Three Roads to Quantum Gravity

by Lee Smolin

  As a result, although it is a unified theory, string theory in its present form makes few predictions about the physics we actually observe. Many different scenarios for what the new, more powerful particle accelerators will find are consistent with one version of string theory or another. Thus, not only does string theory lack experimental confirmation, but it is hard to imagine an experiment that could be done in the next several decades that could definitively confirm or reject it. Nor is there anything special, from the point of view of string theory, about having six out of nine dimensions compactified while the other three are left large. String theory can easily describe a world in which any number of dimensions, from nine down to none at all, are left large.

  String theory thus indicates that the world we see provides only a sparse and narrow sampling of all possible physical phenomena, for if true it tells us that most of the dimensions and most of the symmetry of the world are hidden. Still, many people do believe in it. This is partly because, however incomplete its present formulation may be, string theory remains the one approach that unifies gravity with the other forces consistently at a background dependent level.

  The main problem in string theory, then, is how to see beyond it to a theory which will incorporate the successes of string theory while avoiding its weaknesses. One approach to this problem begins with the following question. What if there were a single theory that unified the different string theories by interpreting each of its solutions as one of the consistent string theories? The different string theories, together with the spacetimes they live in, will not be put in as absolutes. Rather they will all arise from solutions of this new theory. Note that the new theory could not be formulated in terms of any objects moving against a fixed spacetime background, because its solutions would include all the possible background spacetimes. The different solutions of this fundamental theory would be analogous to the different spacetimes which are all solutions to the equations of general relativity.

  Now we can argue by analogy in the following way. Let us take any spacetime which is a solution to the Einstein equations, and wiggle some matter within it. This will generate gravitational waves. These waves move on the original spacetime like ripples moving on the surface of a pond. We can make ripples in the solution of our fundamental theory in the same way. What if these gave rise not to waves moving on the background, but to strings? This may be hard to visualize, but remember that according to the hypothesis of duality strings are just a different way of looking at a field, like the electric field. And if we wiggle a field we get waves. The wiggles in the electric and magnetic field are after all nothing but light. But if duality is true, there must be a way to understand this in terms of the motion of strings through space.

  If this picture is correct, then each string theory is not really a theory in its own right. It is no more than an approximate description of how ripples may move against a background spacetime which itself is a solution to another theory. That theory would be some extension of general relativity, formulated in terms that were relational and background independent.

  This hypothesis would, if true, explain why there are so many different string theories. The solutions to the fundamental theory will define a large number of different possible universes, each described in terms of a different space and time.

  It remains only to construct this single, unifying string theory. This is a project that a few people are working hard on, and I must confess it is something I also am spending a lot of time on. There is presently no agreed upon form of this theory, but at least we have a name for it - we call it M theory. No one knows what the M stands for, which we feel is appropriate for a theory whose existence has so far only been conjectured.

  These days, string theorists spend much of their time looking for evidence that M theory exists. One strategy which has been very successful is to look for relationships between different string theories. A number of cases have been found in which two apparently different versions of string theory turn out to describe exactly the same physical phenomena. (In some cases this is seen directly; in others the coincidence is apparent only certain approximations or from studying simplified versions of the theories in which extra symmetries have been imposed.) These relationships suggest that the different string theories are part of a larger theory. The information about these relationships can be used to learn something about the structure M theory must have, if it exists. For example, it gives us some information about the symmetries that M theory will have. These are symmetries that extend the idea of duality in a major way, which could not be done within any single string theory.

  Another very important question is whether M theory describes a universe in which space and time are continuous or discrete. At first it seems that string theory points to a continuous world, because it is based on a picture of strings moving continuously through space and time. But this turns out to be misleading, for when looked at closely string theory seems to be describing a world in which space has a discrete structure.

  One way to see the discreteness is to study strings on a space that has been wrapped up, so that one dimension forms a circle (Figure 37). The circle which has been wrapped up has radius R. You might think that the theory would get into trouble if we allowed R to get smaller and smaller. But string theory turns out to have the amazing property that what happens when R becomes very small is indistinguishable from what happens when R becomes very large. The result is that there is a smallest possible value for R. If string theory is right, then the universe cannot be smaller than this.

  FIGURE 37

  A cylinder is a two-dimensional space in which one direction is a circle. We see a string wrapped on the circle. This is typical of ideas of how the extra dimensions are hidden; the horizontal direction is typical of the three ordinary directions, while the vertical direction stands for one of the hidden dimensions. Time is not indicated here.

