by John Gribbin
Wheeler decided that Feynman’s next task, in the spring of 1941, should be to give a talk describing the work on direct action at a distance and time-symmetric electrodynamics. Learning how to present work in front of your peers in seminars is an essential part of the training of a research student. Although the first talk is always a fairly informal occasion, for the benefit of people at the student’s home institution, it is a nerve-racking occasion for most students. Feynman not only had to present a highly controversial new idea, but at Princeton in those days, even for an internal seminar, his audience would include Eugene Wigner, one of the leading quantum theorists, Henry Norris Russell, one of the greatest astronomers of the time, John von Neumann, regarded as the smartest mathematician of his generation, Wolfgang Pauli, one of the quantum pioneers who just happened to be at Princeton on a visit from Switzerland, and Albert Einstein, who worked at the nearby Institute for Advanced Study.
Feynman has described his nervous preparations for the talk in Surely You’re Joking, and how at the start of the talk his hands shook as he pulled his notes out from the brown envelope in which he had put them for safekeeping:
But then a miracle occurred, as it has occurred again and again in my life, and it’s very lucky for me: the moment I start to think about the physics, and have to concentrate on what I’m explaining, nothing else occupies my mind – I’m completely immune to being nervous. So after I started to go, I just didn’t know who was in the room. I was only explaining this idea, that’s all.6
After the presentation, Pauli spoke up, saying that he didn’t think the theory could possibly be right, and turned to Einstein to ask if he agreed. ‘No’, replied Einstein, softly. ‘I find only that it would be very difficult to make a corresponding theory for gravitational interaction.’ But he didn’t think that was any reason to reject the Wheeler–Feynman theory, which was a possible way forward.
It was hardly a ringing endorsement, but the theory had stood up to its first test, and Einstein had not dismissed it out of hand. Feynman’s next task was to try to find a way to develop a quantum mechanical version of the theory. At first, he hesitated about getting to grips with this, because Wheeler kept claiming that he was making strides in that direction himself, but Wheeler’s efforts always seemed to take him up a blind alley, leaving the field clear for Feynman. It was his effort to develop a version of quantum theory that did away with fields and simply involved action at a distance that became the topic of his PhD thesis; partly because of his concentration on the thesis in the months that followed, and partly because of the interruption caused by war work, the absorber theory of radiation was only formally published in 1945, in the journal Reviews of Modern Physics, in a paper under the joint names of Wheeler and Feynman, but actually written by Wheeler in a style which Feynman thought unnecessarily complicated.7
Until Feynman came on the scene, the way quantum mechanics had been developed was using the Hamiltonian method, which we described in Chapter 3. This involves a wave function which describes the behaviour of quantum entities, such as electrons and photons, and differential equations which describe how the wave function changes from one instant to the next. The approach is conceptually similar to using the equations of motion based on Newton’s laws to describe the way in which the position of a ball thrown through an upper-storey window changes from one instant to the next as it moves along its path. In classical mechanics, as Feynman had learned back in high school from Abram Bader, you could use the Principle of Least Action to determine the entire path of the ball, from your hand through the window, without calculating how its velocity and other properties changed at each point along the path. This, essentially, is the Lagrangian approach which Feynman had scorned as an undergraduate, perhaps because it seemed too easy for the kind of problems he was working with then. Thinking in terms of particles rather than waves, the key properties involved are the positions and the velocities (strictly speaking, their momenta, but the difference doesn’t matter here) of the particles.
If you are only interested in the state of a system at a particular moment in time, it is relatively easy to set up a description of the action in terms of a function (a mathematical expression) called the Lagrangian, which depends on the velocities and positions of all the particles at that time. Starting with the Lagrangian, it is straight-forward, for those who feel the need, to convert into the Hamiltonian formulation and work out the quantum mechanics in the way that people had already become used to by the early 1940s. But the action involving advanced and retarded interactions (or even retarded interactions alone) brings in the key variables at two different times, simply because when one electron shakes there is a delay before the second electron shakes. It was not at all obvious to Feynman (or anyone else) how to formulate the quantum mechanical version of the appropriate Lagrangian involving two different times.
While Feynman was struggling with this problem in the spring of 1941, one evening he went to a beer party at the Nassau Tavern in Princeton. There, as he recounted in the Nobel lecture, he got talking with a physicist who had recently arrived from Europe, Herbert Jehle. Jehle asked what Feynman was working on, and Feynman told him, ending up by asking, ‘Do you know any way of doing quantum mechanics, starting with action – where the action integral comes into quantum mechanics?’ ‘No’, Jehle replied. But he did know of an obscure paper by Dirac, published eight years previously, which made some use of the Lagrangian. He offered to show it to Feynman the next day.
