Three Roads to Quantum Gravity (15 page)

This raises two profound questions:

These questions have motivated a great deal of research since the mid-1970s. In the next few chapters I shall explain why there is a growing consensus among physicists that the answer to both questions must be ‘yes’.

Both loop quantum gravity and string theory assert that there is an atomic structure to space. In the next two chapters we shall see that loop quantum gravity in fact gives a rather detailed picture of that atomic structure. The picture of the atomic structure one gets from string theory is presently incomplete but, as we shall see in Chapter 11, it is still impossible in string theory to avoid the conclusion that there must be an atomic structure to space and time. In Chapter 13 we shall discover that both pictures of the atomic structure of space can be used to explain the entropy and temperature of black holes.

But even without these detailed pictures there is a very general argument, based simply on what we have learned in the last few chapters, that leads to the conclusion that space must have an atomic structure. This argument rests on the simple fact that horizons have entropy. In previous chapters we have seen that this is common to both the horizons of black holes and to the horizon experienced by an accelerated observer. In each case there is a hidden region in which information can be trapped, outside the reach of external observers. Since entropy is a measure of missing information, it is reasonable that in these cases there is an entropy associated with the horizon, which is the boundary of the hidden region. But what was most remarkable is that the amount of missing information measured by the entropy had a very simple form. It was simply equal to one-quarter of the area of the horizon, in Planck units.

The fact that the amount of missing information depends on the area of the boundary of the trapped region is a very important clue. It becomes even more significant if we put this dependence together with the fact that spacetime can be understood to be structured by processes which transmit information from the past to the future, as we saw in Chapter 4. If a surface can be seen as a kind of channel through which information flows from one region of space to another, then the area of the surface is a measure of its capacity to transmit information. This is very suggestive.

It is also strange that the amount of trapped information is proportional to the area of the boundary. It would seem more
natural for the amount of information that can be trapped in a region to be proportional to its volume, not to the area of its boundary. No matter what is on the other side of the boundary, trapped in the hidden region, it can contain the answer to only a finite number of yes/no questions per unit area of the boundary. This seems to be saying that a black hole, whose horizon has a finite area, can hold only a finite amount of information.

If this is the right interpretation of the results I described in the last chapter, it suffices to tells us that the world must be discrete, since whether a given volume of space is behind a horizon or not depends on the motion of an observer. For any volume of space we may want to consider, we can find an observer who accelerates away from it in such a way that that region becomes part of that observer’s hidden region. This tells us that in that volume there could be no more information than the limit we are discussing, which is a finite amount per unit area of the boundary. If this is right, then no region can contain more than a finite amount of information. If the world really were continuous, then every volume of space would contain an infinite amount of information. In a continuous world it takes an infinite amount of information to specify the position of even one electron. This is because the position is given by a real number, and most real numbers require an infinite number of digits to describe them. If we write out their decimal expansion, it will require an infinite number of decimal places to write down the number.

In practice, the greatest amount of information that may be stored behind a horizon is huge - 10
⁶⁶
bits of information per square centimetre. No actual experiment so far comes close to probing this limit. But if we want to describe nature on the Planck scale, we shall certainly run into this limitation, as it allows us to talk about only one bit of information for every four Planck areas. After all, if the limit were one bit of information per square centimetre rather than per square Planck area, it would be quite hard to see anything because our eyes would then be able respond to at most one photon at a time.

Many of the important principles in twentieth-century
physics are expressed as limitations on what we can know. Einstein’s principle of relativity (which was an extension of a principle of Galileo’s) says that we cannot do any experiment that would distinguish being at rest from moving at a constant velocity. Heisenberg’s uncertainty principle tells us that we cannot know both the position and momentum of a particle to arbitrary accuracy. This new limitation tells us there is an absolute bound to the information available to us about what is contained on the other side of a horizon. It is known as Bekenstein’s bound, as it was discussed in papers Jacob Bekenstein wrote in the 1970s shortly after he discovered the entropy of black holes.

It is curious that, despite everyone who has worked on quantum gravity having been aware of this result, few seem to have taken it seriously for the twenty years following the publication of Bekenstein’s papers. Although the arguments he used were simple, Jacob Bekenstein was far ahead of his time. The idea that there is an absolute limit to information which requires each region of space to contain at most a certain finite amount of information was just too shocking for us to assimilate at the time. There is no way to reconcile this with the view that space is continuous, for that implies that each finite volume can contain an infinite amount of information. Before Bekenstein’s bound could be taken seriously, people had to discover other, independent reasons why space should have a discrete, atomic structure. To do this we had to learn to do physics at the scale of the smallest possible things.

