The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics - Robert Oerter (2006)
Chapter 6. Feynman’s Particles, Schwinger’s Fields
Scientific truth should be presented in different forms, and should be regarded as equally scientific, whether it appears in the robust form and the vivid colouring of a physical illustration, or in the tenuity and paleness of a symbolic expression.
—James Clerk Maxwell, address to the British Association, 15 September 1870
Quantum electrodynamics (QED) is our first example of a complete relativistic quantum field theory. It only describes electrons, positrons, and photons, leaving out most of what makes up normal matter—protons and neutrons. Nonetheless, QED has the characteristics that all relativistic quantum field theories share: particle-antiparticle symmetry, forces carried by intermediate particles, Feynman diagrams, sum-over-paths, renormalization, shielding of charges, and the perturbation expansion.
As the name implies, relativistic quantum field theories are born of quantum mechanics, and from it, they inherit both wave and particle aspects. The previous chapter described QED in terms of particles: electrons, positrons, and photons, and interactions that are possible for those particles. What does QED look like when we describe it in terms of quantum fields?
The Best of All Possible Worlds
There are two ways of doing classical (that is, prequantum) physics. One possibility is to specify the state of the universe (or some part of it) now and give the rules for how to get from now to now-plus-a-little-bit. That is, we say where each particle is, where it’s going, and the forces that influence its motion. Let’s call this the local approach to physics, because what happens to a particle next depends only on influences (other particles, fields) in the particle’s immediate neighborhood. What we’re doing is taking a snapshot of the universe at a particular time, and using that snapshot, plus the laws of physics, to predict what will be happening at a slightly later time.
The other possibility is to look at where the particle (or particles) starts, where it ends up, and say that somehow what it does in between is optimal; it follows the best path, in some sense. Let’s call this the global approach.
For instance, the light scattered off a fish in an aquarium bends (refracts) when it leaves the tank and enters the air. In the local point of view, a ray of light that leaves the fish simply goes straight ahead as along as it is still in water, according to Maxwell’s equations for light. Then, when the water meets the air, another application of Maxwell’s equations tells us that the ray takes a turn, which depends on the ratio of the speed of light in water versus in air. (This is known as Snell’s law.) Finally, the ray again travels in a straight line until it enters your eye.
In the global approach, we look at the entire path from fish to eye, and ask: What is the optimal path? The answer, as discovered by Pierre de Fermat in 1661, is surprisingly simple: The path actually taken by the light ray is the one that takes the least time to get to the eye.
To understand this, think about a lifeguard at the seashore who wants to save a floundering swimmer. He knows he can run faster than he can swim, so the straight-line path is not the fastest, as he would spend too much time in the water.
Instead, he should run along the beach toward the water, then turn and start swimming. The path of least time is given by the same law as in the case of the light ray in the aquarium.
There is a variation of this least time principle that works for particles: It’s called the least action principle. According to this principle, we calculate the difference between the kinetic energy and the potential energy (this difference is called the Lagrangian) for each point along the path. Then add the Lagrangian values for the entire path. This sum gives the action for the path. The least action principle declares that the path actually taken by the particle is the path with the smallest action. In the global approach we look at the whole path from beginning to end, not just what’s happening at one instant. Then, from among all possible paths, we choose the optimal path.
It turns out that the local and global approaches are mathematically identical. Think of the Lagrangian as something that summarizes all the equations of the theory. The least action principle tells how to extract those equations from the Lagrangian. By a careful choice of the Lagrangian, we end up with exactly the same equations as in the local approach. The two approaches, therefore, give the same predictions about how particles and fields will behave. Philosophically, however, they seem totally different. In the local approach, objects only “know about” things that are nearby. Only local fields and objects can influence them. In the global approach, objects seem to “know about” everything they’re going to encounter on their way to the finish line—they somehow see the whole obstacle course and pick the best path through it.
Schrödinger’s equation for quantum mechanics was in the best local tradition. If you know the quantum field for the electron at some time, you can find it for all later times. Apart from some cryptic comments in one of Dirac’s papers, no one had tried to formulate quantum mechanics in a global, least-action perspective before Feynman’s Ph.D. thesis. Feynman set out to treat quantum mechanics using the least action principle, and this led him to the radically different view of particles in his sum-over-paths approach. We can already see a connection: In the classical physics case, the particle “looks ahead” and chooses the path with the smallest action; for Feynman, a quantum particle looks ahead on all possible paths, and decides its probability of being in a certain place based on the result of the sum over all paths.
