Warped Passages (31 page)
Symmetry: The Essential Organizing Principle
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Simple Minds
Athena uncaged three of her owls and let them fly around. Unfortunately for Ike, he had left the top of his convertible down that day and the curious owls flew right in. The most mischievous of the owls pecked at the car’s interior and ended up tearing it a little.
When Ike saw the damage, he stormed into Athena’s room and demanded that she watch her owls more carefully in the future. Athena protested that her owls were almost all well-behaved and she need only keep an eye on the bad one. But by that time the owls were back in their cages, and neither Ike nor Athena could identify which one was guilty
.
The Standard Model works spectacularly well, but only because it is a theory in which quarks, leptons, and the weak gauge bosons—the charged Ws and the Z that communicate the weak force between weakly charged matter—all have mass. The mass of fundamental particles is, of course, critical to everything in the universe; if matter had been truly massless, it wouldn’t form nice, solid objects, and structure and life in the universe as we know it would never have formed. Yet weak gauge bosons and other fundamental particles in
the simplest theory of forces look as if they should be massless and should travel at the speed of light.
You might find it strange that a theory of forces should prefer zero masses. Why shouldn’t any mass be allowed? But the most basic quantum field theory of forces is intolerant in this respect. It ostensibly forbids any nonzero values for the masses of fundamental Standard Model particles. One of the triumphs of the Standard Model is that it shows how to resolve this issue and fashion a theory in which particles have the masses that observations tell us they must have.
In the next chapter we will explore the mechanism by which particles acquire mass—the phenomenon known as the Higgs mechanism. But in this chapter we will discuss the important topic of
symmetry
. Symmetry and symmetry breaking help to determine how the universe goes from an undifferentiated point to the complex structure we now see. The Higgs mechanism is intimately connected with symmetry, and in particular with broken symmetry. Understanding how the elementary particles acquire mass requires some familiarity with these important ideas.
Things That Change but Remain the Same
Symmetry is a sacred word to most physicists. One might conjecture that other communities value symmetry highly as well, since the Christian cross, the Jewish menorah, the Dharma wheel of Buddhism, the crescent of Islam, and the Hindu mandala all exhibit symmetry (see Figure 56). Something has symmetry if you can manipulate it—for example, by rotating it, reflecting it in a mirror, or interchanging its
parts—so that the new configuration is indistinguishable from the initial one. For example, if you were to interchange two identical candles on a menorah, you would see no visible difference. And the mirror image of a cross is identical to the cross itself.
Figure 56.
A menorah, a cross, the Dharma wheel of Buddhism, the crescent of Islam, and the Hindu mandala all exhibit symmetry.
Whether we are talking about mathematics, physics, or the world, we can make transformations that appear to do nothing when there is a symmetry. A system has symmetry if someone could exchange its components, reflect it in a mirror, or rotate it while your back was turned without your noticing any difference when you looked at it again.
Symmetry is often a static property: for example, the symmetry of a cross does not involve time. But physicists often prefer to describe symmetries in terms of imagined
symmetry transformations
—manipulations that one can apply to a system without changing any observable properties. For example, instead of saying that the candles of a menorah are equivalent, I might say instead that a menorah would look the same if I were to interchange two of the candles. I wouldn’t actually have to exchange the candles in order to claim that there was a symmetry. But if, hypothetically, I did interchange the candles, I wouldn’t be able to see any difference. Sometimes I will describe symmetry in this way for simplicity’s sake.
We all are familiar with symmetries not only in science and sacred symbols, but in secular art as well. Symmetry can be found in most paintings, sculpture, architecture, music, dance, and poetry. Islamic art is perhaps the most spectacular in this respect, with its intricate and extensive use of symmetry in architecture and ornamental art, to which anyone who has seen the Taj Mahal can attest. Not only does the building look the same from any side, but when viewed from the edge of the long pool in front, it is perfectly reflected in the calm surface of the water. Even the trees have been planted to preserve the monument’s symmetries. When I was there, I noticed a guide who was pointing out some symmetry points, so I asked him to show me the others. I ended up viewing the building from funny angles and scrambling up rubble on the edge of the site in order to see all the symmetries the monument presents.
In colloquial usage people often equate symmetry with beauty, and certainly some of the fascination with symmetry arises from the
regularity and neatness that it guarantees. Symmetries also help us learn, since repitition, either in time or in space, can create indelible images in our mind. The brain’s programmed response to symmetry and its sheer aesthetic appeal explain in large part why we surround ourselves with it.
But symmetries don’t occur only in art and architecture, but also in nature, without any human intervention. For this reason you often encounter symmetries in physics. The goal of physics is to relate distinct quantities to one another so that we can make predictions based on observations. Symmetry is a natural player in this context. When a physical system has symmetry, you can describe the system on the basis of fewer observations than if the system had no symmetry. For example, if there are two objects with identical properties, I would know the physical laws that govern the behavior of one of the objects if I’ve already measured the behavior of the other. Because the two objects are equivalent, I know that they must behave the same way.