  There is a pretty simple explanation for this, which I hope will at least give you a taste of the kind of reasoning that permeates the study of string theory. The reason why R has a smallest possible value has to do with the fact that there are two different things a string can do when wrapped around a cylinder (it is said to have two degrees of freedom). First, it can vibrate, like a guitar string. Since the radius of the cylinder is fixed there will be a discrete series of modes in which the string can vibrate. But the string has another degree of freedom, because one can vary the number of times it is wrapped around the cylinder. Thus there are two numbers that characterize a string wrapped around a cylinder: the mode number and the number of times it is wrapped.

  It turns out that if one tries to decrease the radius of the cylinder, R, below a certain critical value, these two numbers just trade places. A string in the 3rd mode of vibration wrapped 5 times around a cylinder with R slightly smaller than the critical value becomes indistinguishable from a string wrapped 3 times around a cylinder slightly larger than the critical value, when it is in the 5th mode of vibration. The effect is that every mode of vibration of a string on a small cylinder is indistinguishable from a different mode of a string wrapped on a large cylinder. Since we cannot tell them apart, the modes of strings wrapped around small cylinders are redundant. All the states of the theory can be described in terms of cylinders larger than the critical value.

  Another way to see the discreteness is to imagine a string going by at very nearly the speed of light. It would appear to contain a set of discrete elements, each of which carries a certain fixed amount of momentum. These are called string bits, and they are shown in Figure 38. The more momentum a string has, the longer it is, so there is a limit to the size of an object that can be resolved by looking at it with a string. But since, according to string theory, all the particles in nature are actually made up of strings, then, if the theory is right, there is a smallest size. Just as there is a smallest piece of silver, which is a silver atom, there is a smallest possible process that can propagate, and that is a string bit.
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  FIGURE 38

  A string seen through a Planck-scale magnifying glass is found to consist of discrete bits, rather like a wooden toy snake.

  There turns out to be a simple way to express the fact that there is a minimum size that can be probed in string theory. In ordinary quantum theory the limitations to what can be observed are expressed in terms of the Heisenberg uncertainty principle. This says thatΔx > (h/Δp)

  where Δx is the uncertainty in position, h is Planck’s famous constant and Δp is the uncertainty in momentum. String theory amends this equation toΔx > (h/Δp) + CΔp

  where C is another constant that has to do with the Planck scale. Now, without this new term one can make the uncertainty in position as small as one likes, by making the uncertainty in momentum large. With the new term in the equation one cannot do this, for when the uncertainty in momentum becomes large enough the second term comes in and forces the uncertainty in position to start to increase rather than decrease. The result is that there is a minimum value to the uncertainty in position, and this means that there is an absolute limit to the precision with which any object can be located in space.

  This tells us that M theory, if it exists, cannot describe a world in which space is continuous and one can pack an infinite amount of information into any volume, no matter how small. This suggests that whatever it is, M theory will not be some direct extension of string theory, as it will have to be formulated in a different conceptual language. The present formulation of string theory is likely, then, to be a transitional stage in which elements of a new physics are mixed up with the old Newtonian framework, according to which space and time are continuous, infinitely divisible and absolute. The problem that remains is to separate out the old from new and find a coherent way to formulate a theory using only those principles that are supported by the experimental physics of the twentieth and twenty-first centuries.

  III

  THE PRESENT FRONTIERS

  CHAPTER 12

  THE HOLOGRAPHIC PRINCIPLE

  In Part II we looked at three different approaches to quantum gravity: black hole thermodynamics, loop quantum gravity and string theory. While each takes a different starting point, they all agree that when viewed on the Planck scale, space and time cannot be continuous. For seemingly different reasons, at the end of each of these roads one reaches the conclusion that the old picture according to which space and time are continuous must be abandoned. On the Planck scale, space appears to be composed of fundamental discrete units.

  Loop quantum gravity gives us a detailed picture of these units, in terms of spin networks. It tells us that areas and volumes are quantized and come only in discrete units. String theory at first appears to describe a continuous string moving in a continuous space. But a closer look reveals that a string is actually made of discrete pieces, called string bits, each of which carries a discrete amount of momentum and energy. This is expressed in a simple and beautiful way as an extension of the uncertainty principle, which tells us that there is a smallest possible length.

  Black hole thermodynamics leads to an even more extreme conclusion, the Bekenstein bound. According to this principle the amount of information that can be contained in any region is not only finite, it is proportional to the area of the boundary of the region, measured in Planck units. This implies that the world must be discrete on the Planck scale, for were it continuous any region could contain an infinite amount of information.

  It is remarkable that all three roads lead to the general conclusion that space becomes discrete on the Planck scale. However, the three different pictures of quantum spacetime that emerge seem rather different. So it remains to join these pictures together to make a single picture which, when we understand it, will become the one final road to quantum gravity.