Imagine the Lagrangian forming a trough through which the particle moves:
In classical physics, the particle stays in the very bottom of the trough, in order to end up with the smallest possible action. In quantum mechanics (and also in quantum field theory), the particle follows all paths, including ones that go high up on the sides of the trough and zigzag all over the place. Now, a particle’s quantum field can be positive or negative (to be precise, it is a complex number, but that doesn’t matter as far as we are concerned). As the particle moves along a path, the quantum field oscillates from positive to negative. This is called the phase of the quantum field: whether and by how much it is positive or negative. According to Feynman, the oscillation of the phase is controlled by the Lagrangian, the same quantity that appears in the least action principle. The steeper the slope of the trough, the faster the phase of the particle oscillates. As a result, very small differences in paths on the walls of the trough cause large differences in phase. Then, when we add up all the paths, those on the walls tend to cancel out, because for every path with a positive phase there is a nearby path with a negative phase. In contrast, in the bottom of the trough, phases don’t change very fast, so all nearby paths have positive phase, or else they all have negative phase. Instead of canceling out, they reinforce each other. It is rather like gossip: the more directly it is transmitted to you, the less likely it is to vary from the original source. Gossip that takes a roundabout path through 15 or 16 people is unlikely to be reliable, and in fact, may contradict what you’re hearing from other sources. In that case, you get a cancellation: Since you are hearing contradictory stories, you don’t believe either version.
Now we can ask again a question we have only touched on so far: If I’m made of particles, and my particles behave in such bizarre ways, traveling on every possible path, winking in and out of existence, then why is it that I don’t also behave that way? Why can’t I stop at the bank and the school and the post office simultaneously on my way to work? The answer lies in the fact that the larger an object is, the faster its phase oscillates. My body is made of something like 1028 protons, neutrons, and electrons, so my phase oscillates faster, by a factor of about 1028, than the phase of a single proton. As a result, all paths but the least action path experience cancellations from nearby paths. To say it another way, the trough for me, or for a rock, or a chair, is so narrow and deep that I have to stay at the bottom of it.
From Feynman’s approach, then, we can see how the world can look quantum mechanical when we look at small particles, but still look classical for everyday objects. According to Feynman, we don’t live in “the best of all possible worlds”—we actually live in all possible worlds. But most of these worlds cancel each other out, leaving, for large objects like ourselves, only the “best” world, the one with the smallest action.
Feynman’s diagrammatic approach was tremendously efficient for doing calculations. Feynman, however, was not the first person to calculate an experimentally testable number using relativistic quantum field theory. That honor went to another brilliant young theorist, Julian Schwinger.
“Schwinger was to physics what Mozart was to music,” according to one of his colleagues.1 He started college at the age of 16 and wrote his Ph.D. thesis before he got his bachelor’s degree. In personality, he was a complete contrast to Feynman. Feynman was a casual dresser, a practical joker, and outspoken. Schwinger was shy and introverted. He always worked in his jacket and tie, never loosening the tie in the hottest weather. He would show up at the physics department sometime in the afternoon, have a breakfast of “steak, French fries and chocolate ice cream,” according to a friend, and start work at 7 or 8 P.M. “He would be leaving for home in the morning when other members of the department were arriving.”2 Where Feynman thought in pictures, Schwinger thought in equations. Rather than developing radically new ways to think about a problem, Schwinger was a genius at taking a known formulation and developing it to its fullest extent. His writing style was dense; to some, impenetrable. “Other people publish to show you how to do it, but Julian Schwinger publishes to show you that only he can do it,” said one critic.3 In fact, many later developments of quantum field theory by other people were discovered buried in Schwinger’s earlier published papers. No one at the time had understood what he was doing well enough to make use of it.
Schwinger’s approach to QED was from the field theory point of view. In Chapter 1 we learned that a classical field is simply an arrow at each point of space. A quantum field, according to Schwinger, can be pictured as a quantum harmonic oscillator at each point in space. We know what a quantum harmonic oscillator looks like: a bowl with equally spaced energy levels. When there are no particles anywhere (a state called the vacuum), all of the oscillators are in their lowest energy level. A single particle at one point in space is represented by moving the oscillator at that point up one notch to the next higher energy level:
The difference in energy is exactly mc2, the rest energy of a particle (or quantum) of the field. Bumping the oscillator to the next higher energy level adds an equal amount of energy; another mc2. Therefore, this level represents two particles at that point in space. (For fermions—like the electron, for example—the Pauli exclusion principle prohibits two particles in one spot. For bosons, however, there is no difficulty in having any number of them in one place.) To represent a moving particle, imagine the energy leaping from one oscillator to the next. One oscillator drops down in energy just as a neighboring oscillator jumps up in energy.