In physics, the existence of a symmetry transformation in a system means that there is some definite procedure for rearranging the system that leaves all its measurable physical properties unchanged.
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For example, if a system possesses
rotational
and
translational symmetries
, two well-known examples of symmetries of space, physical laws apply the same way in all directions and in all places. Rotational and translational symmetries tell us, for example, that it doesn’t matter which way you are facing or where you are standing when you swing a baseball bat at a aball: provided you apply the same force, the baseball will behave in exactly the same way. Any experiment should yield the same result if you rotate your setup or if you repeat your measurement in a different room or in a different place altogether.
It is difficult to overstate the importance of symmetry in physical laws. Many physical theories, such as Maxwell’s laws of electrodynamics and Einstein’s theory of relativity, are deeply rooted in symmetry. And by exploiting various symmetries we can usually simplify the task of using theories to make physical predictions. For
example, predictions of the orbital motion of the planets, the gravitational field of the universe (which is more or less rotationally symmetric), the behavior of particles in electromagnetic fields, and many other physical quantities are mathematically simpler once we take symmetry into account.
Symmetries in the physical world are not always completely obvious. But even when they are not readily apparent or when they are merely theoretical tools, symmetries usually greatly simplify the formulation of physical laws. The quantum theory of forces, which will soon be our focus, is no exception.
Internal Symmetries
Physicists generally classify symmetries into different categories. You are probably most familiar with symmetries of space—the symmetry transformations that move or rotate things in the external world. These symmetries, which include the rotational and translational symmetries I just mentioned, tell us that the laws of physics are the same for a system no matter which way the system points and no matter where it is located.
I now want to consider a different kind of symmetry, known as an
internal symmetry
. Whereas spatial symmetries tell us that physics treats all directions and all positions as the same, internal symmetries tell us that physical laws act the same way on distinct, but effectively indistinguishable, objects. In other words, internal symmetry transformations exchange or mix distinct things around in a way that can’t be noticed. In fact, I have already given an example of an internal symmetry—the interchangeability of the candles on a menorah. The internal symmetry says that two candles are equivalent. It is a statement about the candles, not about space.
A traditional menorah, however, has both spatial and internal symmetries. While different candles are equivalent, which means that there is an internal symmetry, a menorah also looks the same if it is rotated 180 degrees about the central candle, which means that it has spatial symmetry as well. But an internal symmetry can exist even when there is no symmetry of space. For example, you can interchange
identical green tiles in a mosaic even when the leaf they combine to portray has an irregular shape.
Another example of an internal symmetry is the interchangeability of two identical red marbles. If you hold one such marble in each hand, it wouldn’t matter which was which. Even if you’d labeled them “1” and “2,” you would never know whether I had somehow managed to interchange the two marbles. Notice that the example of the marbles is not tied to any spatial arrangement in the way that the examples of the menorah and the mosaic were; internal symmetries concern the objects themselves and not their locations in space.
Particle physics deals with somewhat abstract internal symmetries that relate different types of particle. These symmetries treat particles and the fields that create them as interchangeable. Just as two identical marbles behave in exactly the same way when you roll them or bang them against a wall, two particle types that have the same charges and mass obey identical physical laws. The symmetry that describes this is called
flavor symmetry
.
In Chapter 7 we saw that flavors are the three distinct particle types that have identical charges, one in each of the three generations. For example, electrons and muons are two flavors of charged leptons, which means that they have identical charges. Had we lived in a world in which the electron and the muon also had identical masses, the two would have been completely interchangeable. There would then have been a flavor symmetry, according to which the electron and muon would behave identically in the presence of any other particles or forces.
In our world the muon is heavier than the electron, so the flavor symmetry is not exact. But the difference in masses can be insignificant for some physical predictions, so flavor symmetries between light particles with identical charges, such as the muon and electron, are nonetheless often useful for calculations. Sometimes exploiting even slightly imperfect symmetries helps us to compute sufficiently accurate results. For example, the mass difference between particles is often so small (relative to energy or a large mass) that it doesn’t make a measurable difference to predictions.
But the most important type of symmetry for us at this point is the symmetry that is relevant to the theory of forces, which is exact. This
symmetry is also an internal symmetry among particles, but it’s slightly more abstract than the flavor symmetry we just discussed. This particular type of internal symmetry is more analogous to the following example. As you might recall from high school physics, theater, or art class, three spotlights—generally one red, one green, and one blue—can shine together to produce white light. If we were to interchange the positions of three such lights, any of the new setups would still produce white light. It doesn’t matter where any of the individual beams of light originate, so long as we see only the end result, white light. In that case, the internal symmetry transformation that exchanges the different lights would never produce any observable consequences.
We will now see that this symmetry is closely analogous to the symmetries associated with forces, because in both cases you are not able to observe everything. The light setup exhibits symmetry only because we are not allowed to look at everything, only the combined light. If you could see the lights themselves, you would know they had been interchanged. As mentioned earlier, this close analogy between colors and forces is the reason for the terms “color” and “quantum chromodynamics” (QCD) in the description of the strong force.
