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SEVENTH LECTURE - THE THEORY OF EVERYTHING

It would be very difficult to construct a complete unified theory of everything all at one go. So instead we have made progress by finding partial theories. These describe a limited range of happenings and neglect other effects, or approximate them by certain numbers. In chemistry, for example, we can calculate the interactions of atoms without knowing the internal structure of the nucleus of an atom. Ultimately, however, one would hope to find a complete, consistent, unified theory that would include all these partial theories as approximations. The quest for such a theory is known as “the unification of physics.” Einstein spent most of his later years unsuccessfully searching for a unified theory, but the time was not ripe: Very little was known about the nuclear forces. Moreover, Einstein refused to believe in the reality of quantum mechanics, despite the important role he had played in its development. Yet it seems that the uncertainty principle is a fundamental feature of the universe we live in. A successful unified theory must therefore necessarily incorporate this principle.

The prospects for finding such a theory seem to be much better now because we know so much more about the universe. But we must beware of overconfidence. We have had false dawns before. At the beginning of this century, for example, it was thought that everything could be explained in terms of the properties of continuous matter, such as elasticity and heat conduction. The

discovery of atomic structure and the uncertainty principle put an end to that. Then again, in 1928, Max Born told a group of visitors to Göttingen University, “Physics, as we know it, will be over in six months.” His confidence was based on the recent discovery by Dirac of the equation that governed the electron. It was thought that a similar equation would govern the proton, which was the only other particle known at the time, and that would be the end of theoretical physics. However, the discovery of the neutron and of nuclear forces knocked that one on the head, too.

Having said this, I still believe there are grounds for cautious optimism that we may now be near the end of the search for the ultimate laws of nature. At the moment, we have a number of partial theories. We have general relativity, the partial theory of gravity, and the partial theories that govern the weak, the strong, and the electromagnetic forces. The last three may be combined in so-called grand unified theories. These are not very satisfactory because they do not include gravity. The main difficulty in finding a theory that unifies gravity with the other forces is that general relativity is a classical theory. That is, it does not incorporate the uncertainty principle of quantum mechanics. On the other hand, the other partial theories depend on quantum mechanics in an essential way. A necessary first step, therefore, is to combine general relativity with the uncertainty principle. As we have seen, this can produce some remarkable consequences, such as black holes not being black, and the universe being completely self–contained and without boundary. The trouble is, the uncertainty principle means that even empty space is filled with pairs of virtual particles and antiparticles. These pairs would have an infinite amount of energy. This means that their gravitational attraction would curve up the universe to an infinitely small size.

Rather similar, seemingly absurd infinities occur in the other quantum theories. However, in these other theories, the infinities can be canceled out by a process called renormalization. This involves adjusting the masses of the particles and the strengths of the forces in the theory by an infinite amount. Although this technique is rather dubious mathematically, it does seem to work in practice. It has been used to make predictions that agree with observations to an extraordinary degree of accuracy. Renormalization, however, has a serious drawback from the point of view of trying to find a complete theory. When you subtract infinity from infinity, the answer can be anything you want. This means that the actual values of the masses and the strengths of the forces cannot be predicted from the theory. Instead, they have to be chosen to fit the observations. In the case of general relativity, there are only two quantities that can be adjusted: the strength of gravity and the value of the cosmological constant. But adjusting these is not sufficient to remove all the infinities. One therefore has a theory 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. In an attempt to overcome this problem, a theory called “supergravity” was suggested in 1976. This theory was really just general relativity with some additional particles.

In general relativity, the gravitational force can be thought of as being carried by a particle of spin 2 called the graviton. The idea was to add certain other new particles of spin 3/2, 1, 1/2, and 0. In a sense, all these particles could then be regarded as different aspects of the same “superparticle.” The virtual particle/antiparticle pairs of spin 1/2 and 3/2 would have negative energy. This would tend to cancel out the positive energy of the virtual pairs of particles of spin 0, 1, and 2. In this way, many of the possible infinities would cancel out, but it was suspected that some infinities might still remain. However, the calculations required to find out whether there were any infinities left uncanceled were so long and difficult that no one was prepared to undertake them. Even with a computer it was reckoned it would take at least four years. The chances were very high that one would make at least one mistake, and probably more. So one would know one had the right answer only if someone else repeated the calculation and got the same answer, and that did not seem very likely.

