* The result of the calculation is simple. I show it here so you can see how Dirac's spectra work. The possible values of the area A are given in the following formula, where j is a 'half-integer', that is to say, a number which is half of an integer, such as 0, , 1, 32, 2, 52, 3 ...

A is the area that a surface separating two grains of s.p.a.ce can have. 8 is the number eight, nothing special about it. is the Greek pi which we studied at school: the constant which gives the relation between the circ.u.mference and the diameter of any circle, and which appears everywhere in physics, I don't know why. Lp is the Planck length, the extremely small scale at which the phenomena of quantum gravity take place. L2p is the square of Lp, which is the (extremely small) area of a tiny square with sides equal to the Planck length. Therefore 8L2p is simply a 'small' area: the area of a minuscule square with a side which is about a millionth of a billionth of a billionth of a billionth of a centimetre (10-66 cm). The interesting aspect of the formula is the square root and what is within it. The key point is that j is a half-integer, that is to say, it may have only values which are multiples of . For each one of these, the root has a certain value, listed approximately in table 6.1.

Table 6.1 Spin (half-integers) and corresponding value of the area in units of minimal area.

Multiplying the numbers in the right-hand column by the area 8L2p, we obtain the possible values of the area of the surface. These special values are like the ones which appear in the study of the orbits of electrons in atoms, where quantum mechanics allows only certain orbits. The point is that no other areas apart from the values derived by this equation exist. No surface can have an area one tenth of 8L2p.

7. Time Does Not Exist

It must not be claimed that anyone can sense time by itself apart from the movement of things.

Lucretius, De rerum natura1 The alert reader will have realized that in the preceding chapter little attention was given to time. And yet Einstein showed, over a century ago, that we cannot separate time and s.p.a.ce, that we must think of them together as a single whole: s.p.a.cetime. The moment has come to rectify this and bring time back into the picture.

Research on quantum gravity has revolved for years around spatial equations, before having the courage to confront time. In the last fifteen years, a way of thinking about time has begun to emerge. I'll try to explain it.

s.p.a.ce as an amorphous container of things disappears from physics with quantum gravity. Things (the quanta) do not inhabit s.p.a.ce, they dwell one over the other, and s.p.a.ce is the fabric of their neighbouring relations. As we abandon the idea of s.p.a.ce as an inert container, similarly, we must abandon the idea of time as an inert flow along which reality unfurls. Just as the idea of the s.p.a.ce continuum containing things disappears, so, too, does the idea of a flowing continuum 'time' during the course of which phenomena happen.

In a certain sense, s.p.a.ce no longer exists in fundamental theory; the quanta of the gravitational field are not in s.p.a.ce. In the same sense, time no longer exists in the fundamental theory: the quanta of gravity do not evolve in time. Time just counts their interactions. As evidenced with the WheelerDe Witt equation, the fundamental equations no longer contain the time variable. Time emerges, like s.p.a.ce, from the quantum gravitational field.

This was already partially true for cla.s.sical general relativity, where time already appears as an aspect of the gravitational field. But as long as we neglect quantum theory, we can still think of s.p.a.cetime in a rather conventional manner, like the tapestry in which the story of the rest of reality unfolds, even if it is a dynamical, moving tapestry. The moment we take quantum mechanics into account, we recognize that time, too, must have those aspects of probabilistic indeterminacy, granularity and relationality which are common to all of reality. It becomes a 'time' markedly different from all that we have hitherto meant by the word.

This second conceptual consequence of the theory of quantum gravity is more extreme even than the vanishing of s.p.a.ce.

Let's attempt to understand it.

Time is not what we think it is

That the nature of time is different from the common idea which we have of it was already clear over a century ago. Special and general relativity made this explicit. Today, the inadequacy of our common-sense view of time can be easily verified in a laboratory.

Let's reconsider, for example, the first consequence of general relativity, as ill.u.s.trated in Chapter 3. Take two watches, ensure that they mark exactly the same time, place one on the floor and the other on a piece of furniture. Wait for about half an hour and then bring them back next to each other. Will they still tell the same time?

