Covariant quantum fields have become today the best description that we have of the , the apeiron, the primal substance of which everything is formed hypothesized by the man that could perhaps be called the first scientist and the first philosopher, Anaximander.fn45 The separation between the curved and continuous s.p.a.ce of Einstein's general relativity and the discrete quanta of quantum mechanics which dwell in a flat and uniform s.p.a.ce has dissolved. The apparent contradiction is no longer there. Between the s.p.a.cetime continuum and quanta of s.p.a.ce, there is the same relationship as between electromagnetic waves and photons. The waves give an approximate large-scale vision of photons. Photons are the way in which waves interact. Continuous s.p.a.ce and time are an approximate large-scale vision of the dynamic of quanta of gravity. The quanta of gravity are the way in which s.p.a.ce and time interact. The same mathematics coherently describes the quantum gravitational field as other quantum fields.

The conceptual price paid is the relinquishing of the idea of s.p.a.ce, and of time, as general structures within which to frame the world. s.p.a.ce and time are approximations which emerge at a large scale. Kant was perhaps right when he affirmed that the subject of knowledge and its object are inseparable, but he was definitely mistaken when he considered Newtonian s.p.a.ce and time as a priori forms of knowledge, parts of an indispensable grammar for understanding the world. This grammar has evolved, and is still in the process of evolving, with the growth of our knowledge.

General relativity and quantum mechanics are, in the end, not as incompatible as they seemed. On closer inspection, they shake hands and engage in a beautiful dialogue. The spatial relations that weave Einstein's curved s.p.a.ce are the very interactions weaving the relations between the systems of quantum mechanics. The two become compatible and conjoined, two sides of the same coin, as soon as it is recognized that s.p.a.ce and time are aspects of a quantum field, and quantum fields can exist even without being grounded in an external s.p.a.ce.

This rarefied picture of the fundamental structure of the physical world is the vision of reality offered today by quantum gravity.

The main reward of this kind of physics is that, as we shall see in the next chapter, infinity disappears. The infinitely small no longer exists. The infinities which plague conventional quantum field theory, predicated on the notion of a continuous s.p.a.ce, now vanish, because they were generated precisely by the a.s.sumption, physically incorrect, of the continuity of s.p.a.ce. The singularities which render Einstein's equations absurd when the gravitational field becomes too strong also disappear: they are only the result of neglecting the quantization of the field. Little by little, the pieces of the puzzle find their place. In the final sections of this book, I describe some of the physical consequences of this theory.

It may appear strange and difficult to think of discrete elementary ent.i.ties not in s.p.a.ce and time, but weaving s.p.a.ce and time with their relations. But how strange it must have seemed to listen to Anaximander, when he claimed that beneath our feet there was only the same sky that we can see above our heads? Or to Aristarchus, when he tried to measure the distance from the Earth to the Moon and the Sun, discovering that they are extremely distant, and are therefore not the size of little b.a.l.l.s, but gigantic and the Sun is immense compared to the Earth. Or to Hubble, when he realized that the small, diaphanous clouds between stars are vast seas of immensely distant stars ...

For centuries, the world has continued to change and expand around us. We see further, understand it better and are astonished by its variety, by the limitations of the images we had of it. The description we manage to produce to account for it becomes increasingly rarefied, yet simple.

We are akin to small, blind moles underground who know little or nothing about the world. But we continue to learn ...

But all the story of the night told over And all their minds transfigured so together More witnesseth than fancy's images And grows to something of great constancy; But, howsoever, strange and admirable.2

Part Four.

BEYOND s.p.a.cE AND TIME.

I have ill.u.s.trated the basis of quantum gravity, and the image of the world which emerges from it. In the final chapters I describe some consequences of the theory: what the theory tells us about phenomena such as the Big Bang and black holes. I also discuss the current state of possible experiments to test the theory, and what it seems to me nature is telling us in particular with the failure of the expected observation of supersymmetric particles.

I conclude with a few reflections on what is still missing from our understanding of the world: especially thermodynamics, the role of information in a theory without time and s.p.a.ce such as quantum gravity, and the re-emergence of time.

