The Greatest Story Ever Told—So Far

by Lawrence M. Krauss

  But the issue is even more extreme than this statement implies. Knowing has nothing to do with it. As I described in the earlier double-slit experiment example, there is no sense in which the particle has at any time both a specific position and a specific momentum. It possesses a wide range of both, at the same time, until we measure it and thereby fix at least one of them within some small range determined by our measurement apparatus.

  • • •

  Following Heisenberg, the next step in unveiling the quantum craziness of reality was taken by an unlikely explorer, Paul Adrien Maurice Dirac. In one sense, Dirac was the perfect man for the job. As Einstein is reputed to have later said of him, “This balancing on the dizzying path between genius and madness is awful.”

  When I think of Dirac, an old joke comes to mind. A young child has never spoken and his parents go to see numerous doctors to seek help, to no avail. Finally, on his fourth birthday he comes down for breakfast and looks up at his parents and says, “This toast is cold!” His parents nearly burst with happiness, hug each other, and ask the child why he has never before spoken. He answers, “Up to now, everything was fine.”

  Dirac was notoriously laconic, and a host of stories exist about his unwillingness to engage in any sort of repartee, and also about how he seemed to take everything that was said to him literally. Once, while Dirac was writing on a blackboard during one of his lectures, someone in the audience was reputed to have raised his hand and said, “I don’t understand that particular step you have just written down.” Dirac stood silent for the longest while until the audience member asked if Dirac was going to answer the question. To which Dirac said, “There was no question.”

  I actually spoke to Dirac, one day, on the phone—and I was terrified. I was still an undergraduate and wanted to invite him to a meeting I was organizing for undergraduates around the country. I made the mistake of calling him right after my quantum mechanics class, which made me even more terrified. After a rambling request that I blurted out, he was silent for a moment, then gave a simple one-line response: “No, I don’t think I have anything to say to undergraduates.”

  Personality aside, Dirac was anything but timid in his pursuit of a new Holy Grail: a mathematical formulation that might unify the two new revolutionary developments of the twentieth century, quantum mechanics and relativity. In spite of numerous efforts since Schrödinger (who derived his famous wave equation during a two-week tryst in the mountains with several of his girlfriends), and since Heisenberg had revealed the basic underpinning of quantum mechanics, no one had been successful at fully explaining the behavior of electrons bound deep inside atoms.

  These electrons have, on average, velocities that are a fair fraction of the speed of light, and to describe them, we must use Special Relativity. Schrödinger’s equation worked well to describe the energy levels of electrons in the outer parts of simple atoms such as hydrogen, where it provided a quantum extension of Newtonian physics. It was not the proper description when relativistic effects needed to be taken into account.

  Ultimately Dirac succeeded where all others had failed, and the equation he discovered, one of the most important in modern particle physics, is, not surprisingly, called the Dirac equation. (Some years later, when Dirac first met the physicist Richard Feynman, whom we shall come to shortly, Dirac said after another awkward silence, “I have an equation. Do you?”)

  Dirac’s equation was beautiful, and as the first relativistic treatment of the electron, it allowed correct and precise predictions for the energy levels of all electrons in atoms, the frequencies of light they emit, and thus the nature of all atomic spectra. But the equation had a fundamental problem. It seemed to predict new particles that didn’t exist.

  To establish the mathematics necessary to describe an electron moving at relativistic speeds, Dirac had to introduce a totally new formalism that used four different quantities to describe electrons.

  As far as we physicists can discern, electrons are microscopic point particles of essentially zero radius. Yet in quantum mechanics they nevertheless behave like spinning tops and therefore have what physicists call angular momentum. Angular momentum reflects that once objects start spinning, they will not stop unless you apply some force as a brake. The faster they are spinning, or the more massive they are, the greater the angular momentum.

  There is, alas, no classical way of picturing a pointlike object such as an electron spinning around an axis. Spin is thus one of the areas where quantum mechanics simply has no intuitive classical analogue. In Dirac’s relativistic extension of Schrödinger’s equation, electrons can possess only two possible values for their angular momentum, which we simply call their spin. Think of electrons as either spinning around one direction, which we can call up, or spinning around the opposite direction, which we can call down. Because of this, two quantities are needed to describe the configurations of electrons, one for spin-up electrons and one for spin-down electrons.

  After some initial confusion, it became clear that the other two quantities that Dirac needed to describe electrons in his relativistic formulation of quantum mechanics seemed to describe something crazy—another version of electrons with the same mass and spin but with the opposite electric charge. If, by convention, electrons have a negative charge, then these new particles would have a positive charge.

