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.
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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
͢͝͝
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.
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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—
͜͝͞
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
Lawrence Krauss - The Greatest Story Ever Told--So Far Page 12