All of this takes us to the edge of what we know, to the vantage point from which we look upon what we definitely don't know, the immense mystery that surrounds us.

8. Beyond the Big Bang.

The master.

In 1927 a young Belgian scientist, a Jesuit-educated Catholic priest, studies Einstein's equations and realizes just as Einstein had that they predict the universe must expand or contract. But instead of foolishly rejecting the result and stubbornly trying to avoid it, as Einstein did, the Belgian priest believes it and looks for astronomical data to test it.

At the time, galaxies were not called galaxies. They were called nebulae because, seen through a telescope, they looked like small, opalescent clouds among the stars. It was not yet known that they are distant, immense islands of stars like our very own galaxy. But the young Belgian priest understands that the scarce available data on the galaxies were indeed compatible with the possibility that the universe is expanding: nearby galaxies are moving away at great speed, as if they had been launched into the sky; distant galaxies are moving away at even greater speed. The universe is swelling like a balloon.

Two years later, the insight is confirmed, thanks to two American astronomers, Henrietta Leavitt and Edwin Hubble. Leavitt discovers a good technique for measuring the distance of the nebulae, confirming that they are very far away, outside of our own galaxy. Using this technique and the great telescope of the Palomar Observatory, Hubble collects precise data that confirm that the galaxies are moving away, at a speed proportional to their distance.

Figure 8.1 Henrietta Leavitt.

But it is the young Belgian priest who understands, already in 1927, the crucial consequence: if we see a stone flying up, it means that the stone was previously lower down and something has thrown it upwards. If we see the galaxies moving away and the universe expanding, it means that the galaxies were previously much closer and the universe was smaller: and something caused it to start expanding. The young Belgian priest suggests that the universe was originally extremely small and compressed, and started its expansion in a gigantic explosion. He calls this initial state the primordial atom. Today it is known as the Big Bang.

His name was Georges Lematre. In French, this name sounds like le matre meaning 'the master', and few names are more appropriate for the man who first understood the existence of the Big Bang. But in spite of this name, Lematre's character was reserved; he avoided polemics, and never even claimed priority for the discovery of the expansion of the universe, which ended up being attributed to Hubble. Two episodes from his life ill.u.s.trate his profound intelligence. The first involves Einstein, the second a pope.

As mentioned, Einstein was sceptical about the expansion of the universe. He had grown up thinking that the universe is fixed, and had not been able to accept the idea that this was not the case. Even the greatest make mistakes and are prey to preconceived ideas. Lematre met Einstein and tried to dissuade him from his prejudicial view. Einstein resisted, going so far as to answer Lematre: 'Correct calculations, abominable physics.' Later, Einstein was obliged to recognize that Lematre was the one who was actually right. It doesn't fall to everyone to disprove Einstein.

Figure 8.2 Georges Lematre. Copyright Archives Georges Lematre, Louvain.

The same thing happened again. Einstein had introduced the cosmological constant, the small but important modification of his equations I described in Chapter 3, in the (mistaken) hope of rendering the equations compatible with a static universe. When he had to acknowledge that the universe is not static, he turned against the cosmological constant. Lematre, for the second time, tried to persuade him to change his mind: the cosmological constant does not render the universe static, but it is nevertheless right, and there is no reason to take it out. On this occasion, too, Lematre was correct: the cosmological constant produces an acceleration of the expansion of the universe, and this acceleration has recently been measured. Once again, Einstein was wrong and Lematre was right.

When the idea that the universe had emerged from a Big Bang began to be accepted, Pope Pius XII declared in a public address (on 22 November 1951) that the theory confirmed the account of Creation given in Genesis.1 Lematre reacted to this papal position with great concern. He got in touch with the scientific advisor to the pontiff and went to great lengths to persuade the Pope to refrain from making references to links between divine creation and the Big Bang. Lematre was convinced that it was foolish to mix science and religion in this way: the Bible knows nothing about physics, and physics knows nothing about G.o.d.2 Pius XII allowed himself to be persuaded, and the Catholic church never again made public allusion to the subject. It is not given to everyone to disprove the Pope.

