Matvei

Matvei was a younger friend of Lev Landau the scientist who would go on to become the best theoretical physicist of the Soviet Union. Colleagues who knew them both would claim that, of the two, Matvei was the more brilliant. At the point when Heisenberg and Dirac were constructing the bases of quantum mechanics, Landau, wrongly, thought that fields became ill defined due to quanta: quantum fluctuation would prevent us from measuring the value of a component of a field at a point (an arbitrary small region) in s.p.a.ce. The masterly Bohr immediately saw that Landau was wrong, studied the issue in depth and wrote a long and detailed article to show that fields, such as the electric one, remain well defined even when quantum mechanics is brought to bear.1 Landau dropped the issue.

But his young friend Matvei was intrigued, realizing that Landau's intuition, though imprecise, contained something of importance. He repeated the same reasoning by which Bohr had demonstrated that the quantum electric field was well defined at a point of s.p.a.ce, applying it instead to the gravitational field, for which Einstein had just a few years previously written the equations. And here surprise! Landau was right. The gravitational field at a point is not well defined, when taking quanta into account.

There is an intuitive way of understanding what happens. Suppose we want to observe a very, very, very small region of s.p.a.ce. To do this, we need to place something in this area, to mark the point that we wish to consider. Say we place a particle there. Heisenberg had understood that you can't locate a particle at a point in s.p.a.ce for long. It soon escapes. The smaller the region in which we try to locate a particle, the greater the velocity at which it escapes. (This is Heisenberg's uncertainty principle.) If the particle escapes at great speed, it has a great deal of energy. Now let us take Einstein's theory into account. Energy makes s.p.a.ce curve. A lot of energy means that s.p.a.ce will curve a great deal. A lot of energy in a small region results in curving s.p.a.ce so much that it collapses into a black hole, like a collapsing star. But if a particle plummets into a black hole, I can no longer see it. I can no longer use it as a reference point for a region of s.p.a.ce. I can't manage to measure arbitrarily small regions of s.p.a.ce, because if I try to do this these regions disappear inside a black hole.

This argument can be made more precise with a little mathematics. The result is general: quantum mechanics and general relativity, taken together, imply that there is a limit to the divisibility of s.p.a.ce. Below a certain scale, nothing more is accessible. More precisely, nothing exists there.

How small is this minimal region of s.p.a.ce? The calculation is easy: we need only to calculate the minimum size of a particle before it falls into its own black hole, and the result is straightforward. The minimum length is around: Under the sign of the square root there are the three constants of nature we have already encountered: Newton's constant G, discussed in Chapter 2, which sets the strength of gravity; the speed of light c, introduced in Chapter 3 when discussing relativity, which opens up the extended present; and Planck's constant h, found in Chapter 4, which determines the scale of the quantum granularity.fn29 The presence of these three constants confirms the fact that we are looking at something which has to do with gravity (G), relativity (c) and quantum mechanics (h).

The length LP, determined in this fashion, is called the Planck length. It should be called the Brontejn length, but such is the way of the world. In numerical terms, it is equivalent to approximately one millionth of a billionth of a billionth of a billionth of a centimetre (10-33 centimetres). So, that is to say ... small.

It is at this extremely minute scale that quantum gravity manifests itself. To give an idea of the smallness of the scale we are discussing: if we enlarged a walnut sh.e.l.l until it had become as big as the whole observable universe, we would still not see the Planck length. Even after having been enormously magnified thus, it would still be a million times smaller than the actual walnut sh.e.l.l was before magnification. At this scale, s.p.a.ce and time change their nature. They become something different; they become 'quantum s.p.a.ce and time', and understanding what this means is the problem.

Matvei Brontejn understands all of this in the 1930s and writes two short and illuminating articles in which he points out that quantum mechanics and general relativity, taken together, are incompatible with our customary idea of s.p.a.ce as an infinitely divisible continuum.2 There is, however, a problem. Matvei and Lev are sincere communists. They believe in revolution as the liberation of mankind, the construction of a genuinely better society, without injustice, without the immense inequalities which we still see growing systematically throughout the world. They are enthusiastic followers of Lenin. When Stalin a.s.sumes power, they are both perplexed, then critical, then hostile. They write articles which are mildly but openly critical ... This was not the communism they wanted ...

