The Greatest Story Ever Told—So Far

by Lawrence M. Krauss

  Nevertheless, if such a trajectory is possible in the invisible quantum world, then antiparticles must exist in the visible world—particles identical to known particles but with opposite electric charge (which appear in the equations of this theory as if they were particles going backward in time). This also makes it possible for particle-antiparticle pairs to spontaneously appear out of empty space, as long as they annihilate in a time period quickly enough so that their brief existence cannot be measured.

  With this line of reasoning, not only did Feynman give a physical argument for the existence of antiparticles required by the unification of relativity and quantum mechanics, he also demonstrated that at any time we cannot say that only one or two particles are in some region. A potentially infinite number of “virtual” particle-antiparticle pairs—pairs of particles whose existence is so fleeting that they cannot be directly observed—can be appearing and disappearing spontaneously on timescales so short that we cannot measure them.

  This picture sounds so outrageous that you should be incredulous. After all, if we cannot measure these virtual particles directly, how can we claim that they exist?

  The answer is that while we cannot detect the effects of these virtual particle-antiparticle pairs directly, we can indirectly infer their presence because they can indirectly affect the properties of systems we can observe.

  The theory in which these virtual particles are incorporated, along with the electromagnetic interactions of electrons and positrons, called quantum electrodynamics, is the best scientific theory we have so far. Predictions based on the theory have been compared with observations, and they agree to more than ten decimal places. In no other area of science can this level of accuracy be obtained in the comparison between observation and prediction, based on the direct applications of fundamental principles on the most basic scales we can describe.

  But the agreement between theory and observation is only possible if the effects of virtual particles are included. Indeed, the very phenomenon of virtual particles implies that, in quantum theory, forces between particles are always conveyed by the exchange of virtual particles, in a way I shall now describe.

  In quantum electrodynamics, electromagnetic interactions occur by the absorption or emission of the quanta of electromagnetism, namely photons. Following Feynman, we can diagram this interaction as an electron emitting a wavy “virtual” photon (γ) and changing direction:

  Then, the electric interaction between two electrons can be diagrammed as:

  In this case, the electrons interact with each other by exchanging a virtual photon, one that is spontaneously emitted by the electron on the left and absorbed by the other in so short a time that the photon cannot be observed. The two electrons repel each other and move apart after the interaction.

  This also explains why electromagnetism is a long-range force. The Heisenberg uncertainty principle tells us that if we measure a system for some time interval, then there is an associated uncertainty in the measured energy of the system. Moreover, as the time interval gets bigger, the associated uncertainty in energy gets smaller. Because the photon is massless, a virtual massless photon, using Einstein’s relation between mass and energy, can carry an arbitrarily small amount of energy when it is created. This means that it can travel an arbitrarily long time—and therefore an arbitrarily long distance—before being absorbed, and it will still be protected by the uncertainty principle, as the energy it can carry is so small that no visible violation of the conservation of energy will occur. Thus, an electron on Earth can emit a virtual photon that could travel to Alpha Centauri, four light-years away, and that photon can still produce a force on an electron there that absorbs it. If the photon weren’t massless, however, but had some rest mass, m, it would carry with it a minimum energy, given by E = mc2, and could therefore only travel a finite distance (i.e., over a finite time interval) before it would have to be absorbed without producing any visible violation of the conservation of energy.

  These virtual particles have a potential problem, however. If one particle can be exchanged or one virtual particle-antiparticle pair can spontaneously appear out of the vacuum, then why not two or three or even an infinite number? Moreover, if virtual particles must disappear in a time that is inversely proportional to the energy they carry, then what stops particles from popping out of empty space carrying an arbitrarily large amount of energy and existing for an arbitrarily small time?

  When physicists tried to take into account these effects, they encountered infinite results in their calculations.

  The solution? Ignore them.

  Actually not ignore them, but systematically sweep the infinite pieces of calculations under the rug, leaving only finite bits left over. This begs the questions of how one knows which finite parts to keep, and why the whole procedure is justified.

  The answer took quite a few years to get straight, and Feynman was one of the group who figured it out. But for many years after, including up to the time he won the Nobel Prize in 1965, he viewed the whole effort as a kind of trick and figured that at some point a more fundamental solution would arise.

  Nevertheless, a good reason exists for ignoring the infinities introduced by virtual particles with arbitrarily high energies. Because of the Heisenberg uncertainty principle, these energetic particles can propagate only over short distances before disappearing. So how can we be sure that our physical theories, which are designed to explain phenomena at scales we can currently measure, actually operate the same way at these very small scales? Maybe new physics, new forces, and new elementary particles become relevant at very small scales?

  If we had to know all the laws of physics down to infinitesimally small scales in order to explain phenomena at the much larger scales we experience, then physics would be hopeless. We would need a theory of everything before we could ever have a theory of something.