And of course, on this also, it was Lematre who was right: today there is a great deal of talk concerning the possibility that the Big Bang is not a real beginning, that there could have been another universe before it. Imagine in what an embarra.s.sing position the Catholic Church would find itself today, if Lematre had not prevented the Pope from making it official doctrine that that Big Bang and Creation were the same thing. Fiat lux would have to be changed to 'Switch the light back on!'

To contest both Einstein and the Pope, convincing both that they were mistaken, and to be right in both cases, is surely something of a result. 'The master' lived up to his name.

Today confirmations are overwhelming: the universe, in a far-distant past, was extremely hot and extremely compact, and has expanded since. We can reconstruct in detail the history of the universe, starting with its initial hot, compressed state. We know how atoms, elements, galaxies and stars formed and how the universe as we see it today developed. Recent extended observations of the radiation that fills the universe carried out mainly by the Planck satellite once again confirmed in full the theory of the Big Bang. We know with a reasonable degree of certainty what happened on a large scale to our universe in the last 14 billion years, from the time when it was a ball of fire.

And to think that, initially, the phrase 'theory of the Big Bang' was coined by opponents of the theory, to mock an idea that seemed outlandish ... Instead, in the end, we were all persuaded: 14 billion years ago the universe was a compressed ball of fire.

But what happened before this initial hot and compressed state?

Regressing in time, temperature increases, as does the density of matter and energy. There is a point at which they reach the Planck scale: 14 billion years ago. At that point, the equations of general relativity are no longer valid, because it is no longer possible to ignore quantum mechanics. We enter into the realm of quantum gravity.

Quantum cosmology

To understand what happened 14 billion years ago, therefore, quantum gravity is required. What do the loops tell us about the subject?

Consider an a.n.a.logous but simpler situation. According to cla.s.sical mechanics, an electron falling straight into an atomic nucleus would be swallowed by the nucleus and disappear. But this is not what happens in reality. Cla.s.sic mechanics is incomplete, and it is necessary to take quantum effects into account. A real electron is a quantum object and does not follow a precise trajectory: it isn't possible to keep it inside too small a region. The more it is concentrated, the more it slips rapidly away. If we want to stop it around the nucleus, the most we can do is to force it into an orbit of the size of the smallest atomic orbital: it could not stay any closer to the nucleus. Quantum mechanics prevents a real electron from falling into a nucleus. A quantum repulsion pushes away the electron when it gets too close to the centre. Thus, thanks to quantum mechanics, matter is stable. Without it, electrons would fall into nuclei, there would be no atoms and we would not exist.

The same applies to the universe. Let us imagine a universe contracting and becoming extremely small, squashed by its own weight. According to Einstein's equations, this universe would be squashed ad infinitum and at a certain point would disappear altogether, like the electron falling into the nucleus. This is the Big Bang predicted by Einstein's equations, if we ignore quantum theory.

But if we take quantum mechanics into account, the universe cannot be indefinitely squashed. A quantum repulsion makes it rebound. A contracting universe does not collapse down to a point: it bounces back and begins to expand, as if it were emerging from a cosmic explosion (figure 8.3).

Figure 8.3 The Big Bounce of the universe in a graphical representation by Francesca Vidotto, the Italian scientist who first used spinfoams to compute the probability of this process.

The past of our universe may therefore well be the result of just such a rebound. A gigantic rebound known as a Big Bounce instead of Big Bang. This is what seems to emerge from the equations of loop quantum gravity when they are applied to the expansion of the universe.

The image of the bounce must not be taken literally. Going back to the example of the electron, recall that if we want to place an electron as close as possible to an atom, the electron is no longer a particle; we can think of it, instead, as opened up in a cloud of probabilities. An exact position no longer makes sense for the electron. The same for the universe: in the crucial pa.s.sage through the Big Bounce, we can no longer think of a single, although granular, s.p.a.ce and time, but only of a spread-out cloud of probabilities in which time and s.p.a.ce wildly fluctuate. At the Big Bounce, the world is dissolved into a swarming cloud of probabilities, which the equations still manage to describe.