As described in Chapter 3, the answer is no. The watches which we usually wear on our wrists, or have on our mobile phones, are not sufficiently precise to allow us to verify this fact, but in physics laboratories all over the world there are timepieces precise enough to demonstrate the discrepancy which occurs: the watch left on the floor is slow when compared to the one which has been raised above it.

Why? Because time does not pa.s.s in the same way everywhere in the world. In some places, it flows more quickly; in others, more slowly. The closer you get to the Earth, where gravityfn39 is more intense, the slower time pa.s.ses. Remember the twins in Chapter 3, who ended up with different ages as a result of having lived one at sea level and one in the mountains? The effect is very slight: the time gained during a life spent by the sea, with respect to one pa.s.sed in the mountains, consists of fractions of a second but the smallness of the amount does not alter the fact that there is a real difference. Time does not work as we customarily imagine it does.

We must not think of time as if there were a great cosmic clock that marks the life of the universe. We have known for more than a century that we must think of time instead as a localized phenomenon: every object in the universe has its own time running, at a pace determined by the local gravitational field.

But even this notion of a localized time no longer works when we take the quantum nature of the gravitational field into account. Quantum events are no longer ordered by the pa.s.sage of time at the Planck scale. Time, in a sense, ceases to exist.

What does it mean to say that time does not exist?

First, the absence of the variable time from the fundamental equations does not imply that everything is immobile and that change does not happen. On the contrary, it means that change is ubiquitous. Only: elementary processes cannot be ordered along a common succession of instants. At the extremely small scale of the quanta of s.p.a.ce, the dance of nature does not develop to the rhythm kept by the baton of a single orchestral conductor: every process dances independently with its neighbours, following its own rhythm. The pa.s.sing of time is intrinsic to the world, it is born of the world itself, out of the relations between quantum events which are the world and which themselves generate their own time.

In fact, the nonexistence of time does not mean anything particularly complicated. Let's try to understand.

The candle chandelier and the pulse

Time appears in most equations of cla.s.sic physics. It is the variable indicated by the letter t. The equations tell us how things change in time. If we know what has happened in the past, they allow us to predict the future. More precisely, we measure some variables for example, the position A of an object, the angle B of a swinging pendulum, the temperature C of an object and the equations of physics tell us how these variables A, B and C will change with time. They predict the functions A(t), B(t), C(t), and so on, which describe the changing of these variables in time t.

Galileo was the first to understand that the movement of objects on Earth could be described by equations for the functions of time A(t), B(t), C(t) and the first to write explicit equations for these functions. The first law of terrestrial physics found by Galileo, for example, describes how an object falls, that is to say, how its alt.i.tude x varies with the pa.s.sage of time t.fn40 To discover and verify this law, Galileo needed two kinds of measurements. He had to measure the height x of the object and the time t. Therefore, he needed, in particular, an instrument to measure time. He needed a clock.

When Galileo lived there were no accurate clocks. Galileo himself, as a young man, discovered a key to making precise timepieces. He discovered that the oscillations of a pendulum all have the same duration (irrespective of the amplitude). Thus, it is possible to measure time by simply counting the oscillations of a pendulum. It seems such an obvious idea, but it took Galileo to find it; it had not occurred to anyone before him. So it goes, with science.

But things are not really this straightforward.

According to legend, Galileo alighted on the idea in Pisa's marvellous cathedral while watching the slow oscillations of a gigantic candle chandelier, which is still there. (The legend is false, since the chandelier was actually first hung there years after Galileo's death, but it makes for a good story. Perhaps there was another one hanging there at the time.) The scientist was observing the oscillations during a religious service in which he was evidently not particularly absorbed, and he was measuring the duration of each oscillation of the chandelier by counting the beats of his own pulse. With mounting excitement, he discovered that the number of beats was the same for each oscillation: it did not change when the chandelier slowed and oscillated with diminished amplitude. The oscillations all had the same duration.

It's a fine story but, on reflection, it leaves us perplexed and this perplexity goes to the heart of the problem of time. How could Galileo know that his own individual pulse-beats all lasted for the same amount of time?fn41 Not many years after Galileo, doctors began to measure their patients' pulses by using a watch which is nothing, after all, but a pendulum. So we use the beats to a.s.sure ourselves that the pendulum is regular, and then the pendulum to ascertain the regularity of the pulse-beats. Is this not somewhat circular? What does it mean?

