Quantum Man: Richard Feynman's Life in Science (10 page)

Two electrons orbiting in a helium atom, for example, could not normally occupy the same orbit. But they can if they are spinning in two different directions so that they are not in identical quantum mechanical states. If we then consider the next lightest element, lithium, for example, which has three electrons orbiting its nucleus, there is no independent option for the third electron, which therefore must orbit the nucleus in a different, presumably higher-energy, orbit. All of chemistry can be understood to result from the application of this simple principle to predict the energy levels of electrons in atoms.

Similarly, if we bring two identical atoms close together, there is not only an electric repulsion between the negatively charged electrons in one atom and those in the other atom, but the Pauli exclusion principle tells us that there is an additional repulsion because no two electrons can be in the same place in the same quantum state. Thus the electrons in one atom are pushed apart from the electrons in the neighboring atoms so they don’t overlap in the same position in the same orbital configuration. These two effects resulting from the Pauli exclusion principle combine to determine the mechanical properties of all materials that make up the world of our experience.

The Italian physicist Enrico Fermi next explored the statistics of systems of many identical particles with spin ½ such as electrons and demonstrated that the exclusion principle strongly governed the behavior of these many particle states. We now call all such particles with spin
½
,
³
⁄
²
, and so on,
fermions
, after Fermi. Other particles with integral values of spin, including photons—the quanta of electromagnetic fields with spin 1—as well as those particles with no spin whatsoever, are now called
bosons
, after the Indian physicist Satyendra Bose, who, along with Albert Einstein, described the collective behavior of these particles.

B
Y “PLAYING,” AS
he later described it, with the mathematical description of spin ½ particles, in 1928 Dirac was able to derive an equation describing electrons that could account for their spin and was in accord with the requirements for how a theory should behave at relativistic velocities according to Einstein’s theory of relativity. It was a remarkable achievement, and it had an even more remarkable prediction, so remarkable, in fact, that Dirac and most other leading physicists didn’t believe it. The theory predicted that in addition to electrons, there must exist particles just like electrons that were negative energy solutions of the equations. However, since negative energy seemed unphysical—Einstein’s equations always associate positive energy with mass—these particles had to be interpreted somewhat differently.

The interpretation that Dirac came up with reminds me of an old joke I once heard about two mathematicians sitting in a bar in Paris looking at a nearby building. Early on in the lunch they see two people walk into the building. During dessert, they observe three people leaving the building. One mathematician then turns to the other and says, “If one more person goes into that building, it will be empty!”

Similarly, if we interpret negative energy as having less energy than zero, then we might perversely choose to imagine that while an electron has positive energy, and a state with no electrons has zero energy, then a state with negative energy simply has fewer than zero electrons. And a state with a negative energy precisely equal and opposite to the energy of a single electron would thus be described as having one electron less than a state with zero electrons.

While as a formal statement this is consistent, physically it seems ridiculous. What would “one electron less than zero electrons” physically mean? A clue comes from thinking about the charge on an electron: since electrons have negative electric charge, and a state with zero electrons has zero electric charge, then a state with one less than zero electrons would have positive electric charge. Put another way, having a negative number of electrons is equivalent to having a positive number of positively charged particles. Hence, the negative-energy state that appeared in Dirac’s equation could be interpreted as representing a positive-energy particle with a charge equal and opposite to the negative charge on the electron.

But there was at least one major problem with this exotic interpretation.
Only one
particle in nature was known to have a positive charge equal and opposite to the charge on the electron: the proton. But the proton does not resemble the electron at all—for example, it is two thousand times heavier.

Earlier, immediately after deriving his equation, Dirac had recognized another important problem with his negative-energy states. Remember that in quantum mechanics all possible configurations are explored as a system evolves. In particular, as he put it, in his new theory “transitions can take place in which the energy of the electron changes from a positive to a negative value even in the absence of any external field, the surplus energy, at least 2mc
2
in amount, being spontaneously emitted in the form of radiation.” Put in simpler language, an electron could spontaneously decay into the positively charged particle corresponding to the negative-energy state. But this would change the total charge of the system, which is not allowed in electromagnetism. Moreover, if the positively charged particle was the much heavier proton, then such a transition would also violate energy conservation.