Because of this problem, there was a change of opinion in favor of what are called string theories. In these theories the basic objects are not particles that occupy a single point of space. Rather, they are things that have a length but no other dimension, like an infinitely thin loop of string. A particle occupies one point of space at each instant of time. Thus, its history can be represented by a line in space-time called the “world–line.” A string, on the other hand, occupies a line in space at each moment of time. So its history in space–time is a two–dimensional surface called the “world–sheet.” Any point on such a world–sheet can be described by two numbers, one specifying the time and the other the position of the point on the string. The world-sheet of a string is a cylinder or tube. A slice through the tube is a circle, which represents the position of the string at one particular time.

Two pieces of string can join together to form a single string. It is like the two legs joining on a pair of trousers. Similarly, a single piece of string can divide into two strings. In string theories, what were previously thought of as particles are now pictured as waves traveling down the string, like waves on a washing line. 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 corresponds to an H-shaped tube or pipe. String theory is rather like plumbing, in a way. Waves on the two vertical sides of the H correspond to the particles in the sun and the Earth, and waves on the horizontal crossbar correspond to the gravitational force 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 like 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, like in a spider’s web. 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 Joël Scherk and John Schwarz published a paper in which they showed that string theory could describe the gravitational force, but only if the tension in the string were very much higher—about 1039tons. The predictions of the 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 10-33 centimeters. 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. 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 a much higher proposed value of the string tension.

There seemed to have been two reasons for the sudden revival of interest in strings in 1984. 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 a paper by John Schwarz and Mike Green which 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. Whatever the reasons, a large number of people soon began to work on string theory. A new version was developed, the so–called heterotic string. This seemed as if it might be able to explain the types of particle that we observe.

String theories also lead to infinities, but it is thought they will all cancel out in versions like the heterotic string. 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 are almost a necessity. Otherwise, the fact that relativity implies that one cannot travel faster than light means that it would take far too long to get across our own galaxy, let alone to travel to other galaxies. The science fiction idea is that one can take a shortcut through a higher dimension. One can picture this in the following way. Imagine that the space we live in had only two dimensions and was curved like the surface of a doughnut or a torus. If you were on one side of the ring and you wanted to get to a point on the other side, you would have to go around the ring. However, if you were able to travel in the third dimension, you could cut straight across.

Why don’t we notice all these extra dimensions if they are really there? Why do we see only three space and one time dimension? The suggestion is that the other dimensions 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 the three space and one time dimension in which space-time is thoroughly flat. It is like the surface of an orange: if you look at it close up, it is all curved and wrinkled, but if you look at it from a distance, you don’t see the bumps and it appears to be smooth. So it is with space–time. 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 another major problem. 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 three space and one time dimension flatten out, while the other dimensions remained tightly curled up?

One possible answer is the anthropic principle. Two space dimensions do not seem to be enough to allow for the development of complicated beings like us. For example, two–dimensional people living on a one-dimensional Earth would have to climb over each other in order to get past each other. If a twodimensional 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 through its body, it would divide the creature into two separate parts. 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. 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 cause the Earth to spiral away from or into the sun. We would either freeze or be burned up. In fact, the same behavior of gravity with distance would mean that the sun would also be unstable. It would either fall apart or it would collapse to form a black hole. In either case, it would not be 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 the gravitational forces. Thus, the electrons would either escape from the atom altogether or it would spiral into the nucleus. In either case, one could not have atoms as we know them.

It seems clear that life, at least as we know it, can exist only in regions of space-time in which three space and one time dimension are not curled up small. This would mean that one could appeal to the anthropic principle, provided one could show that string theory does at least allow there to be such regions of the universe. And it seems that indeed each string theory does allow such regions. There may well be other regions of the universe, or other universes (whatever that may mean) in which all the dimensions are curled up small, or in which more than four dimensions are nearly flat. But there would be no intelligent beings in such regions to observe the different number of effective dimensions.

Apart from the question of the number of dimensions that space-time appears to have, string theory still has several other problems that must be solved before it can be acclaimed as the ultimate unified theory of physics. We do not yet know whether all the infinities cancel each other out, or exactly how to relate the waves on the string to the particular types of particle that we observe. Nevertheless, it is likely that answers to these questions will be found over the next few years, and that by the end of the century we shall know whether string theory is indeed the long sought-after unified theory of physics.

Can there really be a unified theory of everything? Or are we just chasing a mirage? There seem to be three possibilities:

• There really is a complete unified theory, which we will someday discover if we are smart enough.

• There is no ultimate theory of the universe, just an infinite sequence of theories that describe the universe more and more

accurately.

• There is no theory of the universe. Events cannot be predicted beyond a certain extent but occur in a random and arbitrary manner.

Some would argue for the third possibility on the grounds that if there were a complete set of laws, that would infringe on God’s freedom to change His mind and to intervene in the world. It’s a bit like the old paradox: Can God make a stone so heavy that He can’t lift it? But the idea that God might want to change His mind is an example of the fallacy, pointed out by St. Augustine, of imagining God as a being existing in time. Time is a property only of the universe that God created. Presumably, He knew what He intended when He set it up.