  At first it may not be obvious how to do this. The three different approaches investigate different aspects of the world. Even if there is one ultimate theory of quantum gravity, there will be different physical regimes, in which the basic principles may manifest themselves differently. This seems to be what is happening here. The different versions of discreteness arise from asking different questions. We would find an actual contradiction only if, when we asked the same question in two different theories, we got two different answers. So far this has not happened, because the different approaches ask different kinds of question. It is possible that the different approaches represent different windows onto the same quantum world - and if this is so, there must be a way of unifying them all into a single theory.

  If the different approaches are to be unified, there must be a principle which expresses the discreteness of quantum geometry in a way that is consistent with all three approaches If such a principle can be found, then it will serve as a guide to combining them into one theory. In fact, just such a principle has been proposed in recent years. It is called the holographic principle.

  Several different versions of this principle have been proposed by different people. After a lot of discussion over the last few years there is still no agreement about exactly what the holographic principle means, but there is a strong feeling among those of us in the field that some version of the holographic principle is true. And if it is true, it will be the first principle which makes sense only in the context of a quantum theory of gravity. This means that even if it is presently understood as a consequence of the principles of general relativity and quantum theory, there is a chance that in the end the situation will be reversed and the holographic principle will become part of the foundations of physics, from which quantum theory and relativity may both be deduced as special cases.

  The holographic principle was inspired first of all by the Bekenstein bound, which we discussed in Chapter 8. Here is one way to describe the Bekenstein bound. Consider any physical system, made of anything at all - let us call it The Thing. We require only that The Thing can be enclosed within a finite boundary, which we shall call The Screen (Figure 39). We would like to know as much as possible about The Thing. But we cannot touch it directly - we are restricted to making measurements of it on The Screen. We may send any kind of radiation we like through The Screen, and record whatever changes result on The Screen. The Bekenstein bound says that there is a general limit to how many yes/no questions we can answer about The Thing by making observations through The Screen that surrounds it. The number must be less than one-quarter of the area of The Screen, in Planck units. What if we ask more questions? The principle tells us that either of two things must happen. Either the area of the screen will increase, as a result of doing an experiment that asks questions beyond the limit; or the experiments we do that go beyond the limit will erase, or invalidate, the answers to some of the previous questions. At no time can we know more about The Thing than the limit, imposed by the area of The Screen.

  FIGURE 39

  The argument for the Bekenstein bound. We observe The Thing through The Screen, which limits the amount of information we can receive about The Thing to what can be represented on The Screen.

  What is most surprising about this is not just that there is a limit on the amount of information that can be coded into The Thing - after all, if we believe that the world has a discrete structure then this is exactly what we should expect. It is just that we would normally expect the amount of information that can be coded into The Thing to be proportional to its volume, not to the area of a surface that contains it. For example, suppose that The Thing is a computer memory. If we continue to miniaturize computers more and more, we shall eventually be building them purely out of the quantum geometry in space - and that has to be the limit of what can be done. Imagine that we can then build a computer memory out of nothing but the spin network states that describe the quantum geometry of space. The number of different such spin network states can be shown to be proportional to volume of the world that state describes (The reason is that there are so many states per node, and the volume is proportional to the number of nodes.) The Bekenstein bound does not
dispute this, but it asserts that the amount of information that we outside observers could extract is proportional to the area and not the volume. And the area is proportional not to the number of nodes of the network, but to the number of edges that go through the screen (Figure 40). This tells us that the most efficient memory we could construct out of the quantum geometry of space is achieved by constructing a surface and putting one bit of memory in every region 2 Planck lengths on a side. Once we have done this, building the memory into the third dimension will not help.

  FIGURE 40

  A spin network, which describes the quantum geometry of space, intersects a boundary such as a horizon in a finite number of points. Each intersection adds to the total area of the boundary.

  This idea is very surprising. If it is to be taken seriously, there had better be a good reason for it. In fact there is, for the Bekenstein bound is a consequence of the second law of thermodynamics. The argument that leads from the laws of thermodynamics to the Bekenstein bound is not actually very complicated. Because of its importance I give a form of it in the box on the next page.

  There are at least two more good reasons to believe in the Bekenstein bound. One is that the relationship between Einstein’s theory and the bound can be turned around. In the argument for the Bekenstein bound as I present it in the box, the bound is partly a consequence of the equations of Einstein’s general theory of relativity. But, as Ted Jacobson has shown in a justly famous paper, the argument can be turned on its head so that the equations of Einstein’s theory can be derived by assuming that the laws of thermodynamics and the Bekenstein bound are true. He does this by showing that the area of The Screen must change when energy flows through it, because the laws of thermodynamics require that some entropy flows along with the energy. The result is that the geometry of space, which determines the area of The Screen, must change in response to the flow of energy. Jacobson shows that this actually implies the equations of Einstein’s theory.