But these are harsh times. Landau gets through them, not easily, but he survives. Matvei, the year after having been the first to understand that our ideas on s.p.a.ce and time had to change in a radical way, is arrested by Stalin's police and condemned to death. His execution takes place on the same day as his trial, 18 February, 1938.3 He is thirty years old.

John

After Matvei Brontejn's premature death, many of the century's eminent physicists tried to solve the puzzle of quantum gravity. Dirac dedicated the final years of his life to the problem, opening avenues and introducing ideas and techniques on which a good part of current work on quantum gravity is based. It is thanks to these techniques that we know how to describe a world without time, as I will explain further on. Feynman tried, attempting to adapt the techniques he had developed for electrons and photons to the context of general relativity, but without success: electrons and photons are quanta in s.p.a.ce; quantum gravity is something else: it isn't enough to describe 'gravitons' moving in s.p.a.ce, it is s.p.a.ce itself that has to be quantized.

A few n.o.bel Prizes were awarded to physicists who happened to resolve other problems, almost by mistake, during the course of their attempts to disentangle the puzzle of quantum gravity. Two Dutch physicists, Gerard't Hooft and Martinus Veltman, received the n.o.bel Prize in 1999 for having shown the consistency of the theories which today are used to describe nuclear forces a part of the standard model but their research programme was actually aiming to demonstrate the consistency of a theory of quantum gravity. They were working on the theories of these other forces only as a preliminary exercise. The 'preliminary exercise' earned them a n.o.bel Prize, but they did not succeed in showing the consistency of their version of quantum gravity.

The list could go on and would read like a roll of honour of the century's outstanding theoretical physicists. As well as like a catalogue of failures. Very gradually, though, over the course of decades, ideas were clarified and dead ends explored and usefully closed off; techniques and general ideas were strengthened, and results began to build, one developing from another. To mention here the numerous scientists who have contributed to this gradual, slow-moving, collective construction would require a tedious list of names, each one of whom has added a grain or a stone to the process.

I would like to mention just one, who for years held together the threads of this collective research: the remarkable, eternally youthful Englishman half philosopher and half physicist Chris Isham. It was when reading one of his articles reviewing the question of quantum gravity that I first became enamoured with the problem. The article explained just why it was so difficult, how our conception of s.p.a.ce and time needed to be modified, and gave a lucid overview of all the routes which were being followed at the time, with the results achieved, and difficulties entailed. I was in my third year at university, and the possibility of rethinking s.p.a.ce and time from square one fascinated me. This fascination has never diminished. For, as Petrarch sings, 'The wound does not heal due to the weakening of the bow.'

Figure 5.2 John Wheeler.

The scientist who has most contributed to quantum gravity is John Wheeler, a legendary figure who has traversed the physics of the past century. A pupil of and collaborator with Niels Bohr in Copenhagen; a collaborator with Einstein when Einstein moved to the United States; a teacher who can count among his students figures such as Richard Feynman ... Wheeler was at the heart of the physics of the twentieth century. He was gifted with a fervid imagination. It was he who invented and made popular the term 'black hole'. His name is a.s.sociated with the early extended investigations frequently more intuitive than mathematical into how to think about quantum s.p.a.cetime. Having absorbed Brontejn's lesson that quantum properties of the gravitational field imply a modification of the notion of s.p.a.ce at a small scale, Wheeler looked for novel ideas to help conceive of this quantum s.p.a.ce. He imagined it as a cloud of superimposed geometries, just as we can think of a quantum electron as a cloud of positions.

Imagine that you are looking at the sea from a great height: you perceive a vast expanse of it, a flat, cerulean table. Now you descend and look at it more closely. You begin to make out the great waves swollen by the wind. You descend further, and you see that the waves break up and that the surface of the sea is a turbulent frothing. This is what s.p.a.ce is like, as imagined by Wheeler.fn30 On our scale, immensely larger than the Planck length, s.p.a.ce is smooth. If we move down to the Planck scale, it shatters and foams.

