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As I have noted, in quantum mechanics particles have a wavelike 
character. Thanks to Max Born we recognize that the square of the am-
plitude of the wave associated with a particle at any point—what we 
now call the wave function of the particle, following Schthdinger—
determines the probability of finding the particle at that point. Because 
the amplitude of the oscillating wave above is more or less constant at 
all the peaks, such a wave, if it corresponded to the probability ampli-
tude of finding an electron, would imply a more or less uniform prob-
ability for finding the electron anywhere along the path. 
Now consider what a disturbance would look like if it was the sum of 
two waves of slightly different frequencies (wavelengths), moving along 
the x axis: 
When we combine the two waves, the resulting disturbance will look 
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Because of the slightly different wavelengths of the two waves, the 
peaks and troughs will tend to cancel out, or "negatively interfere" with 
each other everywhere except for the rare places where the two peaks 
occur at the same point (one of these locations is shown in the figure 
above). This is reminiscent of the wave interference phenomenon in the 
Young double-slit experiment I described earlier. 
If we add yet another wave of slightly different wavelength 
the resulting wave then looks like this: 
The interference washes out more of the oscillations aside from the 
position where the two waves line up, making the amplitude of the wave 
at the peak much higher there than elsewhere. 
You can imagine what would happen if I continue this process, con-
tinuing to add just the right amount of waves with slightly different fre-
quencies to the original wave. Eventually the resulting wave amplitudes 
will cancel out more and more at all places except for some small re-
gion around the center of the figure, and at faraway places where all the 
peaks might again line up: 
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The greater the number of slightly different frequencies that I add to-
gether, the narrower will be the width of the largest central peak. Now, 
imagine that this represents the wave function of some particle. The 
larger the amplitude of the central peak, the greater the probability of 
finding the particle somewhere within the width of that peak. But the 
width of that central peak is still never quite zero, so the disturbance 
remains spread out over some small, if increasingly narrow, region. 
Now recall that Planck and Einstein told us that, for light waves, at 
least, the energy of each quantum of radiation, i.e., each photon, is di-
rectly related to its frequency. Not surprisingly, a similar relation holds 
for the probability waves associated with massive particles, but in this 
case it is the momentum of the particle that is related to the frequency 
of the probability wave associated with the particle. 
Hence, Heisenberg's uncertainty relation: If we want to localize a 
particle over a small region, i.e., have the width of the highest peak in 
its wave function as narrow as possible, then we must consider that the 
wave function is made up by adding lots of different waves of slightly 
different frequencies together. But this means that the momentum of 
the particle, which is associated with the frequency of its wave func-
tion, must be spread out somewhat. The narrower the dominant peak in 
space in the particle's wave function, the greater the number of different 
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frequencies (i.e., momenta) that must be added together to make up the 
final wave function. Put in a more familiar way, the more accurately 
we wish to determine the specific position of a particle, the greater the 
uncertainty in its momentum. 
As you can see, there is no restriction here related to actual obser-
vations, or consciousness, or the specific technology associated with any 
observation. It is an inherent property of the fact that, in the quantum 
world, a wave function is associated with each particle, and for particles of 
a fixed specific momentum, the wave function has one specific frequency. 
After discovering this relation, Heisenberg was the first to provide a 
heuristic picture of why this might be the case, which he posed in terms 
of a thought experiment. To measure the position of a particle you have 
to bounce light off the particle, and to resolve the position with great 
precision requires light of a wavelength small enough to resolve this 
position. But the smaller the wavelength, the bigger the frequency and 
the higher the energy associated with the quanta of that radiation. But 
bouncing light with a higher and higher energy off the particle clearly 
changes the particle's energy and momentum. Thus, after the measure-
ment is made, you may know the position of the particle at the time of 
the measurement, but the range of possible energies and momenta you 
have imparted to the particle by scattering light off it is now large. 
For this reason, many people confuse the Heisenberg uncertainty 
relation with the "observer effect," as it has become known, in quan-
tum mechanics. But, as the example l have given should demonstrate, 
inherently the Heisenberg uncertainty principle has nothing to do with 
observation at all. To paraphrase a friend of mine, if consciousness had 
anything to do with determining the results of quantum physics experi-
ments, then in reporting the results of physics experiments we would 
have to discuss what the experimenter was thinking about—for exam-
ple, sex—when performing the experiment. But we don't. The supernova 
explosions that produced the atoms that make up your body and mine 
occurred quite nicely long before our consciousness existed. 
