by John Gribbin
The problem John Slater set Feynman was to work out why quartz expands much less than other substances, such as metals, when it is heated. Feynman quickly became intrigued by the whole idea of how and why things expand at all, and set out to study the way the forces between atoms work in crystals.
In a crystal, the atoms are spaced out at regular intervals in a three-dimensional array, or lattice. They are held in place by electrical forces, but tend to jiggle about a bit. When a crystal is heated, there is more jiggling and the spacing between atoms increases slightly, which is why the crystal expands.
Once he started thinking about how the forces between atoms alter as the crystal expands, Feynman also became intrigued by the behaviour of those forces when the crystal is compressed, so that the atoms are squeezed closer together. He realized that he could treat the forces between pairs of atoms (not just in crystals but also in molecules) as acting like little springs. A spring resists being stretched, but it also resists being compressed. Some of this work covered ground that had already been covered by other people, but Feynman didn’t know that, and worked everything out for himself, from first principles, in his usual way. The basis of Feynman’s approach was that the force on any nucleus in a molecule or in a crystal lattice can be worked out from the distribution of electrical charge on nearby nuclei and in the electron clouds surrounding the nuclei simply from classical electrostatics, once the distribution of the electron cloud is known. You still need quantum mechanics to work out the distribution of electrical charge in the cloud, but once you have done that the rest is, relatively speaking, plain sailing.
After some elegant and sophisticated manipulation of the relevant equations, Feynman was able to prove that the force on each nucleus could be calculated from a relatively simple expression, saving an enormous amount of labour in carrying through these kinds of calculations. His senior thesis, entitled ‘Forces and Stresses in Molecules’ and running to just 30 double-spaced pages of typescript, impressed Slater sufficiently for him to encourage Feynman to write it up in a slightly different form for the Physical Review, where it appeared under the title ‘Forces in Molecules’ later that year (1939). The simplification which so greatly eased the burden of work for chemists trying to calculate the behaviour of atoms in molecules and crystals was also discovered, independently, by another researcher, and is known as the Feynman–Hellmann theorem. It is still used today – not bad for a piece of undergraduate thesis work dating back more than half a century.
One of the most striking things about the senior thesis, though, is that the elegant manipulation of equations is set in a clear, no-nonsense and jargon-free text that carries the authentic Feynman ‘voice’, reading almost like a transcript of a talk. He clearly knew not only how to do physics, but also how to explain physics, at an early age.
By the time Feynman arrived at Princeton, he was more than ready for full-time work in research, and he found in Wheeler exactly the right kind of supervisor to stimulate him in the development of much more original ideas about the way the world works. Early in their relationship, Wheeler set Feynman a few fairly straight-forward problems to investigate. His student’s success at these helped to establish, as if Wheeler had not already realized it, that he was dealing with a special talent. At the same time, Feynman learned how much Wheeler already knew about quantum mechanics. All the while, Feynman was also puzzling over an idea that he had been working on, intermittently, as an undergraduate at MIT. Soon, he was ready to air the puzzle with his new mentor.
Feynman’s jumping-off point, as he stressed when he gave his Nobel lecture in Stockholm in 1965,2 was the conclusion of Dirac’s 1935 book – ‘it seems that some essentially new physical ideas are here needed’. Nowhere was this need for new ideas more obvious than in the puzzle of what was called the ‘self-energy’ of the electron. It all had to do, as far as the undergraduate Feynman could tell, with the concept of a field of force. A charged particle, such as an electron, was supposed to interact with other charged particles by being surrounded by a field of force. The field gets weaker the further away you are from an electron, so it interacts most strongly with nearby charged particles. But, embarrassingly, there is no limit to how strongly the field can interact, provided you get close enough to its source. In fact, the strength of the interaction is inversely proportional to the square of the distance involved. But an electron is a point charge – it has zero radius. So at the electron itself the strength of the field would be one divided by zero, which is infinity. In other words, each electron should have an infinite self-energy – which, among other things, would give it infinite mass, in line with Einstein’s equation E = mc2.
This version of the problem arises even without invoking quantum mechanics; the problem also arises, even more forcefully, in the context of quantum theory. As an undergraduate, Feynman surmised that the answer to the problem must be that the electron does not act on itself at all; from there, it was a small step for him to reject the whole notion of a field, even though the field concept lay at the heart of physics. The prevailing field theories said that if all the charges combine to make a single common field, and the common field interacts with all the charges, there is no way to avoid each charge interacting with itself.
Feynman’s idea was to go back to the older concept of action at a distance, a direct interaction between charges, albeit with a delay.* On this picture, one electron shakes, and a certain time later another electron shakes as a result (the time delay depends on the distance to the second electron and the speed of light). But there is no way for the first electron to interact with itself. This was the state of the idea when Feynman arrived in Princeton. He hadn’t worked out a proper theory along these lines; it was no more than a half-baked idea. But, as Feynman recounted in Stockholm in 1965, he had fallen ‘deeply in love’ with the notion, and ‘I was held to this theory, in spite of all difficulties, by my youthful enthusiasm’ (‘youthful enthusiasm’, of course, sums up Feynman’s approach to all of his work, and life in general, whatever his chronological age). There was, though, a ‘glaringly obvious’ fault with the idea, as he pointed out in that Nobel address. A charged particle such as an electron must, in fact, interact with itself to a certain extent, to account for a phenomenon known as radiation resistance.