Our universe could thus be the result of the collapse of a previous contracting universe pa.s.sing across a quantum phase, where s.p.a.ce and time are dissolved into probabilities.

The word 'universe' becomes ambiguous. If, by 'universe', we mean 'all that there is', then, by definition, there cannot be a second universe. But the word 'universe' has a.s.sumed another meaning in cosmology: it refers to the s.p.a.cetime continuum that we see directly around us, filled with galaxies the geometry and history of which we observe. There is no reason to be certain that, in this sense, this universe is the only one in existence. We can reconstruct the past up to the time when, as in the image by John Wheeler, the spatiotemporal continuum breaks up like sea foam and fragments into a quantum cloud of probabilities, and there is no reason to discard the possibility that beyond this hot foam there could not be another spatiotemporal continuum, similar to the one which we perceive around us.

The probability for a universe to cross the phase of the Big Bounce, pa.s.sing from contraction to expansion, can be computed using the techniques described in the preceding chapter: the s.p.a.cetime boxes. Calculations are made using spinfoams that connect the contracting universe with the expanding one.

All of this is still at an exploratory stage, but what is remarkable in this story is that today we have equations with which to try to describe these events. We are beginning to cast the first few cautious glances, for the moment only theoretically, beyond the Big Bang.

9. Empirical Confirmations?

The appeal of quantum cosmology goes beyond the fascinating theoretical explorations of what there might be beyond the Big Bang. There is another reason for studying the application of the theory to cosmology: it might provide the opportunity to find out whether or not the theory is actually correct.

Science works because, after hypotheses and reasoning, after intuitions and visions, after equations and calculations, we can check whether we have done well or not: the theory gives predictions about things we have not yet observed, and we can check whether these are correct, or not. This is the power of science, that which grounds its reliability and allows us to trust in it with confidence: we can check whether a theory is right or wrong. This is what distinguishes science from other kinds of thinking, where deciding who is right and who is wrong is usually a much thornier question, sometimes even devoid of meaning.

When Lematre defends the idea that the universe is expanding, and Einstein does not believe it, one of the two is wrong; the other right. All of Einstein's results, his fame, his influence on the scientific world, his immense authority, count for nothing. The observations prove him wrong, and it's game over. An obscure Belgian priest is right. It is for this reason that scientific thinking has power.

The sociology of science has shed light on the complexity of the process of scientific understanding; like any other human endeavour, this process is beset by irrationality, intersects with the game of power and is affected by every sort of social and cultural influence. Nevertheless, despite all of this, and in opposition to the exaggerations of a few postmodernists, cultural relativists and the like, none of this diminishes the practical and theoretical efficacy of scientific thinking. Because in the end, in the majority of cases, it is possible to establish with clarity who is right and who is wrong. And even the great Einstein could go on to say (and he did so), 'Ah ... I made a mistake!' Science is the best strategy if we value reliability.

This does not mean that science is just the art of making measurable predictions. Some philosophers of science overly circ.u.mscribe science by limiting it to its numerical predictions. They miss the point, because they confuse the instruments with the objectives. Verifiable quant.i.tative predictions are instruments to validate hypotheses. The objective of scientific research is not just to arrive at predictions: it is to understand how the world functions; to construct and develop an image of the world, a conceptual structure to enable us to think about it. Before being technical, science is visionary.

The verifiable predictions are the sharpened tool which allows us to find out when we have misunderstood something. A theory lacking empirical confirmation is a theory which has not yet pa.s.sed its exams. Exams never end, and a theory is not completely confirmed by one, two or three experiments. But it progressively acquires credibility, stage by stage, as its predictions are revealed to be correct. Theories such as general relativity and quantum mechanics, which initially left many perplexed, earned their credibility gradually, as all of their predictions even the most bizarre were gradually confirmed by experiments and observations.