  Dirac was flummoxed. No such particle had ever been observed. In a moment of desperation, Dirac supposed that perhaps the positively charged particle described by his theory was actually the proton, which, however, has a mass two thousand times larger than that of the electron. He gave some hand-waving arguments for why the positively charged particle might get a heavier mass. The larger weight could be caused by different possible electromagnetic interactions it had with otherwise empty space, which he envisaged might be populated with a possibly infinite sea of unobservable particles. This is actually not as crazy as it sounds, but to describe why would force us toward one of those twists and turns that we want to avoid here. In any case, it was quickly shown that this idea didn’t hold water—first, because the mathematics didn’t support this argument, and the new particles would have to have the same mass as electrons. Second, if the proton and the electron were in some sense mirror images, then they could annihilate each other so that neutral matter could not be stable. Dirac had to admit that if his theory was true, some new positive version of the electron had to exist in nature.

  Fortunately for Dirac, within a year of his resigned capitulation, Carl Anderson found particles in cosmic rays that are identical to electrons but have the opposite charge. The positron was born, and Dirac was heard to say, in response to his unwillingness to accept the implications of his own mathematics, “My equation was smarter than I was!” Much later he reportedly gave another reason for not acknowledging the possibility of a new particle: “Pure cowardice.”

  Dirac’s “prediction,” even if reluctant, was a remarkable milestone. It was the first time that, purely on the basis of theoretical notions arising from mathematics, a new particle was predicted. Think about that.

  Maxwell had “postdicted” the existence of light as a result of his unification of electricity and magnetism. Le Verrier had predicted the existence of Neptune by using observations of anomalies in the orbit of Uranus. But here was a prediction of a new basic feature of the universe based purely on theoretical arguments about nature at its most fundamental scales, with no direct experimental motivation in advance. It may have seemed like a matter of faith, but it wasn’t—after all, the proposer didn’t actually believe it—and while like faith it proposed an unobserved reality, unlike faith it proposed a reality that could be tested, and it could have been wrong.

  The discovery of relativity by Einstein revolutionized our ideas of space and time, and the discoveries by Schrödinger and Heisenberg of the laws of quantum mechanics revolutionized our picture of atoms. Dirac’s first combination of the two provided a new window on the hidden nature of matter at much sma
ller scales. It heralded the beginning of the modern era in particle physics, setting a trend that has continued for almost a century.

  First, if the Dirac equation was applied more generally to other particles, and there was no reason to believe it shouldn’t be, then not only would electrons have “antiparticles,” as they later became known, so would all the other known particles in nature.

  Antimatter has become the stuff of science fiction. Starships such as the USS Enterprise in Star Trek are invariably powered by antimatter, and the possibility of an antimatter bomb was the silliest part of the plot in the recent mystery thriller Angels & Demons. But antimatter is real. Not only was the positron discovered in cosmic rays, but antiprotons and antineutrons were discovered later as well.

  At a fundamental level, antimatter is not so strange. Positrons are just like electrons, after all, only with the opposite charge. They do not, as many people think, fall “up” in a gravitational field. Matter and antimatter can interact and completely annihilate into pure radiation, which seems sinister. But particle-antiparticle annihilation is just one in a host of new possible interactions of elementary particles that can occur once we enter the subatomic realm. Moreover, one would need a large amount of antimatter to actually annihilate enough matter to even light a lightbulb with the energy produced.

  Ultimately, that is why antimatter is strange. It is strange because the universe we live in is full of matter, and not antimatter. A universe made of antimatter would seem identical to ours. And a universe made of equal amounts of matter and antimatter—which would surely seem the most sensible universe to begin with—would, unless something happened in the meantime, be boring because the matter and antimatter would have long ago annihilated each other and the universe would now contain nothing but radiation.

  Why our world is full of matter and not antimatter remains one of the most interesting issues in modern physics. But recognizing that the real reason why antimatter is strange is simply because you never encounter it once caused me to suggest the following analogy. Antimatter is strange in the same sense that Belgians are strange. They are certainly not intrinsically strange, but if you ever ask in a big auditorium full of people, as I have, for the Belgians to raise their hands, almost no one ever does.

  Except when I lectured in Belgium, as I did recently, and where I learned my analogy was not appreciated.

  Chapter 8

  * * *

  A WRINKLE IN TIME

  For you are a mist that appears for a little time and then vanishes.

  —JAMES 4:14

  Each hidden connection in nature revealed by science since the time of Galileo has led physics in new and unexpected directions. The unification of electricity and magnetism revealed the hidden nature of light. Unifying light with Galileo’s laws of motion revealed the hidden connections between space and time embodied in relativity. The unification of light and matter revealed the strange quantum universe. And the unification of quantum mechanics and relativity revealed the existence of antiparticles.

  Dirac’s discovery of antiparticles came as a result of his “guessing” the correct equation to describe the relativistic quantum interactions of electrons with electromagnetic fields. He had little physical intuition to back it up, which is one reason why Dirac himself and others were initially so skeptical of his result. Clarifying the physical imperative for antimatter came through the work of one of the most important physicists of the latter half of the twentieth century, Richard Feynman.

