by Lee Smolin
Let us begin, then, with what we do know about string theory, for these are reasons enough to take it seriously. Quantum theory says that for every wave there is an associated particle. For electromagnetic waves there is the photon. For electrons there is the electron wave (the wave-function). The wave doesn’t even have to be something fundamental. When I strike a tuning fork I set up waves that travel up and down it: these are sound waves travelling in metal. Quantum theory associates a particle with such sound waves; it is called a phonon. Suppose I disturb the empty space around us by making a gravitational wave. This can be done by waving around anything with mass - one of my arms will do, or a pair of neutron stars. A gravitational wave can be understood as a tiny ripple moving against a background, which is the empty space.
The particle associated with gravitational waves is called the graviton. No one has ever observed a graviton. It is hard enough even to detect a gravitational wave, as they interact only very weakly with matter. But as long as quantum theory applies to gravitational waves, gravitons must exist. We know that gravitons must interact with matter, for when anything massive oscillates it produces gravitational waves. Quantum theory says that, just as there are photons associated with light, there must be gravitons associated with gravitational waves.
We know that two gravitons will interact with each other. This is because gravitons interact with anything that has energy, and gravitons themselves carry energy. As with the photon, the energy of a graviton is proportional to its frequency, so the higher the frequency of a graviton, the more strongly it will interact with another graviton. We can then ask what happens when two gravitons interact. We know that they will scatter from each other, changing their trajectories. A good quantum theory of gravity must be able to predict what will happen whenever two gravitons interact. It ought to be able to produce an answer no matter how strong the waves are and no matter what their frequencies are. This is just the kind of question that we know how to approach in quantum theory. For example, we know that photons will interact with any charged particle, such as an electron. We have a good theory of the interactions of photons and electrons, called quantum electrodynamics, QED for short. It was developed by Richard Feynman, Julian Schwinger, Sinitiro Tomonaga and others in the late 1940s. QED makes predictions about the scattering of photons and electrons and other charged particles that agree with experiment to an accuracy of eleven decimal places.
Physics, like the other sciences, is the art of the possible. So I must add a rider here, which is that we do not really understand QED. We know the principles of the theory and we can deduce from them the basic equations that define the theory. But we cannot actually solve these equations, or even prove that they are mathematically consistent. Instead, to make sense of them we have to resort to a kind of subterfuge. We make some assumptions about the nature of the solutions - which, after more than fifty years, are still unproved - and these lead us to a procedure for calculating approximately what happens when photons and electrons interact. This procedure is called perturbation theory. It is very useful in that it does lead to answers that agree very precisely with experiment. But we do not actually know whether the procedure is consistent or not, or whether it accurately reflects what a real solution to the theory would predict. String theory is presently understood mainly in the language of this approximation procedure. It was invented by modifying the approximation procedure, rather than the theory. This is how people were able to invent a theory which is understood only as a list of solutions.
Perturbation theory is actually quite easy to describe. Thanks to Feynman, there is a simple diagrammatic means for understanding it. Picture a world of processes in which three things can happen. An electron may move from point A at one time to point B at another. We can draw this as a line, as in Figure 33. A photon may also travel, which is indicated by a dotted line in the figure. The only other thing that may happen is that an electron and a photon interact, which is indicated by the point where a photon line meets an electron line. To compute what happens when two electrons meet, one simply draws all the things that can happen, beginning with two electrons entering the scene, and ending with two electrons leaving. There are an infinite number of such processes, and we see a few of them in Figure 34. Feynman taught us to associate with each diagram the probability (actually the quantum amplitude, whose square is the probability) of that process. One can then work out all the predictions of the theory.
FIGURE 33
The basic processes in the theory of electrons and photons (called quantum electrodynamics, or QED for short.) Electrons and photons can move freely in spacetime, or they can interact in events in which an electron absorbs or emits a photon.
In the language of these diagrams, now known appropriately as Feynman diagrams, it is very easy to explain what string theory is. The basic postulate of the theory is that there are no particles, only strings moving in space. A string is just a loop drawn in space. It is not made of anything, just as a particle is thought of as a point and nothing else. There is only one kind of string, and the different kinds of particle are postulated to be nothing but different modes of vibration of these loops. So, as shown in Figure 35, photons and electrons are to be thought of just as different ways in which a string can vibrate. When a string moves in time it makes a tube rather than a line (Figure 35). Two strings can also join and merge into one (Figure 36), or one string can split into two. All the interactions that occur in nature, including those of photons and electrons, can be interpreted in terms of these splittings and joinings. We can see from these pictures that string theory gives a very satisfactory unification and simplification of the physical processes represented in Feynman diagrams. Its main virtue is that it gives a simple way of finding theories that make consistent physical predictions.
