A Briefer History of Time (13 page)
Renormalization involves introducing new infinities that have the effect of canceling the infinities that arise in the theory. However, they need not cancel exactly. We can choose the new infinities so as to leave small remainders. These small remainders are called the renormalized quantities in the theory.
Although in practice this technique is rather dubious mathematically, it does seem to work, and it has been used with the theories of the strong, weak, and electromagnetic forces to make predictions that agree with observations to an extraordinary degree of accuracy. Renormalization has a serious drawback from the point of view of trying to find a complete theory, though, because it means that the actual values of the masses and the strengths of the forces cannot be predicted from the theory but have to be chosen to fit the observations. Unfortunately, in attempting to use renormalization to remove the quantum infinities from general relativity, we have only two quantities that can be adjusted: the strength of gravity and the value of the cosmological constant, the term Einstein introduced into his equations because he believed that the universe was not expanding (see Chapter 7). As it turns out, adjusting these is not sufficient to remove all the infinities. We are therefore left with a quantum theory of gravity that seems to predict that certain quantities, such as the curvature of space-time, are really infinite—yet these quantities can be observed and measured to be perfectly finite!
That this would be a problem in combining general relativity and the uncertainty principle had been suspected for some time but was finally confirmed by detailed calculations in 1972. Four years later, a possible solution, called supergravity, was suggested. Unfortunately, the calculations required to find out whether or not there were any infinities left uncanceled in supergravity were so long and difficult that no one was prepared to undertake them. Even with a computer, it was reckoned, it would take many years, and the chances were very high that there would be at least one mistake, probably more. Thus we would know we had the right answer only if someone else repeated the calculation and got the same answer, and that did not seem very likely! Still, despite these problems, and the fact that the particles in the supergravity theories did not seem to match the observed particles, most scientists believed that the theory could be altered and was probably the right answer to the problem of unifying gravity with the other forces. Then in 1984 there was a remarkable change of opinion in favor of what are called string theories.
Before string theory, each of the fundamental particles was thought to occupy a single point of space. In string theories, the basic objects are not point particles but things that have a length but no other dimension, like an infinitely thin piece of string. These strings may have ends (so-called open strings) or they may be joined up with themselves in closed loops (closed strings). A particle occupies one point of space at each moment of time. A string, on the other hand, occupies a line in space at each moment of time. Two pieces of string can join together to form a single string; in the case of open strings they simply join at the ends, while in the case of closed strings it is like the two legs joining on a pair of trousers. Similarly, a single piece of string can divide into two strings.
If the fundamental objects in the universe are strings, what are the point particles we seem to observe in our experiments? In string theories, what were previously thought of as different point particles are now pictured as various waves on the string, like waves on a vibrating kite string. Yet the strings, and the vibrations along it, are so tiny that even our best technology cannot resolve their shape, so they behave, in all of our experiments, as tiny, featureless points. Imagine looking at a speck of dust: up close, or under a magnifying glass, you may find that the fleck has an irregular or even stringlike shape, yet from a distance it looks like a featureless dot.
In string theory the emission or absorption of one particle by another corresponds to the dividing or joining together of strings. For example, the gravitational force of the sun on the earth was pictured in particle theories as being caused by the emission of the force-carrying particles called gravitons by a matter particle in the sun and their absorption by a matter particle in the earth. In string theory, this process corresponds to an H-shaped tube or pipe (string theory is rather like plumbing, in a way). The two vertical sides of the H correspond to the particles in the sun and the earth, and the horizontal crossbar corresponds to the graviton that travels between them.
String theory has a curious history. It was originally invented in the late 1960s in an attempt to find a theory to describe the strong force. The idea was that particles such as the proton and the neutron could be regarded as waves on a string. The strong forces between the particles would correspond to pieces of string that went between other bits of string, as in a spiderweb. For this theory to give the observed value of the strong force between particles, the strings had to be like rubber bands with a pull of about ten tons.
In 1974, Joel Scherk from the École Normale Supérieure in Paris and John Schwarz from the California Institute of Technology published a paper in which they showed that string theory could describe the nature of the gravitational force, but only if the tension in the string was about a thousand million million million million million million tons (1 with thirty-nine zeros after it). The predictions of string theory would be just the same as those of general relativity on normal-length scales, but they would differ at very small distances, less than a thousand million million million million millionth of a centimeter (a centimeter divided by 1 with thirty-three zeros after it). Their work did not receive much attention, however, because at just about that time most people abandoned the original string theory of the strong force in favor of the theory based on quarks and gluons, which seemed to fit much better with observations. Scherk died in tragic circumstances (he suffered from diabetes and went into a coma when no one was around to give him an injection of insulin), so Schwarz was left alone as almost the only supporter of string theory, but now with the much higher proposed value of the string tension.
Feynman Diagrams in String Theory
In string theories, long-range forces are viewed as being caused by connecting tubes rather than the interchange of force-carrying particles.
In 1984, interest in strings suddenly revived, apparently for two reasons. One was that people were not really making much progress toward showing that supergravity was finite or that it could explain the kinds of particles that we observe. The other was the publication of another paper by John Schwarz, this time with Mike Green of Queen Mary College, London. This paper showed that string theory might be able to explain the existence of particles that have a built-in left-handedness, like some of the particles that we observe. (The behavior of most particles would be the same if you changed the experimental setup by reflecting it all in a mirror, but the behavior of these particles would change. It is as if they are left- or right-handed, instead of being ambidextrous.) Whatever the reasons, a large number of people soon began to work on string theory, and a new version was developed that seemed as if it might be able to explain the types of particles that we observe.