  Instead, reasonable physical theories should be ones that are insensitive to any possible new physics occurring at much smaller scales than the scales that the original theories were developed to describe. We call these theories renormalizable, since we “renormalize” the otherwise infinite predictions, getting rid of the infinities and leaving only finite, sensible answers.

  Saying that this is required is one thing, but proving that it can be done is something else entirely. This procedure took a long time to get straight. In the first concrete example demonstrating that it made sense, the energy levels of hydrogen atoms were precisely calculated, which allowed a correct prediction of the spectrum of light emitted and absorbed by these atoms as measured in the laboratory.

  Although Feynman and his Nobel colleagues elucidated the mechanism to mathematically implement this technique of renormalization, the proof that quantum electrodynamics (QED) was a “renormalizable” theory, allowing precise predictions of all physical quantities one could possibly measure in the theory, was completed by Freeman Dyson. His proof gave QED an unprecedented status in physics. QED provided a complete theory of the quantum interactions of electrons and light, with predictions that could be compared with observations to arbitrarily high orders of precision, limited only by the energy and determination of the theorists doing the calculations. As a result, we can predict the spectra of light emitted by atoms to exquisite precision and design laser systems and atomic clocks that have redefined accuracy in measuring distance and time. The predictions of QED are so precise that we can search in experiments for even minuscule departures from them and probe for possible new physics that might emerge as we explore smaller and smaller scales of distance and time.

  With fifty years of hindsight, we now also understand that quantum electrodynamics is such a notable physical theory in part because of a “symmetry” associated with it. Symmetries in physics probe deep characteristics of physical reality. From here on into the foreseeable future, the search for symmetries is what governs the progress of physics.

  Symmetries reflect that a change in the fundamental mathematical q
uantities describing the physical world produce no change in the way the world works or looks. For example, a sphere can be rotated in any direction by any angle, and it still looks precisely the same. Nothing about the physics of the sphere depends on its orientation. That the laws of physics do not change from place to place, or time to time, is of deep significance. The symmetry of physical law with time—that nothing about the laws of physics appears to change with time—results in the conservation of energy in the physical universe.

  In quantum electrodynamics, one fundamental symmetry is in the nature of electric charges. What we call “positive” and “negative” are clearly arbitrary. We could change every positive charge in the universe to negative, and vice versa, and the universe would look and behave precisely the same.

  Imagine, for example, that the world is one giant chessboard, with black and white squares. Nothing about the game of chess would be changed if I changed black into white, and white into black. The white pieces would become black pieces and vice versa, and otherwise the board would look identical.

  Now, precisely because of this symmetry of nature, the electric charge is conserved: no positive or negative charge can spontaneously appear in any process, even due to quantum mechanics, without an equal and opposite charge appearing at the same time. For this reason, virtual particles are only produced spontaneously in empty space in combination with antiparticles. It is also why lightning storms occur on Earth. Electric charges build up on Earth’s surface because storm clouds build up large negative charges at their base. The only way to get rid of this charge is to have large currents flow from the ground upward into the sky.

  The conservation of charge resulting from this symmetry can be understood using my chessboard analogy. That every white square must be located next to a black square means that whenever I switch black and white, the board ultimately looks the same. If I had two black squares in a row, which would mean the board had some net “blackness,” then “black” and “white” would no longer be equivalent arbitrary labels. Black would be physically different from white. In short, the symmetry between black and white on the board would be violated.

  Bear with me now, because I am about to introduce a concept that is much more subtle, but much more important. It’s so important that essentially all of modern physical theory is based on it. But it’s so subtle that without using mathematics, it is hard to describe. It is so subtle that its ramifications are still being unraveled today, more than a hundred years since it was first suggested. So, don’t be surprised if it takes one or two readings to fully get your head around the idea. It has taken physicists much of the past century to get their heads around it.

  This symmetry is called gauge symmetry for an obscure historical reason I shall describe a bit later. But the strange name is irrelevant. It is what the symmetry implies that is important:

  Gauge symmetry in electromagnetism says that I can actually change my definition of what a positive charge is locally at each point of space without changing the fundamental laws associated with electric charge, as long as I also somehow introduce some quantity that helps keep track of this change of definition from point to point. This quantity turns out to be the electromagnetic field.

  Let’s try to parse this using my chessboard analogy. The global symmetry I described before changes black to white everywhere, so when the chessboard is turned by 180 degrees, it looks the same as it did before and the game of chess is clearly not affected.

  Now, imagine instead that I change black to white in one square, and I don’t change white to black in the neighboring square. Then the board will have two adjacent white squares. This board, with two adjacent white squares, clearly won’t look the same as it did before. The game cannot be played as it was before.