FIGURE 34
The processes illustrated in Figure 33 are put together to make Feynman diagrams, which are pictures of the possible ways a process can happen. Shown here are some of the ways in which two electrons can interact simply by absorbing and emitting photons. Each one is a story that is a possible piece of the history of a universe.
FIGURE 35
In string theory there is only one kind of thing that moves, and that is a string - a loop drawn in space. Different modes of vibration of the string behave like the different kinds of elementary particle.
FIGURE 36
All the different kinds of interaction between particles are interpreted in string theory in terms of the splitting and joining of strings.
The trouble with Feynman’s method is that it always leads to infinite expressions. This is because there are loops in the diagrams where particles are created, interact, and are then destroyed. These are called virtual particles because they exist only for a very short time. According to the uncertainty principle, because virtual particles live for a very short time they can have any energy, as the conservation of energy is suspended during their brief lives. This creates big problems. One has to add up all the diagrams to get the overall probability for the process to occur, but if some particles can have any energy between zero and infinity, then the list of possible processes one has to add up will be infinite. This leads to mathematical expressions that are no more than complicated ways of writing the number infinity. As a result, Feynman’s method seems at first to give nonsensical answers to questions about the interactions of electrons and photons.
Quite ingeniously, Feynman and others discovered that the theory was giving silly answers to only a few questions, such as ‘What is the mass of the electron?’ and ‘What is its charge?’ The theory predicts that these are infinite! Feynman figured out that if one simply crosses out these infinite answers wherever they appear, and substitutes the right, finite answer, the answers to all other questions become sensible. All the infinite expressions can be removed if one forces the theory to give the right answer for the mass and the charge of the electron. This procedure is called renormalization. When it works for a theory, that theo
ry is called renormalizable. The procedure works very well for quantum electrodynamics. It also works for quantum chromodynamics, and for the Weinberg-Salam theory, which is our theory of radioactive decay. When this procedure does not work, we say that a theory is not renormalizable - the method fails to give a sensible theory. This is actually the case for most theories; only certain special ones can be made sense of by these methods.
The most important theory that cannot be made sense of in this way is Einstein’s theory of gravity. The reason has to do with the fact that arbitrarily large energies can appear in the particles moving inside the diagrams. But the strength of the gravitational force is proportional to the energy, because energy is mass, from Einstein, and gravity pulls on mass, from Newton. So the diagrams with larger energies give correspondingly larger effects. But according to the theory, the energies inside the diagrams can be arbitrarily large. The result is a kind of runaway feedback process in which we lose all control over what is happening inside the diagrams. No one has ever found a way to describe a gravitational theory in the language of particles moving around Feynman diagrams. But in string theory one can make sense of the effects of gravity. This is one of its great achievements. As with the older theories, there are many string theory variants that lead to infinite expressions for every physical process, and these must be discarded. What is left is a set of theories that have no infinities at all. One does not have to play any games to isolate infinite expressions for masses and throw them out. There are just two possible kinds of string theory: inconsistent and consistent. And all the consistent ones appear to give finite and sensible expressions for all physical quantities.
The list of consistent string theories is very long. There are consistent string theories in all dimensions from one to nine. In nine dimensions there are five different kinds of consistent string theory. When we get down to the three-dimensional world we seem to live in, there are at least hundreds of thousands of different consistent string theories. Most of these theories come with free parameters, so they do not make unique predictions for things like the masses of the elementary particles. Each consistent string theory is very tightly structured. Because all the different kinds of particle arise from vibrations of the same fundamental objects, one is not generally free to choose which particles are described by the theory. There are an infinite number of possible vibrations and hence of possible particles, although most of them will have energies which are too large to observe. Only the lowest modes of vibration correspond to particles with masses we could observe. A remarkable fact is that the particles that correspond to the lowest modes of vibration of a string always include the broad categories of particles and forces we do observe. The other modes of vibration correspond to particles with masses of around 1019 times the mass of the proton. This is the Planck mass, which is the mass of a black hole the size of a Planck length.
However, there still are issues which must be addressed if string theory is to describe our universe. Many string theories predict the existence of particles which have so far not been seen. Many have problems keeping the strength of the gravitational force from varying in space and time. And almost all consistent string theories predict symmetries among their particles beyond those that are seen. The most important of these are supersymmetries.