All objects resist being pushed about – this is the property known as inertia. In a frictionless environment, such as the inside of a spaceship falling freely in orbit around the Earth, any object will sit still (relative to the walls of the spaceship) until it is given a push, then it will keep moving at a steady speed in a straight line (that is, at constant velocity) until it is given another push (perhaps by bouncing off the wall). The key thing is that it takes a force to make something accelerate – which, to a physicist, means to change its speed or its direction of motion, or both. This is encapsulated in Newton’s Laws of Motion, which became the basis of classical mechanics more than 300 years ago, and which still provide an entirely adequate description of the way things work for most everyday purposes, whether that involves designing a bridge that won’t fall down or a spaceship that will fly to the Moon.
Just why things have inertia – where inertia ‘comes from’ – is not explained by Newton’s laws, and Einstein tried to build inertia into his General Theory of Relativity, without entirely succeeding (but see Chapter 14). But that doesn’t matter for now. What does matter is that if you try to accelerate a charged particle, perhaps by shaking it to and fro with a magnetic field, you discover that it has an extra inertia, over and above the inertia you would find for a particle with the same mass but no electric charge. This extra inertia makes it harder to move the charged particle.
Now this is not just some exotic phenomenon only of interest to physicists. The most common reason for shaking electrons to and fro is to make them radiate electromagnetic energy, in line with Maxwell’s equations. This is what goes on in the broadcast antennas of TV and radio stations. It takes energy to make the electrons in t
he antenna oscillate and radiate the signal you want to broadcast, and it takes more energy (requiring a more powerful transmitter) than it would to shake equivalent uncharged particles, which do not radiate, by the same amount. Hence the name radiation resistance. The effect of radiation resistance can be seen in the electricity bills of every TV and radio station.
One curious feature of the classical description of electrons (the same is true for all other charged particles) and electromagnetic fields is that the interaction between each electron and the field (the self-interaction) actually has two components. The first component looks as if it ought to represent ordinary inertia, but is infinite for a point charge. But the second term exactly gives the force of radiation resistance. So the snag with Feynman’s original idea, that an electron could not act on itself at all, was that even if the idea could be made to work it would remove both terms in the expression, getting rid not just of the unwanted infinity but also of the radiation resistance. This was the state of play when he started thinking seriously about the idea once again at Princeton.
Feynman needed some interaction to act back on the electron and give it radiation resistance when it was accelerated, and he wondered whether this back-reaction might come from other electrons (strictly speaking, any other charged particles) rather than from the ‘field’. As physicists do when trying to get to grips with such problems, he considered the simplest possible example – in this case, a universe in which there were only two electrons. When the first charge shakes, it produces an effect on the second charge, which shakes in response (this, of course, is how the receiver in your radio or TV set works, as electrons in it respond to the shaking of the electrons in the broadcast antenna). But now, because the second charge is shaking there must be a back-reaction which shakes up the first charge. Perhaps this could account for radiation resistance. Feynman calculated the size of the effect, but it didn’t work out properly to account for radiation resistance. Baffled, but still in love with the idea, it was at this point that he took it to Wheeler to discuss.
What Feynman didn’t know was that Wheeler had been interested in the idea of action at a distance for some time, and that this had a respectable pedigree as a backwater of physics.3 So the professor didn’t dismiss his student’s idea as crazy, but set out with him to work through the calculations. To Feynman’s embarrassment, Wheeler pointed out the big flaw with his calculation. It takes a certain time for the second electron to respond to the shaking of the first electron, and the same amount of time before the first electron responds to the shaking of the second electron. So the reaction back on the first electron would occur some time after it had been shaken in the first place – not at the right time to cause radiation resistance. What Feynman had actually described and calculated, albeit in an unconventional manner, was simply ordinary reflection of light.
But Wheeler didn’t stop there. Maxwell’s equations, he pointed out, actually have two sets of solutions. One corresponds to a wave moving outward from its source and forward in time at the speed of light; the other (usually ignored) corresponds to a wave converging on its ‘source’ and moving backwards in time at the speed of light (or, if you like, moving forwards at minus the speed of light, –c). This is rather like the way in which the equations of quantum mechanics can be solved to give a solution corresponding to positive-energy electrons and a solution corresponding to negative-energy electrons. Dirac’s equation, published in 1928, was still the crowning glory of quantum mechanics at the beginning of the 1940s, so it did not seem completely crazy for the two young physicists to take the second solution to Maxwell’s equations seriously. The waves corresponding to the usual solution of the equations are called retarded waves, because they arrive somewhere at a later time than they set out on their journey (the journey time is ‘retarded’ by the speed of light); the other solution corresponds to so-called advanced waves, which arrive before they set out on their journey (the journey time is ‘advanced’ by the speed of light). If the back-reaction from the second electron only involved advanced waves, Wheeler realized, its influence on the first electron would arrive exactly at the right time to cause radiation resistance, because it would have travelled the same distance at the same speed, but backwards in time.