The importance of experimental proof, on the other hand, does not mean that, without new experimental data, we cannot make advances. It is often said that science takes steps forward only when there is new experimental data. If this were true, we would have little hope of finding the theory of quantum gravity before measuring something new, but this is patently not the case. Which new data were available to Copernicus? None. He had the same data as Ptolemy. Which new data did Newton have? Almost none. His real ingredients were Kepler's laws and Galileo's results. What new data did Einstein have to discover general relativity? None. His ingredients were special relativity and Newton's theory. It simply isn't true that physics advances only when it is afforded new data.

What Copernicus, Newton, Einstein and many others did was to build upon pre-existing theories which synthesized empirical knowledge across vast fields of nature, and to find a way of combining and rethinking them to improve the general picture.

This is the basis on which the best research on quantum gravity operates. The origin of knowledge, as always in science, is ultimately empirical. But the data on which quantum gravity is built is not new experiments: it is the theoretical edifices which have already structured our knowledge of the world, in forms which are only partly coherent. The 'experimental data' for quantum gravity are general relativity and quantum mechanics. Building on these, trying to understand how a world in which both quanta and curved s.p.a.ce exist may be made coherent, we attempt to look towards the unknown.

The enormous success of the giants who have preceded us in similar situations, such as Newton, Einstein and Dirac, gives us encouragement. We do not presume to be of their stature. But we have the advantage of sitting on their shoulders, and this allows us to look further than they did. One way or another, we cannot but try.

We must distinguish between clues and strong evidence. Clues are what set Sherlock Holmes on the right track, allowing him to solve a mysterious case. Strong evidence is what the judge needs to sentence the guilty. Clues put us on the right path towards a correct theory. Strong evidence is that which subsequently allows us to trust whether the theory we have built is a good one or not. Without clues, we search in the wrong directions. Without evidence, a theory is not reliable.

The same applies to quantum gravity. The theory is in its infancy. Its theoretical apparatus is gaining solidity, and the fundamental ideas are being clarified: the clues are good, and concrete confirmed predictions are still missing. The theory has not yet taken its exams.

Signals from nature

The most studied alternative to the research direction recounted in this book is string theory. The majority of physicists who have worked on string theory, or string-related theories, expected that as soon as the new particle accelerator at CERN in Geneva began to function (the LHC or Large Hadron Collider), particles of a new kind never before observed, but antic.i.p.ated by the theory, would immediately become evident: supersymmetric particles. String theory needs these particles to be consistent: that is why the string theorists eagerly expected them to be found. Loop quantum gravity, on the other hand, is well defined even without supersymmetric particles. The loop theorists were inclined to think that these particles might not exist.

The supersymmetric particles were not observed, to the great disappointment of many. The fanfare that greeted the discovery of the Higgs boson in 2013 also masked this disappointment. The supersymmetric particles are not there at the energy where many string theorists expected them to be. This is not a definitive proof of anything far from it; but nature has given a small clue in favour of the loops.

There have been three major experimental results in fundamental physics in recent years. The first is the revelation of the Higgs boson at CERN in Geneva (figure 9.1). The second is the measurements made by the Planck satellite (figure 9.2), measurements, the data of which were also made public in 2013, confirming the standard cosmological model. The third is the first detection of gravitational waves announced in the first months of 2016. These are the three signals that nature has recently given us.

Figure 9.1 An event at CERN which shows the formation of the Higgs particle.

Figure 9.2 The Planck satellite.

There is something in common between these three results: the complete absence of surprise. This does not diminish their importance: if anything, it makes them even more meaningful. The discovery of the Higgs boson is a rock-hard confirmation of the validity of the ideas behind the standard model of elementary particles, based on quantum mechanics. It is the verification of a prediction made thirty years previously. The Planck measurements are a solid confirmation of the standard cosmological model, based on general relativity with the cosmological constant. The detection of gravitational waves is a spectacular confirmation of general relativity, a theory a hundred years old. The three results, obtained with strenuous technological efforts and extensive collaborations between hundreds of scientists, do nothing other than reinforce the understanding that we already had of the structure of the universe. No real surprises.

But such an absence of surprises was in a sense itself surprising, because many expected to be surprised, that is, to see 'new physics', not yet described by established theories. They expected supersymmetry at CERN, not the Higgs boson. And many expected that Planck would measure discrepancies from the standard cosmological model, discrepancies that would support alternative cosmological theories to general relativity.