In order to address these problems, Dirac made a radical proposal. Remember that electrons are fermions, and therefore only one particle can exist in each different quantum state. Dirac imagined what would happen if empty space actually contained an infinite “sea” of negative-energy electrons, and all available quantum states of these particles could therefore be occupied. There would thus be no available states for real positive-energy electrons to decay into. Moreover, he argued that if by some process a negative-energy state became unoccupied, this would leave a “hole” in the distribution. The hole, corresponding to the absence of a negatively charged electron in the sea, could then be identified with a positively charged particle, which he in turn identified with a proton.

The suspension of disbelief involved in Dirac’s assertion was enormous. It implied first that the vacuum—that is, empty space—somehow contained an infinite number of unobservable particles corresponding to the filled negative-energy levels, and moreover that the odd hole in these filled levels would be observed as a proton, a particle completely unlike the electron except for the magnitude of its electric charge.

As intellectually courageous as it might have been to propose an infinite sea of negative-energy particles, the proposal that holes in this sea represented protons was a rare act of intellectual cowardice for Dirac. The negative-energy states in his equation appeared completely symmetrical to the positive-energy states, suggesting that they had precisely the
same mass
, in manifest contradiction to the fact that the proton is much heavier. Dirac tried to circumvent this apparent problem by supposing that in the filled sea the interactions between particles would be such that the few holes that might appear would receive additional contributions to their mass from these interactions.

Had he had more courage, Dirac could have simply predicted that these holes represented new elementary particles in nature with a mass equal to that of the electron, but with opposite electric charge. But, as he later said, “I just didn’t dare to postulate a new particle at that stage, because the whole climate of opinion at that time was against new particles.”

More charitably, perhaps, Dirac might have hoped to explain all of the known elementary particles at that time, electrons and protons, as resulting from different manifestations of a single particle, the electron. This reflects the spirit of physics, to explain manifestly different phenomena as representing merely two different sides of the same coin. Either way, this confusion did not last long. Other well-known physicists who had examined his theory, including Werner Heisenberg, Herman Weyl, and Robert Oppenheimer, correctly inferred that interactions in the “Dirac sea” would never add mass to the holes and result in the holes having a different mass than the electrons. Ultimately even Dirac was forced to recognize that his theory predicted the existence of a new particle in nature, one he called the
anti-electron
.

Dirac had made his concession in 1931, and he did so just in time. It would only take a year for nature to prove him correct, although there was such skepticism about the possibility of new as-of-yet-unobserved elementary particles that even after finding strong evidence of their existence, the first group to observe the anti-electron, or
positron
as it became known, didn’t believe its own data.

During the 1930s, before particle accelerators were first developed and built, almost all of the information about elementary particles came from observations of the products of nature’s astrophysical accelerators—namely, the cosmic rays that bombard the earth daily, whose origin ranges from as close as our sun, to more energetic sources like exploding stars in distant galaxies at the far reaches of our universe. Two different groups on either side of the Atlantic were examining cosmic ray data in 1932. One group, working in the same laboratory as Dirac at Cambridge in the United Kingdom, under the leadership of Patrick Blackett, told Dirac that they had found evidence of his new particle, but they were too timid to publish their results until they did more tests. In the meantime, perhaps characteristic of a brasher American attitude, Carl Anderson in California published compelling evidence of the existence of the positron in 1933, ultimately leading to the award of a Nobel Prize for his discovery. It is interesting that even after Blackett and his collaborator, Giuseppe Occhialini, were finally induced by Anderson’s discovery to publish their results a year later, they were still hesitant to ascribe this particle to Dirac’s proposal. Ultimately, by the end of 1933 even these experimenters had to admit that if it walks like a duck, and quacks like a duck, it is probably a duck. The properties Dirac predicted agreed strikingly with observations, and like it or not, it appeared that electrons and positrons—the first example of an antiparticle known in nature—could be created in pairs amid the energetic showers produced by cosmic rays bombarding nuclei.

