Fear of Physics (26 page)

Consider, for example, that energy and momentum—direct consequences of two space-time symmetries—together provide a description of motion that is completely equivalent to all of Newton’s Laws describing the motion of objects in the gravitational field of Earth. All the dynamics—force producing acceleration, for example—follow from these two principles. Symmetry even determines the nature of the fundamental forces themselves, as I shall soon describe.

Symmetry determines for us what variables are necessary to describe the world. Once this is done, everything else is fixed. Take my favorite example again: a sphere. When I represent the cow as a sphere I am saying that the
only
physical processes that we need concern ourselves with will depend on the radius of the cow. Anything that depends explicitly upon a specific angular location must be redundant, because all angular locations at a given radius are identical. The huge symmetry of the sphere has reduced a problem with a potentially large number of parameters to one with a single parameter, the radius.

We can turn this around. If we can isolate those variables that are essential to a proper description of some physical process, then, if we are clever, we may be able to work backward to guess what intrinsic symmetries are involved. These symmetries may in turn determine all the laws governing the process. In this sense we are once again following the lead of Galileo. Recall that he showed us that merely learning
how
things move is tantamount to learning
why
things move. The definitions of velocity and acceleration made it clear what is essential for determining the dynamical behavior of moving objects. We are merely going one step further when we assume that the laws governing such behavior are not merely made clear by isolating the relevant variables; rather, these variables alone
determine
everything else.

Let’s return to Feynman’s description of nature as a great chess game being played by the gods, which we are privileged to watch. The rules of the game are what we call fundamental physics, and understanding these rules is our goal. Feynman claimed that understanding these rules is all we can hope for when we claim to “understand” nature. I think now we can claim to go one step further. We suspect that these rules can be completely determined merely by exploring the configuration and
the symmetries of the “board” and the “pieces” with which the game is played. Thus to understand nature, that is, to understand its rules, is equivalent to understanding its symmetries.

This is a very strong claim, and one done in rather grand generality. As I expect that you may be now both confused and skeptical, I want to go through a few examples to make things more explicit. In the process, I hope to give some idea of how physics at the frontier proceeds.

First, let me describe these ideas in the context of Feynman’s analogy. A chess board is a rather symmetrical object. The pattern of the board repeats itself after one space in any direction. It is made of two colors, and if we interchange them, the pattern remains identical. Moreover, the fact that the board has 8 × 8 squares allows a natural separation into two halves, which can also be interchanged without altering the appearance of the board.

Now this alone is not sufficient to determine the game of chess because, for example, the game of checkers can also be played on such a board. However, when I add to this the fact that there are sixteen pieces on each of the two sides of a chess game, eight of which are identical, and of the others there are three sets of two identical pieces plus two loners, things become much more restricted. For example, it is natural to use the reflection symmetry of the board to lay out the three pieces with identical partners—the rook, the knight, and the bishop—in mirror-image pattern reflected about the center of the board. The two colors of the different opponents then replicate the duality of the board. Moreover, the set of moves of all the chess pieces is consistent with the simplest set of movements allowed by the layout of the board. Requiring a piece to move in only one color restricts the movement to diagonals, as is the case for the bishop. Requiring a pawn to capture a piece only when it is on the adjoining space of the same
color requires also that captures be made on the diagonal, and so on. While I do not claim here that this is any rigorous proof that the game of chess is completely fixed by the symmetries of the board and the pieces, it is worth noting that there is only one variation of the game that has survived today. I expect that had there been other viable possibilities, they too would still be in fashion.

You might want to amuse yourself by asking the same question about your favorite sport. Would football be the same if it were not played on a 100-yard field divided into 10-yard sections? More important, how much do the rules depend upon the symmetries of the playing squad? What about baseball? The baseball diamond seems an essential part of the game. If there were five bases arranged on a pentagon, would you need four outs?

But why confine the discussion to sports? How much are the laws of a country determined by the configuration of legislators? And to ask a question that many people worried about military spending in the United States have asked: How much is defense planning dictated by the existence of four different armed forces: air force, army, navy, and marines?

To return to physics, I want to describe how it is that symmetries, even those that are not manifest, can manage to fix the form of known physical laws. I will begin with the one conservation law I have not yet discussed that plays an essential role in physics: the conservation of charge. All processes in nature appear to conserve charge—that is, if there is one net negative charge at the beginning of any process, then no matter how complicated this process is, there will be one net negative charge left at the end. In between, many charged particles may be created or destroyed, but only in pairs of positive and negative charges, so that the total charge at any time is equal to that at the beginning
and
that at the end.

We recognize, by Noether’s theorem, that this universal conservation law is a consequence of a universal symmetry: We could transform all positive charges in the world into negative charges, and vice versa, and nothing about the world would change. This is really equivalent to saying that what we call positive and negative are arbitrary, and we merely follow convention when we call an electron negatively charged and a proton positively charged.

The symmetry responsible for charge conservation is, in fact, similar in spirit to a space-time symmetry that I have discussed in connection with general relativity. If, for example, we simultaneously changed all the rulers in the universe, so that what was 1 inch before might now read 2 inches, we would expect the laws of physics to look the same. Various fundamental constants would change in value to compensate for the change in scale, but otherwise nothing would alter. This is equivalent to the statement that we are free to use any system of units we want to describe physical processes. We may use miles and pounds in the United States, and every other developed country in the world may use kilometers and kilograms. Aside from the inconvenience of making the conversion, the laws of physics are the same in the United States as they are in the rest of the world.