In 1949, Freeman Dyson, who understood both Schwinger’s field theory point of view and Feynman’s sum-over-paths approach, proved that the two approaches were actually identical. It was the same mathematics, the same theory, just two different interpretations of the mathematics. For physicists, this meant they could use the Feynman diagrams that made calculations so convenient, and fall back on Schwinger’s rigorous approach when there were ambiguities, or when the powerful techniques of field theory were called for. For philosophers, it reinforced the answer that quantum mechanics gave to the old question: Wave or particle? The answer: Both, and neither. You could think of the electron or the photon as a particle, but only if you were willing to let particles behave in the bizarre way described by Feynman: going all ways at once, spontaneously popping into existence and disappearing again, interfering with each other and canceling out. You could also think of it as a field, or wave, but you had to remember that the detector always registers one electron, or none—never half an electron, no matter how much the field had been split up or spread out. In the end, is the field just a calculational tool to tell you where the particle will be, or are the particles just calculational tools to tell you what the field values are? Take your pick.
QED survives, nearly unchanged, as a subset of the Standard Model. Other parts of the Standard Model are harder to test, but the QED part has been subjected to many tests and has never yet failed. The first experimentally verified result of the theory, published by Schwinger in 1948, has been called the most accurate prediction of any scientific theory, namely the magnetic properties of the electron. An electron, we know, has a spin like a top. In empty space, a top would always point in the same direction, but here on earth, a top turns or precesses as it spins. An electron, we know, behaves like a tiny magnet. Now, a magnet, when placed in an external magnetic field, precesses like a top.
If the electron were a tiny spinning ball of charge, Maxwell’s equations would let us find the effective magnetic strength, known as the magnetic moment, from the spin and the charge of the electron. But we already know that we can’t think of the electron that way—the spinning ball picture of the electron creates too many problems. Fortunately, the Dirac equation comes to our rescue. The Dirac equation predicts that an electron will precess exactly twice as fast as the spinning ball picture predicts. The fact that the Dirac equation got the factor of 2 correct was a tremendous success, and helped enthrone that equation as the correct description of the electron. By 1947, however, “the sacred Dirac theory was breaking down all over the place,” as Schwinger put it.4 New, more precise experiments revealed that instead of a factor of 2, the ratio was more like 2.002. This small (only one-tenth of one percent) but significant difference was the main impetus behind Schwinger’s and Feynman’s development of QED. Schwinger won the race. The cloud of virtual particles around the electron, like a cloud of gnats around a hiker, altered the motion of the electron by just the right amount. Today, more precise calculations include all Feynman diagrams up to eight interactions—more than 900 diagrams in all—and are done with the help of computers. Currently, the value is 2.0023193044, with an uncertainty of about 2 in the last digit. This agrees precisely with the experimental results (where the uncertainty is much less), making this one of the most accurate theoretical predictions of all time. This is the accuracy you would need to shoot a gun and hit a Coke can—if the can were on the moon!
By the way, when the same magnetic ratio is measured for the proton, it turns out to be nowhere near 2; in fact, it’s about 5.58. Clearly, the Dirac equation, which works so beautifully for the electron, is inadequate to describe the proton. In hindsight, this failure of the Dirac equation was the first hint that the proton is not a point-like particle, as the electron is, but is built of still smaller particles.
Not all experiments can be done with such stunning one-part-in-a-billion precision. Many other aspects of QED have been tested experimentally, however. The increase of mass of particles in accelerators (in accordance with special relativity); the probability a photon will produce an electron-positron pair; the probability a photon will scatter in any given direction off an electron; the detailed explanation of the spectra of atoms; all can be calculated in QED and all have been confirmed (with appropriate allowances for phenomena involving particles other than electrons, positrons, and photons). What’s more, QED has continued to be accurate at smaller and smaller distances. As physicist Robert Serber has pointed out, “Quantum mechanics was created to deal with the atom, which is a scale of 10-8 centimeters,” or one hundred millionth of a centimeter. Then, it was applied to the nucleus of the atom, a hundred thousand times smaller, “and quantum mechanics still worked.... Then after the war, they began building the big machines, the particle accelerators, and they got down to 10-14, 10-15, 10-16 centimeters, and it still worked. That’s an amazing extrapolation—a factor of a hundred million!”5 Not bad for a theory that was thought throughout most of the 1930s and 1940s to be so “crazy” and “ugly” (in Victor Weisskopf’s words)6 that it wasn’t worth working on.