Wheeler sought a way to describe this foaming of s.p.a.ce, this wave of probability of different geometries. In 1966, a young colleague of his who lived in Carolina, Bryce DeWitt, provided the key.4 Wheeler travelled frequently, and met collaborators wherever he could. He asks Bryce to meet at Raleigh Durham airport, in North Carolina, where he had a few hours' wait between connecting flights. Bryce arrives and shows him an equation for 'a wave function of s.p.a.ce', obtained by using a simple mathematical trick.fn31 Wheeler is enthused. From this conversation a type of 'equation of orbitals' for general relativity is born; an equation which should determine the probability of one or another curved s.p.a.ces. For a long time, DeWitt called it Wheeler's equationfn32 while Wheeler called it the DeWitt equation. Everyone else calls it the Wheeler-DeWitt equation.

The idea is very good, and becomes a basis for the attempts to construct the full theory of quantum gravity. But the equation itself is riddled with problems serious ones. In the first place, from a mathematical point of view, the equation is really quite badly defined. If we try to use it to do calculations, we soon obtain results that are infinite, which makes no sense. It must be improved.

But it is also difficult to understand how to interpret this equation, to know what it means. Among its disconcerting aspects is the fact that it no longer contains the time variable. How can it be used to compute the evolution of something which happens in time if it does not include a time variable? Dynamical equations, in physics, always contain the variable t, time. What does a physical theory without a temporal variable signify? For years, research will revolve around such questions, trying to revise the equation in different manners, in order to improve its definition and understand what it might mean.

The first steps of the loops

The fog begins to dissipate towards the end of the 1980s. Surprisingly, some solutions of the WheelerDeWitt equation appear. During these years I found myself first at the University of Syracuse, in New York State, visiting the Indian physicist Abhay Ashtekar, and then in Connecticut, at Yale University, visiting the American physicist Lee Smolin. I remember a period of intense discussions and burning intellectual fervour. Ashtekar had rewritten the WheelerDe Witt equation in a simpler form; and Smolin, together with Ted Jacobson of the University of Maryland in Washington, had been the first to find some of these new strange solutions of the equation.

The solutions had a curious peculiarity: they depended on closed lines in s.p.a.ce. A closed line is a 'loop'. Smolin and Jacobson could write a solution to the WheelerDeWitt equation for every loop: for every line closed on itself. What did this mean? The first works of what will later become known as loop quantum gravity emerge from these discussions, as the meaning of these solutions of the WheelerDeWitt equation gradually clarify. Upon these solutions, little by little, a coherent theory begins to be erected, inheriting the name 'loop theory' from the first solutions studied.

Today there are hundreds of scientists working on this theory, spread throughout the world from China to Argentina, from Indonesia to the United States. What is slowly being erected is the theory now known as loop theory, or loop quantum gravity: the theory to which the following chapters are devoted. It is not the only direction explored in the search for a quantum theory of gravity, but it is the one I consider the most promising.fn33

6. Quanta of s.p.a.ce

The last chapter closed with the solutions of the Wheeler DeWitt equation discovered by Jacobson and Smolin. These solutions depend on lines that close on themselves, or loop. What does it all mean?

Remember Faraday's lines the lines which carry the electric force and which, in Faraday's vision, fill s.p.a.ce? The lines from which the concept of 'field' originates? Well, the closed lines that appear in the solutions of the WheelerDeWitt equation are Faraday lines of the gravitational field.

But two new ingredients are now added to Faraday's ideas.

The first is that we are dealing with quantum theory. In quantum theory, everything is discrete. This implies that the infinitely fine, continuous spiderweb of Faraday's lines now becomes similar to a real spiderweb: it has a finite number of distinct threads. Every single line determining a solution of the WheelerDeWitt equation describes one of the threads of this web.

The second new aspect, the crucial one, is that we are speaking of gravity and, therefore, as Einstein understood, we are not speaking of fields immersed in s.p.a.ce but of the very structure of s.p.a.ce itself. Faraday's lines of the quantum gravitational field are the threads of which s.p.a.ce is woven.