  Feynman could not have been more different from Dirac. While Dirac was taciturn in the extreme, Feynman was gregarious and a charming storyteller. While Dirac rarely, if ever, intentionally joked, Feynman was a prankster who openly enjoyed every aspect of life. While Dirac was too shy to meet women, Feynman, after the death of his first wife, sought out female companions of every sort. Yet, physics breeds strange bedfellows, and Feynman and Dirac will forever be intellectually linked—once again by light. Together they helped complete the description of the long-sought quantum theory of radiation.

  Coming a generation after Dirac, Feynman was in awe of him and spoke of him as one of his physics heroes. Therefore, appropriately, a short 1939 paper that Dirac wrote, in which he suggested a new approach to quantum mechanics, would inspire the work that ultimately won Feynman a Nobel Prize.

  Heisenberg and Schrödinger had explained how systems behave quantum mechanically starting with some initial state of the system and calculating how it evolves over time. But, once again, light provides the key to another way to think about quantum systems.

  We are accustomed to thinking of light as always going in straight lines. But it doesn’t. This is manifest when you view a mirage on a long straight highway on a hot day. The road looks wet way up ahead because light from the sky refracts, bending as it crosses the many successive layers of warm air near the surface of the road, until it heads back up to your eye.

  The French mathematician Pierre de Fermat showed in 1650 another way to understand this phenomenon. Light travels faster in warmer, less dense air than it does in colder air. Because the warmest air is near the surface, the light takes less time to get to your eye if it travels down near the ground and then returns up to your eye than it would if it came directly in a straight line to your eye. Fermat formulated a principle, called the Principle of Least Time, which says that, to determine the ultimate trajectory of any light ray, you simply need to examine all possible paths from A to B and find the one that takes the least time.

  This makes it sound as if light has intentionality, and I resisted the temptation to say light considers all paths and chooses the one that takes the least time because I fully expect that Deepak Chopra would later quote me as implying that light has consciousness. Light does not have consciousness, but the mathematical result makes it appear as if light chooses the shortest distance.

  Now, recall that in quantum mechanics, light rays and electrons do not act as if they take a single trajectory to go from one place to another—they take all possible trajectories at the same time. Each trajectory has a specific probability of being measured, and the classical, least time, trajectory has the largest probability of all.

  In 1939, Dirac suggested a way of calculating all such probabilities and summing them to determine the quantum mechanical likelihood that a particle that starts out at A will end up at B. Richard Feynman, as a graduate student, after learning about Dirac’s paper at a beer party, mathematically derived a specific example demonstrating that this idea worked. By taking Dirac’s hint as a starting point, Feynman derived results that were identical to those that one would derive using the Schrödinger or Heisenberg pictures, at least in simple cases. More important, Feynman could use this new “sum over paths” formula to handle quantum systems that couldn’t easily be described or analyzed by the other methods.

  Eventually Feynman refined his mathematical technique to help push forward Dirac’s relativistic equation for the quantum behavior of electrons and to produce a fully consistent quantum mechanical theory of the interaction between electrons and light. For that work, establishing the theory known as quantum electrodynamics (QED), he shared the Nobel Prize in 1965 with Julian Schwinger and Sin-Itiro Tomonaga.

  Even before completing this work, however, Feynman described an intuitive physical reason why relativity, when combined with quantum mechanics, requires the existence of antiparticles.

  Consider an electron moving along on a possible “quantum” trajectory. What does this mean? An electron takes all possible trajectories between two points as long as I am not measuring it while it travels. Among these are trajectories that are classically not allowed because they would violate rules such as the limitation that objects cannot travel faster than light (arising from relativity). Now the Heisenberg uncertainty principle says that even if I try to measure the electron along its trajectory over some short time interval, some intrinsic uncertainty in the velocity of the electron remains that can never be overcome. Thus even if
I measure the trajectory at various points, I cannot rule out some weird nonclassical behavior during these intervals. Now, imagine the trajectory shown below:

  For the short time in the middle of the time interval shown the electron is traveling faster than the speed of light.

  But Einstein tells us that time is relative, and different observers will measure different intervals between events. And if a particle is traveling faster than light in one reference frame, in another reference frame it will appear to be traveling backward in time, as shown below (this is one of the reasons relativity restricts all observed particles to travel at speeds less than or equal to the speed of light:

  Feynman recognized that in the latter frame this would look like an electron moving forward in time for a little while, then moving backward in time, then moving forward in time. But what does an electron moving backward in time appear like? Since the electron is negatively charged, a negative charge moving backward in time to the right is equivalent to a positive charge moving forward in time to the left. Thus, the picture is equivalent to the following:

  In this picture one starts with an electron moving forward in time, and then sometime later an electron and a particle that appears like an electron but has the opposite charge suddenly appear out of empty space, and the positively charged particle moves to the left, again forward in time, until it encounters the original electron and the two annihilate, leaving only one electron left over to continue moving.

  All of this happens on a timescale that cannot be observed directly, for if it could be, then this strange behavior, violating the tenets of relativity, would be impossible. Nevertheless, you can be assured that inside the paper in the book you are now reading, or behind the screen of your ebook, these kinds of processes are happening all the time.