Suddenly positrons were real. Reflecting on his initial hesitation to accept the conclusions of his theory in predicting the existence of antiparticles, Dirac later said, “My equation was smarter than I was!”

I
T WAS IN
the context of these exotic and revolutionary developments that Richard Feynman, in 1947 and 1948, set to work to invent new “pictures” to incorporate Dirac’s relativistic electrons in his own emerging space-time sum-over-paths picture of quantum mechanics. In doing so, he would find that he needed to reinvent his way of doing physics yet again, even as he tried at the same time to reinvent himself, to sort out the deep emptiness of his personal life.

CHAPTER
8

From Here to Infinity

It therefore seems that I guessed right, that the difficulties of electrodynamics and the difficulties of the hole theory of Dirac, are independent and one can be solved before the other.

—R
ICHARD
F
EYNMAN, IN A
LETTER DATED
1947

P
erhaps it took a man who was willing to break all of the rules to fully tame a theory like quantum mechanics that breaks all of the rules. As Richard Feynman turned his attention once again to QED, he was already cultivating a reputation for scoffing at society’s norms in his job, his love life, and his institutional interactions. Even while at Los Alamos he loved creating havoc—finding holes in security fences, entering through them, and then exiting through the main gate when no record existed of his entering, or picking locks and leaving messages in top-secret safes.

Following Arline’s death and his newfound nihilism after the Trinity bomb explosion, he responded to his inner turmoil by lashing out at convention. From then on, he began to revel in being different. While formerly shy with women, he became a womanizer. Within months after Arline’s death, while still at Los Alamos, he began dating beautiful women at a frenzied pace. Two years later, when his grief had finally surfaced, he was able to write a letter to Arline, exposing his pain: “I’ll bet you are surprised that I don’t even have a girl friend (except you sweetheart) after two years. But you can’t help it, darling, nor can I—I don’t understand it, for I have met many girls and very nice ones . . . but in two or three meetings they all seem ashes.”

The liaisons may have left him feeling empty, but they nevertheless continued. When he first arrived at Cornell, he still looked like a student, and in his loneliness he dated undergraduates he met at freshman dances. His pursuit of women was matched in intensity only by his desire to drop them. In 1947, before he provided final grades for his students, he left on a famous cross-country trip with then graduate student Freeman Dyson. The prime purpose of this expedition was to end an entanglement with a woman in Los Alamos. He had continued an intense long-distance courtship with her, and she was causing another woman in Ithaca to lash out at him in jealousy. Meanwhile yet another woman, one of several who apparently had become pregnant by him and aborted their pregnancies, reacted more stoically in a letter to him, in which she also corrected his misspelling of her name.

All the while he remained in Ithaca he never settled into a single abode. He often stayed with friends, usually married ones, and these visits frequently ended badly as a result of his sexual improprieties. A few years later, when he spent a year in Brazil, he actually devised a set of simple rules for seducing women, including prostitutes, at bars. He became famous for seducing women at conferences abroad.

His attractiveness was understandable. He was brilliant, funny, confident, and charismatic in the extreme. He was tall, and had become more handsome as the years passed. His piercing eyes were mesmerizing, and his energy and enthusiasm were addictive.

But it wasn’t only in matters sexual that he flaunted convention. Everywhere he encountered what he viewed as nonsense, he rebelled, often independent of standard formalities. An episode with several psychiatrists who performed a second draft physical on Feynman in the summer of 1947 was worthy of an Abbott and Costello skit and later became famous. As a result of his eruptions during the psychiatric interview, he was declared mentally unfit to serve, a conclusion that caused both him and Hans Bethe to erupt in nonstop laughter for half an hour when he returned back to work.

Feynman would later cultivate these sorts of anecdotes as part of the Feynman mythology he liked to encourage. But in 1947 he was not yet famous, and the buildup of his unconventional attitudes and behavior coincided with what became the most intense two-year period of creative activity in his life, a time corresponding with experimental discoveries that made solving an otherwise obscure mathematical problem more urgent if physical progress was to be made.