Supersymmetry is an important idea, so it is worth while making a detour here to discuss it. To understand supersymmetry one must know that elementary particles come into two general types: bosons and fermions. Bosons, which include photons and gravitons, are particles whose angular momentum, when measured in units of Planck’s constant, are simple integers. Fermions, which include electrons, quarks and neutrinos, have angular momenta that come in units of one-half. Fermions also satisfy the Pauli exclusion principle, which states that no two of them can be put in the same state. Supersymmetry requires fermions and bosons to come in pairs consisting of one of each, with the same mass. This is definitely not observed in nature. If there were such things as bosonic electrons and quarks, the world would be a very different place, for the Pauli exclusion principle would have no force, and no form of matter would be stable. If supersymmetry is true of our world, then it has been spontaneously broken, which is to say that the background fields must confer a large mass on one member of each pair and not on the other. The only reason to entertain the idea of such a strange symmetry is that it seems to be required for most, if not all, versions of string theory to give consistent answers.
The search for evidence of supersymmetry is a major priority of experiments now under way at particle accelerators. String theorists very much hope that evidence for supersymmetry will be found. If supersymmetry is not found experimentally, it would still be possible to concoct a string theory that agrees with experiment, but this would be a less happy outcome than if experimental support for supersymmetry were forthcoming.
There is obviously something very wonderful about string theory. Among its strong points are the natural way it leads to a unification of all particles and forces, and the fact that there are many consistent string theories that include gravity. String theory is also the perfect realization of the hypothesis of duality discussed in Chapter 9. Also, it cannot be overemphasized that in the language in which it is understood - that of diagrams corresponding to quantum particles moving against a background spacetime - string theory is the only known way of consistently unifying gravity with quantum theory and the other forces of nature.
What is very frustrating is that in spite of this, string theory does not seem to fully incorporate the basic lesson of general relativity, which is that space and time are dynamical rather than fixed, and relational rather than absolute. In string theory, as it has so far been formulated, the strings move against a background spacetime which is absolute and fixed. The geometry of space and time is usually presumed to be fixed for ever; all that happens is that some strings move against this fixed background and interact with one another. But this is wrong, because it replicates the basic mistake of Newtonian physics in treating space and time as a fixed and unchanging background against which things move and interact. As I have already emphasized, the right thing to do is to treat the whole system of relationships that make up space and time as a single dynamical entity, without fixing any of it. This is how general relativity and loop quantum gravity work.
Still, science is not made from absolutes. The progress of science is based on what is possible, which means that it often makes sense to do what is practical, even if it seems to go against established principles. For this reason, even if it is ultimately wrong, it may still be useful to follow the background dependent approach as far as it will go, to see whether there is a consistent picture in which we can answer questions such as what happens when two gravitons moving in empty spacetime scatter from each other. As long as we remember that such a picture can give at best an approximate description this can be an important and necessary step in the discovery of the quantum theory of gravity.
Another main shortcoming of string theory is that is not one theory, but a whole class of theories, so it does not lead to many predictions about the elementary particles. This shortcoming is closely related to the problem of background dependence. Each string theory moves against a different background spacetime, so to define a string theory one must first fix the dimension of space and the geometry of spacetime. In many cases space has more dimensions than the three we observe. This is explained by the hypothesis that in our universe the extra dimensions are curled up too tightly for us to perceive directly. We say that the extra six dimensions have been compactified. Since string theory is simplest if the world has nine spatial dimensions, this leads to a picture in which many of the different consistent string theories in three dimensions can be understood as arising from different ways of choosing the structure of a hidden six-dimensional space.
There are at least hundreds of thousands of ways in which the six extra dimensions may be compactified. Each way corresponds to a different geometr
y and topology for the extra six dimensions. As a result there are at least that many different string theories that are consistent with the basic observation that the world has three large spatial dimensions. Furthermore, each of these theories has a set of parameters that describe the size and other geometric properties of the six compactified dimensions. These turn out to influence the physics that we see in the three-dimensional world. For example, the geometry of the extra dimensions influences the masses and the strengths of the interactions of the elementary particles we observe.
It is most likely irrelevant whether these extra dimensions exist in any literal sense. If one is drawn to a picture of our three-dimensional ‘reality’ embedded in some higher-dimensional realm, then one can believe in the extra dimensions, at least as long as one is working in this background dependent picture. But these extra dimensions can also be seen as purely theoretical devices which are useful for understanding the list of consistent string theories in three dimensions. As long as we stay on the background dependent level, it does not really matter.