It means that we, in reality, never measure time itself; we always measure the physical variables A, B, C ... (oscillations, beats, and many other things) and compare one variable with another, that is to say, we measure the functions A(B), B(C), C(A), and so on. We can count how many beats for each oscillation; how many oscillations for every tick of my stopwatch; how many ticks of my stopwatch between intervals of the clock on the bell-tower ...

The point is that it is useful to imagine that a variable t exists the 'true time' which underpins all those movements, even if we cannot measure it directly. We write the equations for the physical variables with regard to this un.o.bservable t, equations which tell us how things change in t; that is, for instance, how much time it takes for each oscillation, and how long each heart-beat lasts. From this, we can derive how the variables change in relation to each other how many heartbeats there are in one oscillation and compare this prediction with what we observe in the world. If the predictions are correct, we trust that this complicated schema is a sound one and, in particular, that it is useful to employ the variable of time t, even if we cannot measure it directly.

In other words, the existence of the variable time is a useful a.s.sumption, not the result of an observation.

It was Newton who understood all of this: he understood that this was a good way to proceed, and clarified and developed this schema. Newton a.s.serts explicitly in his book that we can't ever measure the true time t but, if we a.s.sume that it exists, we can set up an efficient framework to describe nature.

Having clarified this, we can return to quantum gravity and the meaning of the statement that 'time does not exist'. It simply means that the Newtonian schema no longer works when we are dealing with small things. It was a good one, but only for large things.

If we want to understand the world widely, if we want to understand how it functions in the less familiar situations where quantum gravity matters, we need to abandon this schema. The idea of a time t which flows by itself, and in relation to which all things evolve, is no longer a useful one. The world is not described by equations of evolution in time t. What we must do is simply to enumerate the variables A, B, C ... which we actually observe, and write equations expressing relations between these variables, and nothing else: that is, equations for the relations A(B), B(C), C(A) ... which we observe, and not for the functions A(t), B(t), C(t) ... which we do not observe.

In the example of the pulse and the candle chandelier, we will not have the pulse and the candelabrum evolving in time, but only equations which tell us how the two variables evolve with respect to each other. That is to say, equations which tell us directly how many pulse-beats there are in an oscillation, without mentioning t.

'Physics without time' is physics in which we speak only of the pulse and the chandelier, without mentioning time.

It's a simple change but from a conceptual point of view, it's a huge leap. We must learn to think of the world not as something which changes in time but in some other way. Things change only in relation to one another. At a fundamental level, there is no time. Our sense of the common pa.s.sage of time is only an approximation which is valid for our macroscopic scale. It derives from the fact that we perceive the world in a coa.r.s.e-grained fashion.

The world described by the theory is thus far from the one we are familiar with. There is no longer s.p.a.ce which contains the world, and no longer time during the course of which events occur. There are elementary processes in which the quanta of s.p.a.ce and matter continuously interact with each other. Just as a calm and clear Alpine lake is made up of a rapid dance of a myriad of minuscule water molecules, the illusion of being surrounded by continuous s.p.a.ce and time is the product of a long-sighted vision of a dense swarming of elementary processes.

s.p.a.cetime sushi

How do these general ideas apply to quantum gravity? How can we describe change without the ideas of s.p.a.ce as a container, or time along which the world glides?

Consider a process: for example, the collision of two billiard b.a.l.l.s on a table's green baize. Imagine a red ball played in the direction of a yellow one; it gets close, collides, and the two b.a.l.l.s move away in different directions. This process, like all processes, takes place in a finite zone of s.p.a.ce let's say on a table approximately two metres wide and lasts for a finite interval of time let's say three seconds. To deal with this process in the context of quantum gravity, it is necessary to include s.p.a.ce and time in the process itself (figure 7.1).