The experimental discovery of the positron in 1932 provided remarkable vindication of Dirac’s relativistic QED, representing the first time in history where the existence of a previously unobserved elementary particle was intuited on the basis of purely theoretical reasoning. However, it added, literally, a frustrating new infinite level of confusion for physicists who were trying to make sense of the predictions of the theory. For once the existence of positrons was verified, the horrible complications introduced by the possibility of a Dirac sea and the interactions of both electrons and now these new particles, positrons, with radiation—the very interactions that Feynman had originally hoped to erase from the quantum theory of electromagnetism—confronted physicists squarely in the face.

While the predicted interactions of single electrons with single photons or even with classical electromagnetic light or radio waves were in remarkable agreement with observations, whenever physicists tried to go beyond this simplest approximation, by including multiple quantum interactions or even attempting to address the long-standing problem of an electron interacting with itself—the very problem that Feynman first tried to tackle in graduate school—their answers remained infinite and thus physically untenable. This inability to make sense of a theory that was clearly correct at some deep level could safely be ignored at the time in almost all practical applications, but it wore on a select group of ambitious physicists like a raw and exposed nerve. A sense of the desperation dominating the scene can be gleaned from statements of several among the great theoretical physicists of the time. Heisenberg wrote in 1929 that he was frustrated trying to understand Dirac’s ideas and that he was concerned that he might be “forever irritated by Dirac.” Wolfgang Pauli wrote in 1929 about his concerns (which presciently reflected concerns that many physicists, including Feynman, later voiced about more recent developments in physics): “I am not very satisfied. . . . In particular, the self-energy of the electron makes much bigger difficulties than Heisenberg had thought at the beginning. Also the new results to which our theory leads to are very suspect and the risk is very great that the entire affair loses touch with physics and degenerates into pure mathematics.”

Heisenberg in turn wrote to Pauli in 1935, “With respect to QED . . . we know that everything is wrong. But in order to find the direction in which we should depart . . . we must know the consequences of the existing formalism much better than we do.” He later added in a subsequent paper, “The present theory of the positron and QED must be considered provisional.” Even Dirac said of QED in 1937, “Because of its extreme complexity, most physicists will be glad to see the end of it.”

The concerns were so great that these physicists, and in particular the great Danish physicist Neils Bohr, worried that perhaps quantum mechanics itself might be at the root of the problem and might have to be replaced by different physics. Bohr wrote to Dirac in 1930: “I have been thinking a good deal of the relativity problems lately and believe firmly that the solution of the present troubles will not be reached without a revision of our general physical ideas still deeper than that contemplated in the present quantum mechanics.”

Even Pauli suggested, in 1936, that quantum mechanics might have to be revised when dealing with systems, like Dirac’s hole theory, that allow an infinite number of particles to be present in empty space. For beyond the well-known infinity associated with the self-energy of the electron due to its interaction with its own electromagnetic field, Dirac’s introduction of antiparticles created another new class of infinite interactions that further muddied the quantum waters. These new interactions involved, not the intermediate photons that Feynman and Wheeler had worked so hard to get rid of, but pairs of “virtual” electrons and positrons.

Since physicists now knew that particles and antiparticles can annihilate into pure radiation, and that the reverse process—the complete conversion of energy into mass, could also, in principle, occur. There are constraints on this conversion, however. For example, an electron and its antiparticle, a positron, cannot annihilate into a single particle of radiation (a photon) for the same reason that when a bomb explodes, all the pieces do not fly off in a single direction. If the electron and positron come at each other with equal and opposite velocities, then their total momentum is zero. If a single photon were produced in their annihilation, it would fly off at the speed of light in some direction, carrying nonzero momentum. Thus, at least two photons must be produced when an electron and positron annihilate, so the two emitted particles can fly off in equal and opposite directions as well. Similarly, a single photon cannot suddenly convert into a positron and electron pair. Two photons must come together to produce the final pair.