Next day the two physicists went to the Princeton University library, dug out the relevant bound volume of the Physikalische Zeitschrift der Sowjetunion (hardly the place Feynman would have looked without being pointed in the right direction!) and went through Dirac’s paper together. It was exactly the kind of thing Feynman was looking for. Under the title ‘The Lagrangian in Quantum Mechanics’, Dirac pointed out that quantum mechanics had been developed by analogy with the Hamiltonian approach to classical mechanics, went on to say that the Lagrangian approach seemed to be more fundamental, and pointed out the desirability of finding the counterpart in quantum mechanics to the Lagrangian in classical mechanics – just what Feynman was trying to do. What Dirac then described in that paper was a way of carrying the wave function description of a quantum system forwards in time by a tiny step, an infinitesimal amount. This doesn’t sound much in itself, but physicists are used to dealing with infinitesimals, which appear in differential equations which can then be integrated up to deal with much larger (macroscopic) steps in time or space. Dirac hadn’t done that; he had only found a way of taking the wave function forwards in time by infinitesimal steps. But what caught Feynman’s eye was the way Dirac repeatedly said in the paper that the function he was using was ‘analogous’ to the Lagrangian in classical mechanics. It was an imprecise use of language that obscured, rather than clarified, what Dirac was driving at.
‘What does he mean?’, Feynman asked Jehle. ‘What does that mean, analogous? What is the use of [a word like] that?’ Jehle didn’t know. ‘You Americans!’, he laughed, ‘you always want to find a use for everything!’ Feynman decided that perhaps Dirac meant that the two expressions were equivalent to one another, but Jehle disagreed. To find out, Feynman tried setting the expressions equal to one another and working through the simplest version of the resulting equations from Dirac’s paper. It didn’t quite work; he had to put a constant in as well, making the two expressions proportional to one another, not exactly equal. When he did so, though, everything fell into place, and at the end of the calculation he came out with the familiar Schrödinger equation of quantum mechanics. So he turned from the blackboard and said, ‘Well, you see Professor Dirac meant that they were proportional.’ To a physicist such as Feynman, the terms ‘equal’ and ‘proportional’ are more or less interchangeable, because if two mathematical expressions are proportional to one another, one expression is simply the other expression multiplied by a constant, and the effect of a constant on the equations is so trivial tha
t it can largely be ignored.
Professor Jehle’s eyes were bugging out – he had taken out a little notebook and was rapidly copying it down from the blackboard, and said, ‘No, no, this is an important discovery. You Americans are always trying to find out how something can be used. That’s a good way to discover things!’ So I thought I was finding out what Dirac meant, but, as a matter of fact, I had made the discovery that what Dirac thought was analogous was, in fact, equal. I had then, at last, the connection between the Lagrangian and quantum physics, but still with wave functions and infinitesimal times.8
In 1946, Feynman had a chance to find out what Dirac had really meant by that word ‘analogous’. Both Dirac and Feynman were present at the Princeton bicentennial celebration in the autumn of 1946, and Feynman took the opportunity of mentioning the 1933 paper. Did Dirac remember the paper? ‘Yes’, he replied. Feynman described the functions involved. ‘Did you know that they are not just analogous, they are equal, or rather proportional.’ Dirac replied, ‘Are they?’ Feynman said, ‘Yes.’ Dirac said, ‘Oh, that’s interesting.’ He really had not known, until Feynman told him, that the quantity he had described in the 1933 paper was indeed the Lagrangian required as the basis for a new understanding of quantum physics.9
It was only a couple of days after going through Dirac’s old paper with Jehle that Feynman had the flash of insight which allowed him to use this Lagrangian to solve problems involving paths through space and time joining events a finite distance apart, instead of only an infinitesimal distance apart. These four-dimensional trajectories are known as world lines, and can be represented in two-dimensional graphs by imagining all of the three dimensions of space compressed into a single direction ‘across the page’ with the passage of time denoted by separation ‘up the page’ (see Figure 6). A line on such a diagram represents the history of a particle as it takes a certain amount of time to move from A to B, and beyond. The insight Feynman had, while lying in bed one night, unable to sleep,10 was that you had to consider every possible way in which a particle could go from A to B – every possible ‘history’. The interaction between A and B is conceived as involving a sum made up of contributions from all of the possible paths that connect the two events.
Figure 6. By representing time ‘up the page’ and space ‘across the page’ physicists can describe in simple geometric terms how particles move through spacetime. Particle 1 sits in one place, moving only in time (getting older). Particle 2 goes on a journey which takes it past the points A and B at different times. This is a spacetime diagram.
For obvious reasons, this became known as the ‘sum over histories’, or ‘path integral’ approach to quantum mechanics. A useful, if slightly imprecise, way to think of this is that the least action idea in effect gives you the integral (or sum) along a single trajectory, while the path integral approach extends this to include all possible trajectories, summing up (integrating) the paths themselves together, not just integrating along one path. We will explain the technique more fully in Chapter 6, in the context of Feynman’s later work. As Feynman put it in his Nobel lecture, ‘the connection between the wave function of one instant and the wave function of another instant a finite time later could be obtained by an infinite number of integrals’; and this kind of infinity is no problem to the mathematicians, because (unlike dividing by zero) it is just the kind of thing that calculus is set up to handle, so that the equations give you a finite answer after you add up (integrate) the infinite number of infinitesimally small steps. ‘At last’, said Feynman, ‘I had succeeded in representing quantum mechanics directly in terms of the action.’ And although the representation had been set up using the idea of the wave equation as a guide, once the structure was in place that scaffolding could be removed without a trace, leaving a completely new description of quantum mechanics.