Can we really believe a theory that is this crazy? After all, my body is made of protons, neutrons, and electrons. Am I to believe that every time I walk from the couch to the refrigerator, my electrons make virtual trips to Hawaii, the Bahamas, and Mars? As Richard Feynman put it:
It is not a question of whether a theory is philosophically delightful, or easy to understand, or perfectly reasonable from the point of view of common sense. The theory of quantum electrodynamics describes nature as absurd from the point of view of commonsense. And it agrees fully with experiment. So I hope you can accept nature as she is—absurd.7
The Great Synthesis
The full theory of QED finally came together in 1948 at an amazing conference of the top theoretical physicists that met at Pocono Manor Inn in Pennsylvania. Many of the participants had been doing applied work during the war years, such as the atomic bomb project, and they were eager to return to real physics, fundamental physics. Schwinger gave a five-hour talk in which he showed how all of the infinities that kept cropping up in calculations could be removed. His methods, convoluted but rigorous and precise, left heads spinning. Then, Feynman got his chance. He introduced his diagrams along with other unconventional techniques, he made arbitrary assumptions, he invented rules out of thin air when he got into difficulty, and waved away the audience’s objections. He got the same results as Schwinger, but he convinced no one that his grab bag of techniques would work. After the conference, Robert Oppenheimer received a letter from a Japanese physicist, Sin-itiro Tomonaga, who had independently come to the same conclusion as Schwinger and Feynman. All the infinities could be avoided; QED was a consistent and usable theory. Feynman, Schwinger, and Tomonaga received the Nobel Prize in physics in 1965.
QED describes only the interactions of electrons, positrons, and photons. It has nothing to say about nuclear forces, or about the multitude of new particles that started showing up in experiments. For the first time, though, physicists had a theory that brought together quantum mechanics and special relativity. QED was the first complete relativistic quantum field theory. Other theories would follow as physicists tried to understand the nuclear forces and the new particles, but again and again relativistic quantum field theory provided the theoretical structure underpinning the solution.
In nineteenth-century physics, the universe contained two things: particles and fields. Particles were tiny and hard, like small billiard balls. Fields were elastic and spread throughout space. Particles produced fields according to their electric charge and their motion, and particles responded to the fields of the particles around them. Relativistic quantum field theory completely eliminates the distinction between particles and fields. Matter (the electrons and positrons) and forces (the photons) are both described in the same way—by quantum fields. Quantum fields combine particle nature and field nature in a single entity. The quantum field spreads out through space, just like a classical field, but it is quantized: When you try to measure the field, you always find a whole particle, or two particles, or none. You never detect half an electron or a fractional photon. If fields were peanut butter, classical fields would be smooth and quantum fields would be chunky.
This dual nature of quantum fields is reflected in the two descriptions physicists employ when working with them. Feynman diagrams picture the field as a dense mesh of tiny interacting particles. Schwinger instead pictured the field as a continuous collection of harmonic oscillators, spread throughout the universe. Both pictures are useful in their own way. Schwinger’s field picture makes the symmetries of the theory apparent and emphasizes the continuous nature of the quantum field. Feynman diagrams are easy to visualize, they emphasize the particle aspect of the theory, and they greatly simplify the process of calculation. We will need to call both pictures into play in the coming chapters.
Special relativity is built into the structure of relativistic quantum field theory in a fundamental way. We are thus guaranteed that (real, rather than virtual) massless particles will always travel at the speed of light, and that particles with mass will never exceed it. The twists and flips we can do with Feynman diagrams, the fact that an electron moving backward in time is identical to a positron, and the ability to create matter and antimatter from pure (photon) energy, these are all consequences of the intimate connection with special relativity.
Relativistic quantum field theory harmonizes the formerly discordant strains of fields, relativity, and quantum mechanics. In achieving that harmony, physicists had to throw out much of what they thought they knew. Physicists began, back in the days of Galileo, by describing the behavior of everyday objects, falling rocks and rolling spheres. As they moved farther from everyday experience into the world of atoms and molecules, electrons, protons, and neutrons, they carried the tools they had developed: the concept of a particle as an ideal mathematical point, and the concept of a wave in an ideal continuous medium. In hindsight, it is perhaps not so surprising that these concepts, abstracted from experience with large objects, should fail when confronted with the world of the very small. The new concept, the quantum field, was an attempt to describe a phenomenon so far beyond our experience that analogies to everyday objects simply didn’t work anymore. The microworld was a much stranger place than anyone could have imagined. Even physicists trained in the mathematical techniques of field theory need help to develop physical intuition about this new world. Here the two sets of mental images, field and particle, are of immeasurable assistance.
Relativistic quantum field theory is a milestone in intellectual history: a theory that goes beyond what can be represented in simple pictures and analogies, where the mathematics must be the guide and the mental images of wave and particle are of secondary importance. The images can help shape a physicist’s intuition, but the mathematics is the ultimate arbiter. Relativistic quantum field theory finally answered the long-standing question of the nature of light: Is it particle or wave? The answer: Light was something completely new—a quantum field, neither particle nor wave, but with aspects of both. But the revolution in thought goes deeper than that. Matter, too, can be described by a quantum field. Electrons are just as ineffable and evanescent as photons. If electrons, then why not protons and neutrons, too? Could all matter be described by quantum fields? Was the world of solid objects going to dissolve into an amorphous sea of probabilities and virtual particles? The answers would only come with the development of the Standard Model.