Fear of Physics (25 page)

Now, inventing a hitherto undetected particle is not something to be taken lightly, but Pauli was not to be taken lightly either. He had made important contributions to physics, notably the “Pauli exclusion principle,” which governs the way electrons in atoms can behave. This Austrian-born genius also had an intimidating personality. He was famous for his habit of jumping up while listening to lectures and grabbing the chalk out of the speaker’s hand if he thought the speaker was spouting nonsense. Moreover, the idea of giving up momentum and energy conservation, which had worked so well everywhere else in physics, seemed much more radical—in the spirit of creative plagiarism I discussed earlier—than what he was proposing. So the neutrino became an established part of physics long before it was experimentally observed in 1956, and before it became a standard part of astrophysics.

Today, of course, we would be more hesitant to give up momentum conservation, even at these small scales, because we recognize it as a consequence of a truly fundamental symmetry of nature. Unless we expect new dynamic laws of nature to depend somehow upon position, we can count on momentum conservation to be valid. And, of course, it doesn’t apply only on subatomic
scales. It is a fundamental part of understanding human-scale activities such as baseball, skating, driving, or typing. Whenever one finds an isolated system, with no external forces acting upon it, the momentum of this system is conserved, namely, it remains constant for all time.

Where does one find such isolated systems? The answer is anywhere you choose! There is a well-known cartoon that shows two scientists at a blackboard filled with equations, with one saying to the other: “Yes, but I don’t think drawing a box around it makes it a Unified Theory.” This may be true, but all you have to do to define a system is to draw an imaginary box around it. The trick lies in choosing the right box.

Consider the following example. You run into a brick wall with your car. Now draw a box around the car, and call that a system. Originally you were moving along at a constant speed; the momentum of the car was constant. Suddenly the wall comes along and stops you. Since your momentum changes to zero by the time you are at rest, the wall must exert a force on your system, the car. The wall will have to exert a certain force to stop you, depending upon your initial velocity.

Next, draw a box around the car and the wall. In this new system, no external forces are apparently at work. It would seem that the only thing acting on you is the wall, and the only thing acting on the wall is you. From this vantage point, what happens when you hit the wall? Well, if no external forces are acting in this system, momentum must be conserved—that is, must remain constant. Initially, you were moving along and had some momentum, and the wall was at rest, with zero momentum. After the crash, both you and the wall are apparently at rest. What happened to the momentum? It had to go somewhere. The fact that it seems to have disappeared here is merely a signal that the
box is still not enough, namely, the system of you and the wall is not really isolated. The wall is anchored to the Earth. It is then clear that momentum can be conserved in this collision only if the Earth itself takes up the momentum that was initially carried by your car. The truly isolated system is thus made up of you, the wall, and the Earth. Since the Earth is much more massive than your car, it doesn’t have to move very much to take up this momentum, but nevertheless its motion must change! So, the next time someone tells you the Earth moved, rest assured that it did!

The search for symmetry is what drives physics. In fact, all the hidden realities discussed in the last chapter have to do with exposing new symmetries of the universe. Those that I have described related to energy and momentum conservation are what are called space-time symmetries, for the obvious reason that they have to do with those symmetries of nature associated with space and time, and to distinguish them from those that don’t. They are thus integrally related to Einstein’s Theory of Special Relativity. Because relativity puts time on an equal footing with space, it exposes a new symmetry between the two. It ties them together into a new single entity, space-time, which carries with it a set of symmetries not present if space and time are considered separately. Indeed, the invariance of the speed of light is itself a signal of a new symmetry in nature connecting space and time.

We have seen just how motion leaves the laws of physics invariant by providing a new connection between space and time. A certain four-dimensional space-time “length” remains invariant under uniform motion, just as a standard three-dimensional spatial length remains invariant under rotations. This symmetry of
nature is possible only if space and time are tied together. Thus, purely spatial translations and purely time translations, which are themselves responsible for momentum and energy conservation, respectively, must be tied together. It is a consequence of special relativity that energy and momentum conservation are not separate phenomena. Together they are part of a single quantity called “energy-momentum.” The conservation of this single quantity—which, in fact, requires some redefinition of both energy and momentum separately, as they are traditionally defined in the context of Newton’s Laws—then becomes a single consequence of the invariances of a world in which space and time are tied together. In this sense, special relativity tells us something new: Space-time is such that we cannot have energy conservation without momentum conservation and vice versa.

There is one more space-time symmetry that I have so far alluded to only obliquely. It is related to the symmetry that results in energy-momentum conservation in special relativity, but it is much more familiar because it has to do with three dimensions and not four. It involves the symmetry of nature under rotations in space. I have described how different observers might see different facets of an object that has been rotated, but we know that fundamental quantities such as its total length remain unchanged under such a rotation. The invariance of physical laws as I rotate my laboratory to point in some different direction is a crucial symmetry of nature. We do not expect, for example, nature systematically to prefer some specific direction in space. All directions should be identical as far as the underlying laws are concerned.