But what if I choose to change the length of rulers by different amounts from point to point? What happens then? Well, Einstein told us that there is nothing wrong with the procedure. It merely implies that the laws governing the motion of particles in such a world will be equivalent to those resulting from the presence of some gravitational field.

What general relativity tells us, then, is that there is a general symmetry of nature that allows us to change the definition of length from point to point only if we also allow for the existence
of such a thing as a gravitational field. In this case, we can compensate for the local changes in the length by introducing a gravitational field. Alternatively, we might be able to find a global description in which length remains constant from point to point, and then there need be no gravitational field present. This symmetry, called general-coordinate invariance, completely specifies the theory we call general relativity. It implies that the coordinate system we use to describe space and time is itself arbitrary, just like the units we use to describe distance are arbitrary. There is a difference, however. Different coordinate systems may be equivalent, but if the conversion between them varies
locally
—that is, standard lengths vary from point to point—this conversion will also require the introduction of a gravitational field for certain observers in order for the predicted motion of bodies to remain the same. The point is this: In the weird world in which I choose to vary the definition of length from point to point, the trajectory of an object moving along under the actions of no other force will appear to be curved, and not straight. I earlier described this in my example of a plane traveling around the world as seen by someone looking at the projection on a flat map. I can account for this, and still be in agreement with Galileo’s rules, only if I allow for an apparent force to be acting in this new frame. This force is gravity. The form of gravity, remarkably, can then be said to result from the general-coordinate invariance of nature.

This is not to imply that gravity is a figment of our imagination. General relativity tells us that mass
does
curve space. In this case, all coordinate systems we can choose will account for this curvature one way or another. It may be that locally one might be able to dispense with a gravitational field—that is, an observer falling freely will not “feel” any force on him- or herself, just as the astronauts orbiting Earth are freely falling and therefore feel no gravitational
pull. However, the trajectories of different free-falling observers will bend with respect to one another—a sign that space is curved. We can choose whatever reference frame we wish. Fixed on Earth, we will experience a gravitational force. Freely falling observers may not. However, the dynamics of particles in both cases will reflect the underlying curvature of space, which
is
real and which is caused by the presence of matter. A gravitational field may be “fictional” in the sense that an appropriate choice of coordinates can get rid of it everywhere, but this is possible only if the underlying space in this case is completely flat, meaning that there is no matter around. One such example is a rotating coordinate system, such as you might find if you were standing against the wall in one of those carnival rides where a large cylindrical room turns and you get pushed to the outside. Inside the turning room, you might imagine that there is a gravitational field pulling you outward. There is, in fact, no mass that acts as a source of such a field, such as Earth does for our gravitational field. Those spectators watching recognize that what you may call gravitational field is merely an accident of a poor choice of coordinate system, one fixed to the rotating room. Curvature is real; a gravitational field is subjective.

I started out talking about electric charge and ended up talking about gravity. Now I want to do for electric charge what I did for length in space-time. Is there a symmetry of nature that
locally
allows me arbitrarily to choose my convention of the sign of electric charge and still keep the predictions of the laws of physics the same? The answer is yes, but only if there exists another field in nature acting on particles that can “compensate” for my local choice of charge in the same way that a gravitational field “compensates” for an arbitrarily varying choice of coordinate system.

The field that results from such a symmetry of nature is not the electromagnetic field itself, as you might imagine. Instead, this field plays the role of space-time curvature. It is always there if a charge is nearby, just as curvature of space is always there if a mass is around. It is not arbitrary. Instead, there is another field, related to the electromagnetic field, that plays a role analogous to the gravitational field. This field is called a
vector potential
in electromagnetism.

This weird symmetry of nature, which allows me locally to change my definition of charge or length at the expense of introducing extra forces, is called a gauge symmetry, and I referred briefly to it in an earlier chapter. Its presence, in different forms, in general relativity and electromagnetism was the reason Hermann Weyl introduced it and tried to unify the two. It turns out to be far more general, as we shall see. What I want to stress is that such a symmetry (1)
requires
the existence of various forces in nature, and (2) tells us what are those quantities that are truly “physical” and what are the quantities that are just artifacts of our particular reference frame. Just as the angular variables on a sphere are redundant if everything depends only on the radius of the sphere, so in some sense the gauge symmetry of nature tells us that electromagnetic fields and space-time curvature are physical and that gravitational fields and vector potentials are observer-dependent.

The exotic language of gauge symmetry would be mere mathematical pedantry if it were used only to describe things after the fact. After all, electromagnetism and gravity were understood well before gauge symmetry was ever proposed. What makes gauge symmetry important is its implications for the rest of physics. We have discovered in the last twenty-five years that all the known forces in nature result from gauge symmetries. This in turn has allowed us to build a new understanding of things we
did not before understand. The search for a gauge symmetry associated with these forces has allowed physicists to distinguish the relevant physical quantities that underly these forces.

It is a general property of a gauge symmetry that there must exist some field which can act over long distances, associated with the ability to compensate for the freedom to vary the definition of certain properties of particles or space-time from point to point over long distances without changing the underlying physics. In the case of general relativity, this is manifested by the gravitational field; in electromagnetism, it is manifested by electric and magnetic fields (which themselves result from vector potentials). But the weak interaction between particles in nuclei acts only over very short distances. How can it be related to an underlying gauge symmetry of nature?