We must not, in other words, describe only the two b.a.l.l.s, but also all that is around them: the table and any other material objects and the s.p.a.ce in which they are immersed during the time that elapses between the start of the shot and the end of the process. s.p.a.ce and time are the gravitational field, Einstein's 'mollusc': we are also including the gravitational field, that is to say, a piece of the mollusc, in the process. Everything is immersed in Einstein's great mollusc: here, imagine that you are slicing a small, finite portion of it, like a piece of sushi, which encompa.s.ses the collision and what surrounds it.

Figure 7.1 A region of s.p.a.ce in which a black ball hits a stationary white ball, propels it and rebounds. The box is the region of s.p.a.cetime. Within it are drawn the trajectories of the b.a.l.l.s.

What we obtain from this is a s.p.a.cetime box (as in figure 7.1): a finite portion of s.p.a.cetime a few cubic metres in dimension by a few seconds of time. This process does not occur 'in' time. The box is not in s.p.a.cetime, it includes s.p.a.cetime. It isn't a process in time, in the same way in which grains of s.p.a.ce are not in s.p.a.ce. The pa.s.sage of time is only the measure of the process itself, just as quanta of gravity are not in s.p.a.ce, as they themselves const.i.tute s.p.a.ce.

The key to understanding how quantum gravity works lies in considering not solely the physical process given by the two b.a.l.l.s but rather the entire process defined by the whole box with all that it entails, including the gravitational field.

Now let us return to Heisenberg's original insight: quantum mechanics does not tell us what happens during the course of a process, but the probability which ties together the different initial and final states of the process. In our case, the initial and final states are given by all that happens at the border of the s.p.a.cetime box.

What the equations of loop quantum gravity give us is the probability a.s.sociated with a given possible boundary of the box the probability that the b.a.l.l.s will come out of the box in one particular configuration or another, if they have entered it in another.

How is this probability computed? Recall Feynman's sum over paths, which I described when speaking about quantum mechanics. Probabilities, in quantum gravity, can be calculated in the same way. By considering all the possible 'trajectories' that have the same boundary. Since we are including the dynamics of s.p.a.cetime, this means considering all possible s.p.a.cetimes which have the same boundary as the box.

Quantum mechanics a.s.sumes that between the initial boundary, where the two b.a.l.l.s enter, and the final boundary where they exit, there is no definite s.p.a.cetime nor definite trajectory of the b.a.l.l.s. There is a quantum 'cloud' in which all the possible s.p.a.cetimes and all possible trajectories exist together. The probability of seeing the b.a.l.l.s going out in one way or another can be computed by summing over all possible s.p.a.cetimes.

Spinfoam

If quantum s.p.a.ce has the structure of a spin network, what structure will s.p.a.cetime have? What will one of the s.p.a.cetimes previously alluded to in the calculation be like?

It must be a 'history' of a spin network. Imagine that you take the graph of the spin network and move it: every node in the web draws a line, like the b.a.l.l.s in figure 7.1, and every line of the graph, moving, draws a surface (for example, a moving segment draws a rectangle). But there is more: a node can open up into two or more nodes, just as a particle can split into two or more particles. Conversely, two or more nodes can combine into a single one. In this way, a graph which evolves draws an image like the one in figure 7.2.

Figure 7.2 An evolving spin-network: three nodes combine into a single node, and then separate again. On the right, the spinfoam representing this process.

The image portrayed on the right of figure 7.2 is a 'spinfoam'. 'Foam' because it is made of surfaces which meet on lines, which in turn meet on vertices, resembling a foam of soap bubbles (figure 7.3). 'Spinfoam' because the faces of the foam carry spins, as do the links of the graphs whose evolution they describe.

Figure 7.3 The foam of soap bubbles.

To compute the probability of a process, one must sum up over all the possible spinfoams within the box which have the same boundary as that process. The boundary of a spinfoam is a spin network and the matter on it.

The equations of loop quantum gravity express the probability of a process in terms of sums over spinfoams with given boundaries. In this way it is possible to compute, in principle, the probability of any physical event.fn42 Figure 7.4 A vertex of spinfoam. Courtesy of Greg Egan.