At first, the research was focused on these lines and how they could 'weave' our three-dimensional physical s.p.a.ce. Figure 6.1 represents an early attempt to give an intuitive idea of the discrete structure of s.p.a.ce which would result from this.

Soon after, thanks to the intuitions and mathematical ability of young scientists such as the Argentine Jorge Pullin and the Pole Jurek Lewandowski, it became clear that the key to understanding the physics of these solutions lies in the points where these lines intersect. These points are called nodes, and the lines between nodes are called links. A set of intersecting lines forms what is called a graph, that is to say, a combination of nodes connected by links, as in figure 6.3.

A calculation, in fact, demonstrates that, without nodes, physical s.p.a.ce has no volume. In other words, it is in the nodes of the graph, not in the lines, that the volume of s.p.a.ce 'resides'. The lines 'link together' individual volumes sitting at the nodes.

Getting to a full clarification of the resulting picture of quantum s.p.a.cetime took years. It was necessary to transform the ill-defined mathematics of the WheelerDe Witt equation into a structure sufficiently well defined to be able to compute with. With this, it became possible to achieve precise results. The key technical result which clarifies the physical meaning of the graphs is the calculation of the spectra of volume and of area.

Figure 6.1 The quantum version of Faraday's lines of force, which weave s.p.a.ce like a three-dimensional mesh of interlinked rings (loops).

Figure 6.2 The spectrum of the volume: the volumes of a regular tetrahedron that are possible in nature are limited in number. The smallest, at the bottom, is the smallest volume in existence.

Spectra of volume and area

Take any region of s.p.a.ce, for example, the room in which you are reading this, if you are in a room. How big is this room? The size of the s.p.a.ce of the room is measured by its volume. Volume is a geometrical quant.i.ty which depends on the geometry of s.p.a.ce, but the geometry of s.p.a.ce as Einstein understood, and as I recounted in Chapter 3 is the gravitational field. Volume is therefore a property of the gravitational field, expressing how much gravitational field there is between the walls of the room. But the gravitational field is a physical quant.i.ty and, like all physical quant.i.ties, is subject to the laws of quantum mechanics. In particular, like all physical quant.i.ties, volume may not a.s.sume arbitrary values but only certain particular ones, as I described in Chapter 4. The list of all possible values is called, if you remember, the spectrum. Hence there should exist a 'spectrum of the volume' (figure 6.2) Dirac provided us with the formula with which to compute the spectrum of every variable. The calculation took time, first to formulate it and then to complete it, and made us suffer. It was completed in the mid-1990s, and the answer, as expected (Feynman used to say that we should never do a calculation without first knowing the result), is that the spectrum of the volume is discrete. That is, the volume can only be made up of 'discrete packets'. These are somewhat similar to the energy of the electromagnetic field, which is also formed of discrete packets: photons.

The nodes of the graph represent the discrete packets of volume and, as in the case of photons, can only have certain sizes, which can be computed using Dirac's general quantum equation.fn34 Every node n in the graph has its own volume vn: one of the numbers in the spectrum of the volume. The nodes are the elementary quanta of which physical s.p.a.ce is made. Every node of the graph is a 'quantum particle of s.p.a.ce'. The structure that emerges is the one ill.u.s.trated in figure 6.3.

Figure 6.3 On the left, a graph formed by nodes connected by links. On the right, the grains of s.p.a.ce which the graph represents. The links indicate the adjacent particles, separated by surfaces.

A link is an individual quantum of a Faraday line. Now we can understand what it represents: if you imagine two nodes as two small 'regions of s.p.a.ce', these two regions will be separated by a small surface. The size of this surface is its area. The second quant.i.ty, after the volume, which characterizes the quantum webs of s.p.a.ce, is the area a.s.sociated with each line.fn35 The area, just as in the case of the volume, is a physical variable, and has a spectrum which may be calculated using Dirac's equation.* Area is not continuous, it is granular. There is no such thing as an arbitrarily small area.

s.p.a.ce appears continuous to us only because we cannot perceive the extremely small scale of these individual quanta of s.p.a.ce. Just as when we look closely at the cloth of a T-shirt, we see that it is woven from small threads.