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The Heisenberg uncertainty principle epitomizes in many ways the 
complete demise of our classical worldview of nature. Independent of 
any technology we might someday develop, nature puts an absolute 
limit on our ability to know, with any degree of certainty, both the mo-
mentum and position of any particle. 
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 re-
puted 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, some-
one in the audience was reputed to have raised his hand and said, 1 
don't understand that particular step you have just written down? Dirac 
stood silent for the longest while until the audience member asked if 
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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: sNo, 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 Schthdinger 
(who derived his famous wave equation during a two-week tryst in the 
mountains with several of his girlfriends), and since Heisenberg had re-
vealed the basic underpinning of quantum mechanics, no one had been 
successful at fully explaining the behavior of electrons bound deep in-
side 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. 
Schrodinger's equation worked well to describe the energy levels of elec-
trons in the outer parts of simple atoms such as hydrogen, where it pro-
vided 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 
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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 fundamen-
tal problem. It seemed to predict new particles that didn't exist. 
To establish the mathematics necessary to describe an electron mov-
ing 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 nev-
ertheless 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 Schthdinger'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 direc-
tion, 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 for-
mulation 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, 
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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 pos-
sibility 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 spostdicted" 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 funda-
mental scales, with no direct experimental motivation in advance. It 
may have seemed like a matter of faith, but it wasn't—after all, the pro-
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poser didn't actually believe it—and while like faith it proposed an un-
observed 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 Schthdinger 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 hid-
den nature of matter at much smaller 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 par-
ticles, 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 sup" in a gravitational field. Matter and an-
timatter 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 
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equal amounts of matter and antimatter—which would surely seem the 
most sensible universe to begin with—would, unless something hap-
pened 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 en-
counter 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. 
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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 direc-
tions. 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 physi-
cists of the latter half of the twentieth century, Richard Feynman. 
Feynman could not have been more different from Dirac. While 
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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 intel-
lectually 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 ap-
proach to quantum mechanics, would inspire the work that ultimately 
won Feynman a Nobel Prize. 
Heisenberg and Schrodinger 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 ikso 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 
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the ground and then returns up to your eye than it would if it came di-
rectly 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 an-
other—they take all possible trajectories at the same time. Each trajec-
tory 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 gradu-
ate student, after learning about Dirac's paper at a beer party, mathemati-
cally 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 Schrodinger or Heisen-
berg 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, establish-
ing the theory known as quantum electrodynamics (QED), he shared 
the Nobel Prize in 1963 with Julian Schwinger and Sin-Itiro Tomonaga. 
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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" trajec-
tory. 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 uncer-
tainty 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: 
time 
For the short time in the middle of the time interval shown the elec-
tron 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 travel-
ing 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: 
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Ttime 
Feynman recognized that in the latter frame this would look like an 
electron moving forward in time for a little while, then moving back-
ward 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.. 
7%
7
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 for-
ward in time, until it encounters the original electron and the two an-
nihilate, 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 rela-
tivity, would be impossible. Nevertheless, you can be assured that inside 
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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. 
Nevertheless, if such a trajectory is possible in the invisible quan-
tum 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 an-
nihilate 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 di-
rectly 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 sci-
ence can this level of accuracy be obtained in the comparison between 
observation and prediction, based on the direct applications of funda-
mental principles on the most basic scales we can describe. 
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But the agreement between theory and observation is only possible if 
the effects of virtual particles are included. Indeed, the very phenome-
non 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 (y) and changing direction: 
Then, the electric interaction between two electrons can be dia-
grammed as: 
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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 mea-
sured 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 en-
ergy will occur. Thus, an electron on Earth can emit a virtual photon 
that could travel to Alpha Centauri, four light-years away, and that pho-
ton 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 = mo, 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 disap-
pear 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 en-
countered 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 ef-
fort 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 intro-
duced by virtual particles with arbitrarily high energies. Because of the 
Heisenberg uncertainty principle, these energetic particles can propa-
gate only over short distances before disappearing. So how can we be 
sure that our physical theories, which are designed to explain phenom-
ena at scales we can currently measure, actually operate the same way 
at these very small scales? Maybe new physics, new forces, and new el-
ementary 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 insensi-
tive 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 in-
finite 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 
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THE GREATEST STORY EVER TOLD-SO FAR 
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 ab-
sorbed by these atoms as measured in the laboratory. 
Although Feynman and his Nobel colleagues elucidated the mecha-
nism 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 char-
acteristics 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 
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