T
he first approach to quantum gravity that yielded a detailed description of the atomic structure of space and spacetime was loop quantum gravity. The theory offers more than a picture: it makes precise predictions about what would be observed were it possible to probe the geometry of space at distances as short as the Planck scale.

According to loop quantum gravity, space is made of discrete atoms each of which carries a very tiny unit of volume. In contrast to ordinary geometry, a given region cannot have a volume which is arbitrarily big or small - instead, the volume must be one of a finite set of numbers. This is just what quantum theory does with other quantities: it restricts a quantity that is continuous according to Newtonian physics to a finite set of values. This is what happens to the energy of an electron in an atom, and to the value of the electric charge. As a result, we say that the volume of space is predicted to be quantized.

One consequence of this is that there is a smallest possible volume. This minimum volume is minuscule - about 10
⁹⁹
of them would fit into a thimble. If you tried to halve a region of this volume, the result would not be two regions each with half that volume. Instead, the process would create two new regions which together would have more volume than you started with. We describe this by saying that the attempt to measure a unit of volume smaller than the minimal size alters the geometry of the space in a way that allows more volume to be created

Volume is not the only quantity which is quantized in loop
quantum gravity. Any region of space is surrounded by a boundary which, being a surface, will have an area, and that area will be measured in square centimetres. In classical geometry a surface can have any area. In contrast, loop quantum gravity predicts that there is a smallest possible area. As with volume, the theory limits the possible areas a surface can have to a finite set of values. In both cases the jumps between possible values are very small, of the order of the square and cube of the Planck length. This is why we have the illusion that space is continuous.

These predictions could be confirmed or refuted by measurements of the geometry of things made on the Planck scale. The problem is that because the Planck scale is so small, it is not easy to make these measurements - but it is not impossible, as I shall describe in due course.

In this chapter and the next I shall tell the story of how loop quantum gravity developed from a few simple ideas into a detailed picture of space and time on the shortest possible scales. The style of these chapters will be rather more narrative than the others, as I can describe from personal experience some of the episodes in the development of the theory. I do this mainly to illustrate the complicated and unexpected ways in which a scientific idea can develop. This can only be communicated by telling stories, but I must emphasize that there are many stories. My guess is that the inventors of string theory have better stories, with more human drama. I must also stress that I do not intend these chapters to be a complete history of loop quantum gravity. I am sure that each of the people who worked on the theory would tell the story in a different way. The story I tell is sketchy and leaves out many episodes and steps in the theory’s development. Worse, it leaves out many of the people who at one time or another have contributed something important to the theory.

The story of loop quantum gravity really begins in the 1950s with an idea that came from what would seem to be a totally different subject - the physics of superconductors. Physics is like this: the few really good ideas are passed around from field to field. The physics of materials such as metals and
superconductors has been a very fertile source of ideas about how physical systems might behave. This is undoubtedly because in these fields there is a close interaction between theory and experiment which makes it possible to discover new ways for physical systems to organize themselves. Elementary particle physicists do not have access to such direct probes of the systems they model, so it has happened that on several occasions we have raided the physics of materials for new ideas.

Superconductivity is a peculiar phase that certain metals can be put into in which their electrical resistance falls to zero. A metal can be turned into a superconductor by cooling it below what is called its critical temperature. This critical temperature is usually very low, just a few degrees above absolute zero. At this temperature the metal undergoes a change of phase something like freezing. Of course, it is already a solid, but something profound happens to its internal structure which liberates the electrons from its atoms, and the electrons can then travel through it with no resistance. Since the early 1990s there has been an intensive quest to find materials that are superconducting at room temperature. If such a material were to be found there would be profound economic implications, as it might greatly reduce the cost of supplying electricity. But the set of ideas I want to discuss go back to the 1950s, when people first understood how simple superconductors work. A seminal step was the invention of a theory by John Bardeen, Leon Cooper and John Schrieffer, known as the BCS theory of superconductivity. Their discovery was so important that it has influenced not only many later developments in the theory of materials, but also developments in elementary particle physics and quantum gravity.

You may remember a simple experiment you did at school with a magnet, a piece of paper and some iron filings. The idea was to visualize the field of the magnet by spreading the filings on a piece of paper placed over the magnet. You would have seen a series of curved lines running from one pole of the magnet to the other (
Figure 19
). As your teacher may have told you, the apparent discreteness of the field lines is an illusion. In nature they are distributed continuously; they only appear to be a discrete set of lines because of the finite size of the iron
filings. However, there is a situation in which the field lines really are discrete. If you pass a magnetic field through a superconductor, the magnetic field breaks up into discrete field lines, each of which carries a fundamental unit of magnetic flux (
Figure 20
). Experiments show that the amount of magnetic flux passing through a superconductor is always an integer multiple of this fundamental unit.

Field lines between two poles of an ordinary magnet, in air.