initial pair of electrons can make their way to X and Y. The other pictures represent some of those ways. The crucial idea is that for each possible way that the electrons can get to X and Y we are to identify a quantum clock – C1 is the first in a long list of clocks.2 When we’ve got all of the clocks, we are to add them all together and obtain one ‘master’ clock. The size of that clock (squared) tells us the probability of finding the pair of electrons at X and Y. So once again we are to imagine that the electrons make their way to X and Y not by one particular route, but rather by scattering off each other in every possible way. If we look at the final few pictures in the figure, we can see a variety of more elaborate ways for the electrons to scatter. The electrons not only exchange photons, they can emit and reabsorb a photon themselves, and in the final two figures something very odd is happening. These pictures include the scenario where a photon appears to emit an electron which ‘goes in a circle’ before ending up where it started out – we shall have more to say about that in a little while. For now, we can simply imagine a series of increasingly complicated diagrams corresponding to cases where the electrons emit and absorb huge numbers of photons before finally ending up at X and Y. We’ll need to contemplate the multifarious ways that the electrons can end up at X and Y, but there are two very clear rules: electrons can only hop from place to place and emit or absorb a single photon. That’s really all there is to it; electrons can hop or they can branch. Closer inspection should reveal that none of the pictures we have drawn contravenes those two rules because they never involve anything more complicated than a junction involving two electrons and a photon. We must now explain how to go about computing the corresponding clocks, one for each picture in Figure 10.1.
Let’s focus on the uppermost picture and explain how to determine what the clock associated with it (clock C1) looks like. Right at the start of the process, there are two electrons sitting there, and they will each have a clock. We should start out by multiplying them together according to the clock multiplication rule to get a new, single clock, which we will denote by the symbol C. Multiplying them makes sense because we should remember that the clocks are actually encoding probabilities, and if we have two independent probabilities then the way to combine them is to multiply them together. For example, the probability that two coins will both come up heads is ½ × ½ = ¼. Likewise, the combined clock, C, tells us the probability to find the two electrons at their initial locations.
The rest is just more clock multiplication. The upper electron hops to A, so there is a clock associated with that; let’s call it P(1,A) (i.e. ‘particle 1 hops to A’). Meanwhile the lower electron hops to B and we have a clock for that too; call it P(2,B). Likewise there are two more clocks corresponding to the electrons hopping to their final destinations; we shall denote them by P(A,X) and P(B,Y). Finally, we also have a clock associated with the photon, which hops from A to B. Since the photon is not an electron, the rule for photon propagation could be different for the rule for electron propagation so we should use a different symbol for its clock. Let’s denote the clock corresponding to the photon hop L(A,B).3 Now we simply multiply all the clocks together to produce one ‘master’ clock: R = C × P(1, A) × P(2, B) × P(A, X) × P(B, Y) × L(A, B). We are very nearly done now, but there remains some additional clock shrinking to do because the QED rule for what happens when an electron emits or absorbs a photon says that we should introduce a shrinking factor, g. In our diagram, the upper electron emits the photon and the lower one absorbs it – that makes for two factors of g, i.e. g2. Now we really are done and our final ‘clock 1’ is obtained by computing C1 = g2 × R.
The shrinking factor g looks a bit arbitrary, but it has a very important physical interpretation. It is evidently related to the probability that an electron will emit a photon, and this encodes the strength of the electromagnetic force. Somewhere in our calculation we had to introduce a connection with the real world because we are calculating real things and, just as Newton’s gravitational constant G carries all the information about the strength of gravity, so g carries all the information about the strength of the electromagnetic force.4
If we were actually doing the full calculation, we’d now turn our attention to the second diagram, which represents another way that our original pair of electrons can make their way to the same points, X and Y. The second diagram is very similar to the first in that the electrons start out from the same places, but now the photon is emitted from the upper electron at a different point in space and at a different time and it is absorbed by the lower electron at some other new place and time. Otherwise things run through in precisely the same way and we’ll get a second clock, ‘clock 2’, denoted C2.
Then, on we’d go, repeating the entire process again and again for each and every possible place where the photon can be emitted and each and every possible place where it can be absorbed. We should also account for the fact that the electrons can start out from a variety of different possible starting positions. The key idea is that each and every way of delivering electrons to X and Y needs to be considered, and each is associated with its own clock. Once we have collected together all of the clocks, we ‘simply’ add them all together, to produce one final clock whose size tells us the probability of finding one electron at X and a second at Y. Then we are finished – we will have figured out how two electrons interact with each other because we can do no better than compute probabilities.
What we have just described really is the heart of QED, and the other forces in Nature admit a satisfyingly similar description. We will come on to those shortly, but first we have a little more to discover.
Firstly, a paragraph describing two small, but important, details. Number 1: we have simplified matters by ignoring the fact that electrons have spin and therefore come in two types. Not only that, photons also have spin (they are bosons) and come in three types. This just makes the calculations a little mor
e messy because we need to keep track of which types of photon and electron we are dealing with at every stage of the hopping and branching. Number 2: if you have been reading carefully then you may have spotted the minus signs in front of a couple of the pictures in Figure 10.1. They are there because we are talking about identical electrons hopping their way to X and Y and the two pictures with the minus sign correspond to an interchange of the electrons relative to the other pictures, which is to say that an electron which started out at one of the upper cluster of points ends up at Y whilst the other, lower, electron ends up at X. And as we argued in Chapter 7, these swapped configurations get combined only after an extra 6-hour wind of their clocks – hence the minus sign.
You may also have spotted a possible flaw in our plan – there are an infinite number of diagrams describing how two electrons can make their way to X and Y, and summing an infinite number of clocks might seem onerous to say the least. Fortunately, every appearance of a photon–electron branching introduces another factor of g into the calculation, and this shrinks the size of the resultant clock. This means that the more complicated the diagram, the smaller the clock it will contribute and the less important it will be when we come to add all the clocks up. For QED, g is quite a small number (it’s around 0.3), and so the shrinking is pretty severe as the number of branchings increases. Very often, it is enough to consider only diagrams like the first five in the figure, where there are no more than two branchings, and that saves lots of hard work.