The fact that physical laws are invariant under rotations implies that there is an associated quantity that is conserved. Momentum is related to the invariance of nature under spatial translations, while this new quantity is related to the invariance
of nature under translations by an angle. For this reason, it is called
angular momentum.
Like momentum, angular momentum conservation plays an important role in processes ranging from atomic scales to human scales. For an isolated system, the angular momentum must be conserved. Indeed, for every example of momentum conservation, one can replace the words “distance” by “angle” and “velocity” by “angular velocity” to find some example of angular momentum conservation. It is a prime manifestation of creative plagiarism.

Here’s one example. When my car hits another car that was at rest, and the bumpers lock so that the two move off together, the combination will move more slowly than my car alone was originally moving. This is a classic consequence of momentum conservation. The momentum of the combined system of the two cars must be the same after the collision as it was before. Since the combined mass of the system is larger than the mass of the original object that was moving, the combined object must move slower to conserve momentum.

On the other hand, consider a figure skater rotating very fast with her arms held tight by her side. When she spreads her arms outward, her rotation slows right down, as if by magic. This is a consequence of angular momentum conservation, just as the previous example was a consequence of momentum conservation. As far as rotations and angular velocities are concerned, an object with a bigger radius acts just like an object with a bigger mass. Thus, the act of raising her arms increases the radius of the skater’s rotating body. Just as the two cars move together more slowly than one car as long as no external force acts on the system, so, too, a skater with increased radius will rotate more slowly than she will when her radius is smaller, as long as no external force is acting upon her. Alternatively, a skater who starts
herself rotating slowly with her arms outstretched can then vastly increase her rotation speed by pulling her arms in. Thus are Olympic medals won.

There are other conserved quantities in nature that arise from symmetries other than those of space-time, such as electric charge. I will return to these later. For the moment, I want to continue with one strange facet of the rotational invariance of nature, which will allow me to introduce a ubiquitous aspect of symmetry that isn’t always manifest. For example, even though the underlying laws of motion are rotationally invariant—that is, there is no preferred direction picked out by laws governing dynamics—the world isn’t. If it were, then we should find it impossible to give directions to the grocery store. Left looks different than right; north is different than south; up is different than down.

It is easy for us to regard these as mere accidents of our circumstances, because that is exactly what they are. Were we somewhere else, the distinctions between left and right, and north and south, might be totally different. Nevertheless, the very fact that an accident of our circumstances can hide an underlying symmetry of the world is one of the most important ideas directing modern physics. To make progress, and to exploit the power of such symmetries, we have to look beneath the surface.

Many of the classic examples of hidden realities I discussed in the last chapter are related to this idea that symmetry can be masked. This idea goes by the intimidating name
spontaneous symmetry breaking,
and we have already encountered it in a number of different guises.

A good example comes from the behavior of the microscopic magnets in a piece of iron, which I discussed at the end of the last chapter. At low temperature, when there is no external magnetic
field applied to these magnets, it is energetically favorable for them all to line up in some direction, but the direction they choose is random. There is nothing in the underlying physics of electromagnetism that picks out one direction over another, and there is no prediction in advance that can precisely determine which direction they will choose. Once they have chosen, however, that direction becomes very special. An insect, sensitive to magnetic fields, living inside such a magnet would grow up assuming there was something intrinsically different about “north,” that being the direction in which the microscopic magnets were aligned.

The trick of physics is to rise above the particular circumstances that may be attached to our own existence and attempt to peer beyond them. In every case I know of, this implies searching for the true symmetries of the world. In the case I just described, it would mean discovering that the equations governing the magnets were invariant under rotations and that north could be rotated to be south and the physics would still be the same.

The prototypical example of this is the unification of the weak and electromagnetic interactions. There, the underlying physics makes no distinction between the massless photon and the very massive Z particle. In fact, there is a symmetry of the underlying dynamics under which a Z can be turned into a photon and everything will look exactly the same. In the world in which we live, however, this same underlying physics has produced a specific realization, a solution of the equations—the “condensate” of particles occupying otherwise empty space—inside which the photon and the Z behave quite differently.

Mathematically, these results can be translated to read: A particular solution of a mathematical equation need not be invariant under the same set of transformations under which the underlying equation is invariant. Any specific realization of an underlying
mathematical order, such as the realization we see when we look around the room, may break the associated underlying symmetry. Consider the example invented by the physicist Abdus Salam, one of those who won the Nobel Prize for his work on the unification of electromagnetism and the weak interaction: When you sit down at a circular dinner table, it is completely symmetrical. The wineglass on the right and the wineglass on the left are equivalent. Nothing but the laws of etiquette (which I can never remember) specifies which is yours. Once you choose a wineglass—say, the one on the right—everyone else’s choice is fixed, that is, if everyone wants a wineglass. It is a universal fact of life that we live in one particular realization of what may be an infinity of possibilities. To paraphrase Rousseau: The world was born free, but it is everywhere in chains!

Why should we care so much about symmetries in nature, even those that are not manifest? Is it just some peculiar aesthetic pleasure that physicists derive, some sort of intellectual masturbation? Perhaps in part, but there is also another reason. Symmetries,
even
those that are not directly manifest, can completely determine the physical quantities that arise in the description of nature
and
the dynamical relations between them. In short, symmetry may
be
physics. In the final analysis, there may be nothing else.