But no. What nature is confirming is simple: general relativity, quantum mechanics and, within quantum mechanics, the standard model.

Many theoretical physicists are today looking for new theories by picking arbitrary hypotheses. 'Let us imagine that ...' I don't think that this way of doing science has ever produced good results. Our fantasy is too limited to 'imagine' how the world may be made, unless we search for inspiration in the traces we have at our disposal. The traces that we have our clues are either the theories which have been successful, or new experimental data, nothing else. It is in this data and in these theories that we must try to uncover what we have been unable yet to imagine. This is how Copernicus, Newton, Maxwell and Einstein proceeded. They never tried to 'guess' a new theory unlike, in my opinion, the way in which too many theoretical physicists are trying to do today.

The three recent experimental results I mentioned speak with the voice of Nature itself: 'Stop dreaming of new fields and strange particles; supplementary dimensions, other symmetries, parallel universes, strings, and whatever else. The pieces of the puzzle are simpler: general relativity, quantum mechanics and the standard model. The next step forward may be "only" a question of combining them in the correct manner.' It's rea.s.suring advice for the loop quantum gravity community, because these are the hypotheses of the theory: general relativity, quantum mechanics and compatibility with the standard model, nothing else. The radical conceptual consequences the quanta of s.p.a.ce, the disappearance of time are not bold hypotheses: they are the rational consequences that follow from taking the basic insights of our best theories seriously.

Once again, these are not definitive proofs. Supersymmetric particles might finally exist, perhaps, at a scale still not reached, and could exist even if loop theory is correct. Supersymmetry failed to show up where expected, and string theorists are a little downcast, loop theorists are buoyant, but it is still a matter of clues; there is no strong evidence at all.

To find more concrete confirmation of the theory, we need to look elsewhere. The primordial universe could open the window to predictions capable of confirming the theory. In a not too distant future, we hope. Or they could prove the theory wrong.

A window on to quantum gravity

If we have the equations that describe the transition of the universe across the quantum phase, we can compute effects of quantum phenomena upon the universe which we observe today. The universe is filled with cosmic radiation: a sea of photons remained in the cosmos since the early hot phase, the residual glare of the early high temperature.

The electromagnetic field in the immense s.p.a.ce between galaxies trembles like the surface of the sea after a big storm. This quivering, disseminated throughout the universe, is called the cosmic background radiation. It has been studied in the past few years by satellites such as COBE, WMAP and, most recently, Planck. An image of the minute fluctuations of this radiation is given in figure 9.3. The details of the structure of this radiation tell us the history of the universe and, hidden in the folds of these details, there could be footprints of the quantum beginning of our universe.

One of the most active sectors of research in loop quantum gravity is studying how the quantum dynamic of the primordial universe is reflected in this data. The results are preliminary, but encouraging. With more calculations and more precise measurements, it should be possible to arrive at a test of the theory.

In 2013 Abhay Ashtekar, Ivan Agullo and William Nelson published an article in which they calculate that, under certain hypotheses, the statistical distribution of the fluctuations of this source of cosmic radiation should reveal the effect of the initial bounce: the wide-angle fluctuations should be different from those predicted by the theory that does not take quanta into account. The current state of the measurement is described in figure 9.4, where the black line represents the prediction by Ashtekar, Agullo and Nelson, and the grey dots the measured data. For now, these are not sufficient to evaluate whether the upward bend of the black line predicted by the three authors is correct or not. But measurements are getting more precise. The situation is still fluid. But those who, like myself, have spent their lives seeking to understand the secrets of quantum s.p.a.ce are following with close attention, anxiety and hope the continuous honing of our capacity to make observations, to measure and to calculate and are awaiting the moment in which nature will tell us whether we are right or not.

Figure 9.3 The fluctuations of the cosmic background radiation. This is the image of the oldest object in the universe available to us. These fluctuations were produced 14 billion years ago. In the statistics of such fluctuations we hope to find confirmation of the predictions of quantum gravity.