  But hold on for a moment. What if I have a guidebook that tells me what game pieces should do every time they encounter adjacent squares where one color has been changed but not the next. Then the rules of the game can remain the same, as long as I consult the guidebook each time I move. This guidebook therefore allows the game to proceed as if nothing were changed.

  In mathematics, a quantity that ascribes some rule associated with each point on a surface like a chessboard is called a function. In physics, a function defined at every point in our physical space is called a field, such as, for example, the electromagnetic field, which describes how strong electric and magnetic forces are at each point in space.

  Now here’s the kicker. The properties that must characterize the form of the necessary function (which allows us to change our definition of electric charge from place to place without changing the underlying physics governing the interaction of electric charges) are precisely those that characterize the form of the rules governing electromagnetic fields.

  Put another way, the requirement that the laws of nature remain invariant under a gauge transformation—namely some transformation that locally changes what I call positive or negative charge—identically requires the existence of an electromagnetic field that is governed precisely by Maxwell’s equations. Gauge invariance, as it is called, completely determines the nature of electromagnetism.

  This presents us with an interesting philosophical question. Which is more fundamental, the symmetry or the physical equations that manifest the symmetry? In the former case, where this gauge symmetry of nature requires the existence of photons, light, and all the equations and phenomena first discovered by Maxwell and Faraday, then God’s apparent command “Let there be light” becomes identical with the command “Let electromagnetism have a gauge symmetry.” It is less catchy, perhaps, but nevertheless true.

  Alternatively, one could say that the theory is what it is, and the discovery of a mathematical symmetry in the underlying equations is a happy accident.

  The difference between these two viewpoints seems primarily semantic, which is why it might interest philosophers. But nature does provide some guidance. If quantum electrodynamics were the only theory in nature that respected such a symmetry, the latter view might seem more reasonable.

  But every known theory describing nature at a fundamental scale reflects some type of gauge symmetry. As a result, physicists now tend to think of symmetries of nature as fundamental, and the theories that then describe nature as being restricted in form to respect these symmetries, which in turn then reflect some key underlying mathematical features of the physical universe.

  Whatever one might think of regarding this epistemological issue, what matters in the end to physicists is that the discovery and application of this mathematical symmetry, gauge symmetry, has allowed us to discover more about the nature of reality at its smallest scales than any other idea in science. As a result, all attempts to go beyond our current understanding of the four forces of nature, electromagnetism, the two forces associated with atomic nuclei, the strong and weak forces, which we shall meet shortly, and gravity—including the attempt to create a quantum theory of gravity—are built on the mathematical underpinnings of gauge symmetry.

  • • •

  That gauge symmetry has such a strange name has little to do with quantum electrodynamics and is an anachronism, related to a property of Einstein’s General Theory of Relativity, which, like all other fundamental theories, also possesses gauge symmetry. Einstein showed that we are free to choose any local coordinate system we want to describe the space around us, but the function, or field, that tells us how to connect these coordinate systems from point to point is related to the underlying curvature of space, determined by the energy and momentum of material in space. The coupling of this field, which we recognize as the gravitational field, to matter, is precisely determined by the invariance of the geometry of space under the choice of different coordinate systems.

  The mathematician Hermann Weyl was inspired by this symmetry of General Relativity to suggest that the form of electromagnetism might also reflect an underlying symmetry associated with physical changes in length scales. He called these different “gaug
es,” inspired by the various track gauges of railroads. (Einstein, and Sheldon on The Big Bang Theory, aren’t the only physicists who have been inspired by trains.) While Weyl’s guess turned out to be incorrect, the symmetry that does apply to electromagnetism became known as gauge symmetry.

  Whatever the etymology of the name, gauge symmetry has become the most important symmetry we know of in nature. From a quantum perspective—in the quantum theory of electromagnetism, quantum electrodynamics—the existence of gauge symmetry becomes even more important. It is the essential feature that ensures that QED is sensible.

  If you think about the nature of symmetry, then it begins to make sense that such a symmetry might ensure that quantum electrodynamics makes sense. Symmetries tell us, for example, that different parts of the natural world are related, and that certain quantities remain the same under various types of transformations. A square looks the same when we rotate it ninety degrees because the sides are all the same length and the angles at each corner are the same. So, symmetry can tell us that different mathematical quantities that result from physical calculations, such as the effects of many virtual particles, and many virtual antiparticles, for example, can have the same magnitude. They may also have opposite signs so that they might cancel exactly. The existence of this symmetry is what can require such exact cancellations.

  In this way, one might imagine that in quantum electrodynamics the nasty terms that might otherwise give infinite results can cancel with other potentially nasty terms, and all the nastiness can disappear. And this is precisely what happens in QED. The gauge symmetry ensures that any infinities that might otherwise arise in deriving physical predictions can be isolated in a few nasty terms that can be shown by the symmetry to either disappear or to be decoupled from all physically measurable quantities.