At first sight, this way for making calculations in quantum gravity, based on spinfoams, seems very different from the usual ways in which things are computed in theoretical physics. There is no given s.p.a.ce, no given time, and spinfoams seem objects quite remote from, say, the particles of the standard model. But in fact there are strong similarities between the spinfoam technique and the calculation techniques used in the standard model. In fact, even more than this, the spinfoam technique is actually a beautiful merging of the two main calculation techniques used in the context of the standard model: Feynman diagrams and the lattice approximation.

Feynman diagrams are used, for instance, to compute processes dominated by electromagnetic or weak forces. A Feynman diagram represents a sequence of elementary interactions among particles. An example is in figure 7.5, which represents two particles, or two quanta of the field, interacting. The particle on the left splits into two particles, one of which splits in turn into two particles, which then reunite, converging with the particle on the right. The graph portrays a history of the field's quanta.

Figure 7.5 A Feynman diagram.

The lattice approximation is used when the forces are strong and the particle picture is no longer effective for describing physics, for instance in computing the strong forces between quarks inside the nucleus of an atom. The lattice technique entails approximating a continuous physical s.p.a.ce by means of a lattice, or a grid, as in figure 7.6. This grid is not a.s.sumed to be a faithful description of s.p.a.ce, but only an approximation, as when engineers calculate the resistance of a bridge by approximating the concrete with a finite number of elements. These two methods of making calculations Feynman diagrams and the lattice are the two most efficient techniques of quantum field theory.

Figure 7.6 A grid approximating physical s.p.a.cetime.

In quantum gravity, something beautiful occurs: the two methods of making calculations become one and the same. The s.p.a.cetime foam represented in figure 7.2, used to compute a physical process in quantum gravity, may be interpreted either as a Feynman diagram or as a lattice calculation.fn43 Therefore, the two calculation techniques used for the standard model turn out to be particular cases of a common technique: summing over the spinfoams of quantum gravity.

Earlier, I set out Einstein's equations. Again, I can't resist including here the complete collection of the equations of loop theory, even if the reader will obviously not be able to decipher them not before undertaking the study of a good deal of mathematics. Someone once claimed that a theory isn't credible if its equations cannot be summarized on a T-shirt. Here is that T-shirt for loop quantum gravity (figure 7.7).

Figure 7.7 The equations of loop quantum gravity, summarized on a T-shirt.

These equationsfn44 are the mathematical version of the picture of the world I have given in the last two chapters. We are not at all sure if they are the correct equations but, in my opinion, they are the best account of quantum gravity we have at present.

s.p.a.ce is a spin network whose nodes represent its elementary grains, and whose links describe their proximity relations. s.p.a.cetime is generated by processes in which these spin networks transform into one another, and these processes are described by sums over spinfoams. A spinfoam represents a history of a spin network, hence a granular s.p.a.cetime where the nodes of the graph combine and separate.

This microscopic swarming of quanta, which generates s.p.a.ce and time, underlies the calm appearance of the macroscopic reality surrounding us. Every cubic centimetre of s.p.a.ce, and every second that pa.s.ses, is the result of this dancing foam of extremely small quanta.

What is the world made of?

The backdrop of s.p.a.ce has disappeared, time has disappeared, cla.s.sic particles have disappeared, along with the cla.s.sic fields. So what is the world made of?

The answer now is simple: the particles are quanta of quantum fields; light is formed by quanta of a field; s.p.a.ce is nothing more than a field, which is also made of quanta; and time emerges from the processes of this same field. In other words, the world is made entirely from quantum fields (figure 7.8).

These fields do not live in s.p.a.cetime; they live, so to speak, one on top of the other: fields on fields. The s.p.a.ce and time that we perceive in large scale are our blurred and approximate image of one of these quantum fields: the gravitational field.

Fields that live on themselves, without the need of a s.p.a.cetime to serve as a substratum, as a support, and which are capable by themselves of generating s.p.a.cetime, are called 'covariant quantum fields'. The substance of which the world is made has been radically simplified in recent years. The world, particles, light, energy, s.p.a.ce and time all of this is nothing but the manifestation of a single type of ent.i.ty: covariant quantum fields.

Figure 7.8 What is the world made of? Of only one ingredient: covariant quantum fields.