This new picture of the world was based on the idea of amplitudes. For each possible way that a particle can go from one point to another in spacetime there is a number, which Feynman called the amplitude. This number involves the action multiplied by a certain constant involving the mathematical i, the square root of –1, so it is called a complex number.
The important thing about a complex number is that it has two parts, which can be thought of in terms of little arrows. An arrow has a certain length, and it points in a certain direction. That’s all there is to complex numbers, a length and a direction, which are kept apart by attaching this number i to one of them. Each little arrow represents a complex number. You add up the little arrows by placing them head to tail, but with each successive arrow turned through the appropriate angle (so it points in the right direction), and then drawing a new arrow from the tail of the first arrow to the head of the last arrow, which gives you both a length and a direction for the arrow that is the sum of all the little arrows. The direction of the arrow is also related to the phase of a wave. If you imagine the arrow attached to the spoke of a wheel, rotating around a fixed point, the direction in which the tip of the arrow points as the wheel turns goes up and down like a wave. When the arrow is pointing straight up, it corresponds to a peak in the wave. After the wheel has turned by 90 degrees, the arrow is horizontal. After another 90 degrees, it is pointing straight down, and that corresponds to a trough in the wave. After a further 90 degrees, it is horizontal again, and after the final 90 degrees needed to make up a circle it is back where it started. So the extent to which two waves are in phase with one another – how closely they match in step – can be described in terms of two little arrows, pointing in slightly (or considerably) different directions, or in terms of complex numbers.
The probability of a particle following a particular history is given by the square of the amplitude, and the probability of it getting from A to B at all is given by adding up all the amplitudes first, and then squaring the result.
In spite of Jehle’s comments about the attitude of ‘you Americans’, Feynman did not immediately try to find a use for any of this. He did not apply his discoveries to practical problems in his thesis, but wrote down the general principles of his new approach to quantum mechanics, and developed the mathematical formalism. As we have seen, by this time he had plenty of other things on his mind, including both Arline’s state of health and his war work, and at the time the thesis was written up, in the spring of 1942, his main concern was simply to get enough down on paper to satisfy the PhD examiners. Because of all these factors, it wouldn’t be until 1948 that the path integral approach to quantum mechanics was published in the journal Reviews of Modern Physics, making it widely available to anyone who was interested. And it was only in the late 1940s, as well, that the new approach was triumphantly successful in solving the problems of quantum electrodynamics, as we discuss in Chapter 6. Perhaps for these reasons, the importance of Feynman’s PhD thesis11 itself is sometimes overlooked, and it’s worth emphasizing its value here before we go on to look at all the events which helped to delay the completion of the theory of quantum electrodynamics.
One of the strange features of quantum mechanics is that right from the moment it was invented (or discovered) in the mid-1920s, there were two completely different descriptions of the quantum world. One was Schrödinger’s approach, based on waves; the other was Heisenberg’s approach, based on particles.12 Both versions of quantum theory had been shown to be exactly equivalent to one another (by Dirac, among others), but most physicists worked with the wave equation (and still do), because it seemed comforting and familiar to people who had been brought up on wave equations. Now, Feynman had found a third approach to quantum mechanics, based on the action; arguably, that alone is enough to rank him with Schrödinger, Heisenberg and Dirac in the physicists’ pantheon. It gave the same answers as the other two versions of the theory everywhere that they could be compared, and it could even handle problems that could not be solved using the wave function approach. It is also both relatively simple to use and clearly tied, through the Lagrangian, to the understanding of classical mec
hanics developed since the time of Newton. Wheeler has gone so far as to say that Feynman’s PhD thesis marked the moment ‘when quantum theory became simpler than classical theory’.13
This isn’t just hindsight. In the same reminiscence, Wheeler tells how before Feynman had even completed his PhD Wheeler was visiting Einstein one day and couldn’t resist telling him the news:
Feynman has found a beautiful picture to understand the probability amplitude for a dynamical system to go from one specified configuration at one time to another specified configuration at a later time. He treats on a footing of absolute equality every conceivable history that leads from the initial state to the final one, no matter how crazy the motion in between. The contributions of these histories differ … in phase. And the phase is nothing but the classical action integral, apart from the Dirac factor, ħ. This prescription reproduces all of standard quantum theory. How could one ever want a simpler way to see what quantum theory is all about!
Indeed, as we shall see in Chapter 6, Feynman’s path integral approach also works just as well in describing classical mechanics – so much so that Wheeler himself introduced Feynman’s idea in the graduate course in classical mechanics that he was teaching that year. The point is not so much that quantum mechanics became simpler than classical mechanics, but that they became part of the same system – the same world view. Using Feynman’s path integral approach, based on the Principle of Least Action, there is no longer any difference between classical mechanics and quantum mechanics, except for a trivial adjustment to the mathematics. Using the sum over histories approach, it is, in fact, possible to teach classical mechanics from the beginning (right back in school) in such a way that quantum mechanics follows on as a straightforward and logical development from familiar ideas.
