Lawrence Krauss - The Greatest Story Ever Told--So Far

by Why Are We Here (pdf)

  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?

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  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.

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  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 quantities describing the physical world produce no

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  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

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  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.

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  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.

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  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—

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  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 “gauges,” 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 appl
y 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.

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  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.

  This profoundly important result, proven by decades of work by

  some of the most creative and talented theoretical physicists in the

  world, established QED as the most precise and preeminent

  quantum theory of the twentieth century.

  Which made it all the more upsetting to discover that, while this

  mathematical beauty indeed allowed a sensible understanding of one

  of nature’s fundamental forces—electromagnetism—other nastiness

  began when considering the forces that govern the behavior of

  atomic nuclei.

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  C h a p t e r 9