With the advent of quantum mechanics, we have come to realize that events cannot be predicted with complete accuracy but that there is always a degree of uncertainty. If one liked, one could ascribe this randomness to the intervention of God. But it would be a very strange kind of intervention. There is no evidence that it is directed toward any purpose. Indeed, if it were, it wouldn’t be random. In modern times, we have effectively removed the third possibility by redefining the goal of science. Our aim is to formulate a set of laws that will enable us to predict events up to the limit set by the uncertainty principle.

The second possibility, that there is an infinite sequence of more and more refined theories, is in agreement with all our experience so far. On many occasions, we have increased the sensitivity of our measurements or made a new class of observations only to discover new phenomena that were not predicted by the existing theory. To account for these, we have had to develop a more advanced theory. It would therefore not be very surprising if we find that our present grand unified theories break down when we test them on bigger and more powerful particle accelerators. Indeed, if we didn’t expect them to break down, there wouldn’t be much point in spending all that money on building more powerful machines.

However, it seems that gravity may provide a limit to this sequence of “boxes within boxes.” If one had a particle with an energy above what is called the Planck energy, 1019 GeV, its mass would be so concentrated that it would cut itself off from the rest of the universe and form a little black hole. Thus, it does seem that the sequence of more and more refined theories should have some limit as we go to higher and higher energies. There should be some ultimate theory of the universe. Of course, the Planck energy is a very long way from the energies of around a GeV, which are the most that we can produce in the laboratory at the present time. To bridge that gap would require a particle accelerator that was bigger than the solar system. Such an accelerator would be unlikely to be funded in the present economic climate.

However, the very early stages of the universe are an arena where such energies must have occurred. I think that there is a good chance that the study of the early universe and the requirements of mathematical consistency will lead us to a complete unified theory by the end of the century—always presuming we don’t blow ourselves up first.

What would it mean if we actually did discover the ultimate theory of the universe? It would bring to an end a long and glorious chapter in the history of our struggle to understand the universe. But it would also revolutionize the ordinary person’s understanding of the laws that govern the universe. In Newton’s time it was possible for an educated person to have a grasp of the whole of human knowledge, at least in outline. But ever since then, the pace of development of science has made this impossible. Theories were always being changed to account for new observations. They were never properly digested or simplified so that ordinary people could understand them.You had to be a specialist, and even then you could only hope to have a proper grasp of a small proportion of the scientific theories.

Further, the rate of progress was so rapid that what one learned at school or university was always a bit out of date. Only a few people could keep up with the rapidly advancing frontier of knowledge. And they had to devote their whole time to it and specialize in a small area. The rest of the population had little idea of the advances that were being made or the excitement they were generating.

Seventy years ago, if Eddington is to be believed, only two people understood the general theory of relativity. Nowadays tens of thousands of university graduates understand it, and many millions of people are at least familiar with the idea. If a complete unified theory were discovered, it would be only a matter of time before it was digested and simplified in the same way. It could then be taught in schools, at least in outline. We would then all be able to have some understanding of the laws that govern the universe and which are responsible for our existence.

Einstein once asked a question: “How much choice did God have in constructing the universe?” If the no boundary proposal is correct, He had no freedom at all to choose initial conditions. He would, of course, still have had the freedom to choose the laws that the universe obeyed. This, however, may not really have been all that much of a choice. There may well be only one or a small number of complete unified theories that are self-consistent and which allow the existence of intelligent beings.

We can ask about the nature of God even if there is only one possible unified theory that is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the question of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing? Is the unified theory so compelling that it brings about its own existence? Or does it need a creator, and, if so, does He have any effect on the universe other than being responsible for its existence?

And who created Him?

Up until now, most scientists have been too occupied with the development of new theories that describe what the universe is, to ask the question why. On the other hand, the people whose business it is to ask why—the philosophers—have not been able to keep up with the advance of scientific theories. In the eighteenth century, philosophers considered the whole of human knowledge, including science, to be their field. They discussed questions such as: Did the universe have a beginning? However, in the nineteenth and twentieth centuries, science became too technical and mathematical for the philosophers or anyone else, except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein, the most famous philosopher of this century, said, “The sole remaining task for philosophy is the analysis of language.” What a comedown from the great tradition of philosophy from Aristotle to Kant.

However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all be able to take part in the discussion of why the universe exists. If we find the answer to that, it would be the ultimate triumph of human reason. For then we would know the mind of God.