When we say that the volume of a room is, for example, 100 cubic metres, we are in effect counting the grains of s.p.a.ce the quanta of the gravitational field which it contains. In a room, this number has more than a hundred digits. When we say that the area of this page is 200 square centimetres, we are actually counting the number of links in the web, or loops, which traverse the page. Across the page of this book, there is a number of quanta with more or less seventy digits.

The idea that measuring length, area and volume is a question of counting individual elements had been proposed in the nineteenth century by Riemann himself. The mathematician who had developed the theory of continuous curved mathematical s.p.a.ces was already aware that a discrete physical s.p.a.ce is, ultimately, more reasonable than a continuous one.

To summarize, the theory of loop quantum gravity, or loop theory, combines general relativity with quantum mechanics in a rather conservative way, because it does not employ any other hypothesis apart from those of the two theories themselves, suitably rewritten to render them compatible. But the consequences are radical.

General relativity taught us that s.p.a.ce is something dynamic, like the electromagnetic field: an immense, mobile mollusc in which we are immersed, which stretches and bends. Quantum mechanics teaches us that every field of this sort is made of quanta, that is to say, it has a fine, granular structure. It follows that physical s.p.a.ce, being a field, is made of quanta as well. The same granular structure characterizing the other quantum fields also characterizes the quantum gravitational field, and therefore s.p.a.ce. We expect s.p.a.ce to be granular. We expect quanta of gravity, just as there are quanta of light, quanta of the electromagnetic field, and as particles are quanta of quantum fields. But s.p.a.ce is the gravitational field, and the quanta of the gravitational field are quanta of s.p.a.ce: the granular const.i.tuents of s.p.a.ce.

The central prediction of loop theory is therefore that s.p.a.ce is not a continuum, it is not divisible ad infinitum, it is formed of 'atoms of s.p.a.ce'. A billion billion times smaller than the smallest of atomic nuclei.

Loop theory describes this atomic and granular quantum structure of s.p.a.ce in a precise mathematical form. It is obtained by applying the general equations of quantum mechanics written by Dirac to Einstein's gravitational field.

In particular, loop theory specifies that volume (for example, the volume of a given cube) cannot be arbitrarily small. A minimum volume exists. No s.p.a.ce smaller than this minimum volume exists. There is a minimum quantum of volume: an elementary atom of s.p.a.ce.

Atoms of s.p.a.ce

Remember Achilles chasing after the tortoise? Zeno observed that there is something difficult to accept in the idea that Achilles has to cover an infinite number of distances before reaching the slow-moving creature. Mathematics had found a possible answer to this difficulty, showing how an infinite number of progressively smaller intervals could nevertheless amount to a finite total interval.

But is this what truly happens in nature? Are there intervals between Achilles and the tortoise that can be arbitrarily short? Does it really make sense to talk of a billionth of a billionth of a billionth of a millimetre, and then to think of dividing it again further innumerable times?

The calculation of the quantum spectra of geometric quant.i.ties indicates that the answer is negative: arbitrarily small chunks of s.p.a.ce do not exist. There is a lower limit to the divisibility of s.p.a.ce. It is at a very small scale indeed, but it is there. This is what Matvei Brontejn had intuited in the 1930s. The calculation of the spectra of volume and area confirms Brontejn's idea and frames it in a mathematically precise manner.

Achilles does not need an infinite number of steps to reach the tortoise because, in a s.p.a.ce made of grains of finite size, infinitely small steps do not exist. The hero will come ever closer to the creature until, in the end, he reaches it in a single quantum leap.

But, on reflection, was this not precisely the solution proposed by Leucippus and by Democritus? They spoke of the granular structure of matter, and we are rather unsure as to what, precisely, they said about s.p.a.ce. Unfortunately, we do not have their texts and must make do with the spa.r.s.e fragments in the citations of others. It is like trying to reconstruct Shakespeare's plays from a list of Shakespeare quotes.fn36 Democritus's argument on the incongruity of the continuum as a collection of points, reported by Aristotle, may be applied to s.p.a.ce. I imagine that if we could ask Democritus if it makes sense to split a s.p.a.ce interval ad infinitum, his reply could only be to repeat that divisibility must have a limit. For the philosopher of Abdera, matter is made of atoms that cannot be divided. Having once understood that s.p.a.ce is very much like matter s.p.a.ce, as he had said himself, has its own nature, 'a certain physics' I suspect he would not have hesitated to deduce that s.p.a.ce, too, can only be made of elementary chunks that cannot be divided. We are perhaps just following in the footsteps of Democritus.