String theories also lead to infinities, but it is thought that in the right version they will all cancel out (though this is not yet known for certain). String theories, however, have a bigger problem: they seem to be consistent only if space-time has either ten or twenty-six dimensions, instead of the usual four! Of course, extra space-time dimensions are a commonplace of science fiction. Indeed, they provide an ideal way of overcoming the normal restriction of general relativity that one cannot travel faster than light or back in time (see Chapter 10). The idea is to take a shortcut through the extra dimensions. You can picture this in the following way. Imagine that the space we live in has only two dimensions and is curved like the surface of an anchor ring or doughnut. If you were on the inside edge of the ring and you wanted to get to a point across the ring on the other side, you would have to move in a circle along the inner edge of the ring until you reached the target point. However, if you were able to travel in the third dimension, you could leave the ring and cut straight across.
Why don’t we notice all these extra dimensions if they are really there? Why do we see only three space dimensions and one time dimension? The suggestion is that the other dimensions are not like the dimensions we are used to. They are curved up into a space of very small size, something like a million million million million millionth of an inch. This is so small that we just don’t notice it: we see only one time dimension and three space dimensions, in which space-time is fairly flat. To picture how this works, think of the surface of a straw. If you look at it closely, you see the surface is two-dimensional. That is, the position of a point on the straw is described by two numbers, the length along the straw and the distance around the circular dimension. But its circular dimension is much smaller than its dimension of length. Because of that, if you look at the straw from a distance, you don’t see the thickness of the straw and it looks one-dimensional. That is, it appears that to specify the position of a point you need only to give the length along the straw. So it is with space-time, string theorists say: on a very small scale it is ten-dimensional and highly curved, but on bigger scales you don’t see the curvature or the extra dimensions.
If this picture is correct, it spells bad news for would-be space travelers: the extra dimensions would be far too small to allow a spaceship through. However, it raises a major problem for scientists as well: why should some, but not all, of the dimensions be curled up into a small ball? Presumably, in the very early universe all the dimensions would have been very curved. Why did one time dimension and three space dimensions flatten out, while the other dimensions remain tightly curled up?
One possible answer is what is called the anthropic principle, which can be paraphrased as "We see the universe the way it is because we exist." There are two versions of the anthropic principle, the weak and the strong. The weak anthropic principle states that in a universe that is large or infinite in space and/or time, the conditions necessary for the development of intelligent life will be met only in certain regions that are limited in space and time. The intelligent beings in these regions should therefore not be surprised if they observe that their locality in the universe satisfies the conditions that are necessary for their existence. It is a bit like a rich person living in a wealthy neighborhood not seeing any poverty.
Some go much further and propose a strong version of the principle. According to this theory, there are either many different universes or many different regions of a single universe, each with its own initial configuration and, perhaps, with its own set of laws of science. In most of these universes the conditions would not be right for the development of complicated organisms; only in the few universes that are like ours would intelligent beings develop and ask the question, "Why is the universe the way we see it?" The answer is then simple: if it had been different, we would not be here!
Few people would quarrel with the validity or utility of the weak anthropic principle, but there are a number of objections that one can raise to the strong anthropic principle as an explanation of the observed state of the universe. For instance, in what sense can all these different universes be said to exist? If they are really separate from each other, what happens in another universe can have no observable consequences in our own universe. We should therefore use the principle of economy and cut them out of the theory. If, on the other hand, they were just different regions of a single universe, the laws of science would have to be the same in each region, because otherwise we could not move continuously from one region to another. In this case the only difference between the regions would be their initial configurations, so the strong anthropic principle would reduce to the weak one.
The anthropic principle gives one possible answer to the question of why the extra dimensions of string theory curled up. Two space dimensions do not seem to be enough to allow for the development of complicated beings like us. For example, two-dimensional animals living on a circle (the surface of a two-dimensional earth) would have to climb over each other in order to get past each other. And if a two-dimensional creature ate something it could not digest completely, it would have to bring up the remains the same way it swallowed them, because if there were a passage right through its body, it would divide the creature into two separate halves: our two-dimensional being would fall apart. Similarly, it is difficult to see how there could be any circulation of the blood in a two-dimensional creature.
There would also be problems with more than three space dimensions. The gravitational force between two bodies would decrease more rapidly with distance than it does in three dimensions. (In three dimensions, the gravitational force drops to one-quarter as you double the distance. In four dimensions it would drop to one-eighth, in five dimensions to one-sixteenth, and so on.) The significance of this is that the orbits of planets, like the earth, around the sun would be unstable: the least disturbance from a circular orbit (such as would be caused by the gravitational attraction of other planets) would result in the earth spiraling away from or into the sun. We would either freeze or be burned up. In fact, the same behavior of gravity with distance in more than three space dimensions means that the sun would not be able to exist in a stable state, with pressure balancing gravity. The sun would either fall apart or collapse to form a black hole. In either case, it would not be of much use as a source of heat and light for life on earth. On a smaller scale, the electrical forces that cause the electrons to orbit around the nucleus in an atom would behave in the same way as gravitational forces. Thus the electrons would either escape from the atom altogether or would spiral into the nucleus. In either case, there would be no atoms as we know them.