Wheeler set Feynman the task of calculating what mixture of advanced and retarded waves would be required to produce the correct form of radiation resistance. Between them, Wheeler and Feynman also proved that in the real Universe, full of charged particles, all the interactions would cancel out in the right way to produce the same radiation resistance that they had calculated for the simple case.
A key ingredient of their model is that a wave has both a magnitude and a ‘phase’ – if two waves are the same size, but one is precisely out of step with the other, so that the first wave produces a peak where the second wave produces a trough, they are out of phase, and cancel each other. If the two waves march precisely in step, so that the two peaks are on top of each other, they are in phase, and produce a combined wave twice as big as either wave on its own. As a result, it turned out that you need a mixture of exactly half advanced waves and half retarded waves generated by each charge every time it shakes (see Figure 5), using the solution of Maxwell’s equations that is completely symmetrical in time.
Figure 5. The Wheeler–Feynman theory of radiation describes the interaction between two charged particles in terms of waves moving forwards and backwards through time. Because of a phase change at the charged particles, the waves exactly cancel out everywhere except in the region of spacetime between the particles, where they reinforce one another. See also Figure 3, p. 40.
Wheeler discovered that the Dutch physicist Adriaan Fokker had reached a similar conclusion in a series of papers published between 1929 and 1932; but Feynman’s version was much more straight-forward and easier to understand, while Fokker had never developed his ideas further. The half wave which is retarded goes out from the first electron forwards in time, while the half wave that is advanced goes out backwards in time. When the second electron shakes in response, it produces another half retarded wave which is exactly out of step with the first wave, and so precisely cancels out the remaining half retarded wave for all later times, and a half advanced wave which goes back down the track of the first wave to the original electron, in step with that wave, reinforcing the original half wave to make a full wave matching the usual solution to Maxwell’s equations. This half advanced wave arrives at the first electron, of course, at the moment it starts to shake, and causes the radiation resistance. Then it continues back into the past, cancelling out the original half advanced wave from the first electron. The result is that between the two electrons there is a single wave exactly matching the conventional solution to Maxwell’s equations, but everywhere else the wave cancels out, and radiation resistance emerges automatically from the equations, while the infinite self-energy never appears.
‘If we assume all actions are via half-advanced and half-retarded solutions of Maxwell’s equations and assume that all sources are surrounded by material absorbing all the light that is emitted, then we could account for radiation resistance as a direct action of the charges of the absorber acting back by advanced waves on the source.’4 Because of this essential role of the absorber in determining the way radiation is emitted, it is sometimes called the ‘absorber theory’ of radiation.
It took months to work all this out, in the autumn of 1940. In its initial form, as we have described it here, the theory still involved electromagnetic waves, an echo of the field that Feynman had been trying to do away with; but the two researchers also found, to Feynman’s delight, that the whole thing could indeed be described without using Maxwell’s equations at all, but directly in terms of the motion of the particles involved and a suitable time delay, using the Principle of Least Action, without any vestige of a field at all. This only works if the interaction is half advanced and half retarded, when it turns out that interactions can only occur after delays correspo
nding to influences which travel at the speed of light. All of conventional electrodynamics could be written in this new and mathematically simple way, without involving electromagnetic waves or fields at all, provided you were open-minded enough to accept the reality of interactions that travelled backwards in time – that when one electron shakes, another electron may shake as a result before the first electron shakes. And, as Feynman pointed out in his Nobel lecture, aside from gravity, electrodynamics ‘is essentially all of classical physics’. It was another example of the way in which fundamental features of physics could be described in quite different ways to give the same answers.
While the two researchers were working on all this in the autumn of 1940, one day Feynman received a telephone call from Wheeler. He said, ‘Feynman, I know why all electrons have the same charge and the same mass.’ When Feynman asked, ‘Why?’, he replied, ‘Because they are all the same electron!’ And he explained his latest bright idea, that a positron could be regarded as an electron going backwards in time, and that all the electrons and all the positrons in the Universe were really a kind of cross-section through a complicated zig-zag path in which a single particle traversed the Universe, through both space and time, in a complicated knot. When the first flush of his enthusiasm had worn off, Wheeler found that the idea couldn’t really be made to work, not least because there would have to be the same number of positrons in the Universe as there are electrons, since for every zig forwards in time there must be a corresponding zag backwards in time. In fact, there don’t seem to be any positrons in the Universe except ones created in particle interactions, that soon meet up with electrons and annihilate. But Wheeler’s bright idea contained the germ of an important concept which Feynman would later develop in a different way – the idea that changing the direction in which an electron is moving through time is equivalent to changing the sign of its charge, so that an electron going forwards in time is a positron going backwards in time, and vice versa. Positrons could simply be represented, in all quantum mechanical calculations, as electrons going from the future to the past, like the advanced waves in the usually neglected solution to Maxwell’s equations: yet another example of how the same thing could be described in different ways.5