I certainly don't mean to imply that the physics of two millennia was useless, that experiments and mathematics are pointless and that Democritus could be as convincing as modern science. Obviously not. Without experiments and mathematics, we would never have understood what we have understood. Yet we develop our conceptual schema for understanding the world by exploring new ideas but also by building on the powerful intuitions of giant figures from the past. Democritus is one of them, and we discover the new sitting on his t.i.tanic shoulders.

But let us return to quantum gravity.

Spin networks

The graphs which describe the quantum states of s.p.a.ce are characterized by a volume v for every node and a half-integer j for every line. A graph with this additional information is called a spin network (figure 6.4). (Half-integers in physics are called 'spin' because they appear in the quantum mechanics of spinning objects.) A spin network represents a quantum state of the gravitational field: a quantum state of s.p.a.ce; a granular s.p.a.ce in which area and volume are discrete. Fine-mesh grids are used elsewhere in physics to approximate continuous s.p.a.ce. Here, there is no s.p.a.ce continuum to approximate: s.p.a.ce is genuinely granular.

The crucial difference between photons (the quanta of the electromagnetic field) and the nodes of the graph (the quanta of gravity) is that photons exist in s.p.a.ce, whereas the quanta of gravity const.i.tute s.p.a.ce themselves. Photons are characterized by 'where they are'.fn37 Quanta of s.p.a.ce have no place to be in, because they are themselves that place. They have only one piece of information which characterizes them spatially: information about which other quanta of s.p.a.ce they are adjacent to, which one is next to which other. This information is expressed by the links in the graph. Two nodes connected by a link are two nodes in proximity. They are two grains of s.p.a.ce in contact with each other: this 'touching' constructs the structure of s.p.a.ce.

Figure 6.4 A spin network The quanta of gravity, that is, are not in s.p.a.ce, they are themselves s.p.a.ce. The spin networks which describe the quantum structure of the gravitational field are not immersed in s.p.a.ce; they do not inhabit a s.p.a.ce. The location of single quanta of s.p.a.ce is not defined with regard to something else but only by the links and the relation these express.

If I step from grain to grain along the links until I complete a circuit and return to the grain from which I started, I will have made a 'loop'. These are the original loops of the loop theory. In Chapter 4 I showed that the curvature of s.p.a.ce may be measured by looking at whether an arrow transported across a closed circuit returns pointing in the same direction, or turned. The mathematics of the theory determines this curvature for every closed circuit on a spin network, and this makes it possible to evaluate the curvature of s.p.a.cetime, and hence the force of the gravitational field, from the structure of a spin network.fn38 Now, quantum mechanics is more than granularity. There is also the fact that evolution is probabilistic the way in which the spin networks evolve is random. I'll speak about this in the next chapter, devoted to time.

And there is the fact that what matters is not how things are, but rather how they interact. Spin networks are not ent.i.ties; they describe the effect of s.p.a.ce upon things. Just as an electron is in no place but diffused in a cloud of probability in all places, s.p.a.ce is not actually formed by a single specific spin network but rather by a cloud of probabilities over the whole range of all possible spin networks.

At an extremely small scale, s.p.a.ce is a fluctuating swarm of quanta of gravity which act upon each other, and together act upon things, manifesting themselves in these interactions as spin networks, grains interrelated with each other (figure 6.5).

Figure 6.5 At a minute scale, s.p.a.ce is not continuous: it is woven from interconnected finite elements.

Physical s.p.a.ce is the fabric resulting from the ceaseless swarming of this web of relations. The lines themselves are nowhere; they are not in a place but rather create places through their interactions. s.p.a.ce is created by the interaction of individual quanta of gravity.

This is the first step towards understanding quantum gravity. The second concerns time. And, to time, the next chapter is devoted.