But remember that with virtual particles all bets are off, and energy and momentum need not be conserved as long as the virtual particle disappears in a time sufficiently short so that it cannot be measured directly. Thus, a virtual photon can spontaneously transform into an electron-positron pair, as long as the electron-positron pair then annihilates and transforms back into the single virtual photon on a short timescale.

The process involving a photon momentarily splitting up into an electron-positron pair is called
vacuum polarization
. It gets this name because in a real medium such as any solid object made of atoms, which contains both positive and negative charges, if we turn on a large external electric field, we can “polarize” the medium by separating charges of different types—negative charges will be pushed in one direction by the field, while positive charges will be pulled in the other direction. Thus a neutral material will remain neutral, but the charges of different signs will spatially separate. This is what momentarily happens in empty space as a photon splits temporarily into a negatively charged electron and its positively charged antiparticle, a positron. Thus, empty space gets briefly polarized.

Whatever we call it, an electron, which previously had to be thought of as having a cloud of virtual photons around it, now had to be thought of as being surrounded by a cloud of virtual photons
plus
electron-positron pairs. In some sense, this picture is just another way of thinking about Dirac’s interpretation of positrons as “holes” in an infinite sea of electrons in the vacuum. Either way, once we include relativity, and the existence of positrons, the theory of a single electron turns into a theory of an infinite number of electrons and positrons.

Moreover, just as emission and absorption of virtual photons by a single electron produced an infinite electron self-energy in calculations, the production of virtual particle-antiparticle pairs produced a new infinite correction in QED calculations. Recall once again that the electric force between particles can be thought of as being due to the interchange of virtual photons between those particles. If the photons could now split up into electron-positron pairs, this process would change the strength of the interaction between particles and would thus shift the calculated energy of interaction between an electron and a proton in an atom like hydrogen. The problem was, the calculated shift was infinite.

The frustrating fact was that the Dirac theory produced very accurate predictions for the energy levels of electrons in atoms as long as only the exchange of single photons was considered and the annoying higher-order effects that produced infinities were not. Morever Dirac’s prediction of positrons had been vindicated by experimental data. Were it not for these facts, many physicists, as Dirac implied, would have preferred to simply dispense with QED altogether.

I
T TURNED OUT
that what was needed to resolve all of these difficulties was neither a wholesale disposal of quantum mechanics, nor dispensing with all these virtual particles, but rather developing a deeper understanding of how to implement the basic principles of quantum theory in the context of relativity. It would take a long circuitous route, and the guidance brought by key experiments, before this fact, hidden in a mire of crushingly complex calculations, would become clear, both to Feynman and to the rest of the world.

The process of discovery began slowly and confusedly, as it usually does. After his
Reviews of Modern Physics
paper was completed, Feynman turned his attention once again to Dirac’s theory. He had decided physics was fun again, and in spite of his otherwise unsettled personal situation, his focus never strayed far from the problem that had obsessed him since he had been an undergraduate—the infinite self-energy of an electron. It was a puzzle he hadn’t yet solved, and it was contrary to his nature to let it go.

He began with a warm-up problem. Since the spin of the electron makes sense only quantum mechanically, Feynman began by trying to understand whether he could account for spin directly within his sum-over-paths formalism. One complication of Dirac’s theory is that a single equation had four separate pieces: one to describe spin-up electrons, one to describe spin-down electrons, one to describe spin-up positrons, and one to describe spin-down positrons. Since the normal conception of spin requires three dimensions (a two-dimensional plane to spin in and a perpendicular axis to spin around), Feynman reasoned that he could make the problem simpler if he first tried to consider a world with just one spatial dimension and one time dimension, where the different sorts of allowed paths were also trivial. Paths would just involve travel back and forth in the one space dimension, namely, a line.

He was able to derive the simplified version of the Dirac equation appropriate for such a two-dimensional world if every time an electron “turned around” from rightward motion to leftward motion, the probability amplitude for that path got multiplied by a “phase factor,” which in this case was a “complex number,” an exotic number that involved the square root of –1. Complex numbers can appear in probability amplitudes, and actual probabilities depend on the square of these numbers, so that only real numbers appear in the final result.