Notice that an inhabitant of the southern hemisphere is in a certain sense 'surrounded' by the northern hemisphere, since in whichever direction she goes to exit her hemisphere, she will always arrive in the other one. But the contrary is obviously true as well: each hemisphere 'surrounds', and is surrounded by, the other. A 3-sphere may be represented in a similar fashion, but with everything given an additional dimension: two b.a.l.l.s stuck together all along their edges (figure 3.12).

When we leave one ball we enter into the other, just as when we leave one of the discs in the representation of the globe we enter into the other disc. Each ball surrounds and is surrounded by the other. Einstein's idea is that s.p.a.ce could be a 3-sphere: something with a finite volume (the sum of the volume of the two b.a.l.l.s), but without borders.fn17 The 3-sphere is the solution which Einstein proposes in his work of 1917 to the problem of the border of the universe. This article initiates modern cosmology, the study of the entire visible universe, studied at the grandest scale. From it will arise the discovery of the expansion of the universe; the theory of the Big Bang; the problem of the birth of the universe, and much else besides. I speak about all of this in Chapter 8.

Figure 3.12 A 3-sphere can be represented as two b.a.l.l.s joined together.

There is one more observation which I would like to make about Einstein's 3-sphere. However incredible it might seem, the same idea had already been conceived by another genius, from an entirely different cultural universe: Dante Alighieri, Italy's greatest poet. In the Paradiso, the third part of his major poem, the Commedia, Dante offers a grandiose vision of the medieval world, calqued on the world of Aristotle, with the spherical Earth at its centre, surrounded by the celestial spheres (figure 3.13).

Figure 3.13 Traditional representation of Dante's universe.

Accompanied by his shining loved one, Beatrice, Dante ascends these spheres in the course of a fantastic, visionary journey up to the outermost sphere. When he reaches it, he contemplates the universe below him with its rotating heavens and the Earth, very far down, at its centre. But then he looks even higher and what does he see? He sees a point of light surrounded by immense spheres of angels, that is to say, by another immense ball, which, in his words 'surrounds and is at the same time surrounded by' the sphere of our universe! Here are Dante's verses from Canto XXVII of the Paradiso. Questa altre parte dell'Universo d'un cerchio lui comprende si come questo li altri: 'This other part of the universe surrounds the first in a circle like the first surrounds the others.' And in the next canto, still on the last 'circle', parendo inchiuso da quel ch'elli inchiude: 'appearing to be to be enclosed by those that it encloses'. The point of light and the sphere of angels are surrounding the universe, and at the same time they are surrounded by the universe! It is an exact description of a 3-sphere!

The usual representations of Dante's universe common in Italian schoolbooks (such as figure 3.13) place the angelic spheres separate from the celestial ones. But Dante writes that the two b.a.l.l.s 'surround and are surrounded by' each other. Dante has a clear geometrical intuition of a 3-sphere.fn18 The first to notice that the Paradiso describes the universe as a 3-sphere was an American mathematician, Mark Peterson, in 1979. In general, scholars of Dante are not very familiar with 3-spheres. Today, every physicist and mathematician could easily recognize the 3-sphere in Dante's description of the universe.

How is it possible that Dante had an idea that sounds so modern? I think it was possible, in the first place, due to the profound intelligence of Italy's finest poet. This intelligence is one of the reasons why the Commedia is so fascinating. But it is also due to the fact that Dante was writing well before Newton convinced everyone that the infinite s.p.a.ce of the cosmos was the flat one of Euclidean geometry. Dante was free of the restraints upon our intuition we have as a result of our Newtonian schooling.

Dante's scientific culture was based princ.i.p.ally on the teachings of his mentor and tutor, Brunetto Latini, who has left us a small, enchanting treatise, Li tresor, which is a sort of encyclopaedia of medieval knowledge, written in a delightful combination of old French and Italian. In Li tresor, Brunetto explains in detail the fact that the Earth is round. But he does so, curiously to the eyes of a modern reader, in terms of 'intrinsic' rather than 'extrinsic' geometry. That is to say, he does not write, 'the Earth is like an orange', as the Earth would look if seen from the outside, but writes instead, 'Two knights who could gallop sufficiently far in opposite directions would meet up on the other side.' And: 'If he were not impeded by the seas, a man who set out to walk for ever would return to the point on the Earth from which he departed.' In other words, he adopts an internal, not an external, point of view: the perspective of someone who walks the Earth, not of someone who looks at it from afar. At first glance it might seem like a pointless, complicated way of explaining that the Earth is a ball. Why doesn't Brunetto simply say that the Earth is like an orange? But, on reflection: if, say, an ant walks on an orange, it will at some point find itself upside down, and must keep itself attached by means of the tiny suction pads on its legs, to avoid falling off. And yet a traveller who walks the Earth never finds himself upside down, and needs no suction pads on his legs. Brunetto's description is not so quaint after all.

Now, think about it. For someone who has learned from his teacher that the form of the surface of our planet is such that by walking always in a straight line we return to the point we started from, it is perhaps not so difficult to take the next obvious step, and imagine that the form of the entire universe is such that, flying always in a straight line, we return to the same point of departure: a 3-sphere is a s.p.a.ce in which 'two winged knights that could fly in opposite directions would meet up on the other side'. In technical terms, the description of the geometry of the Earth offered by Brunetto Latini in Li tresor is given in terms of intrinsic geometry (seen from the inside) rather than extrinsic (seen from the outside), and this is exactly the description that is suitable to generalize the notion of 'sphere' from two dimensions to three. The best way of describing a 3-sphere is not to try to 'see it from the outside', but rather to describe what happens when moving within it.

The method developed by Gauss to describe curved surfaces, and generalized by Riemann to describe the curvature of s.p.a.ces in three or more dimensions, basically, amounts to Brunetto Latini's way. That is to say, the idea is to describe a curved s.p.a.ce not as 'seen from the outside', stating how it curves in an external s.p.a.ce, but instead in terms of what may be experienced by somebody within that s.p.a.ce, who is moving and always remaining within it. For instance, the surface of an ordinary sphere, as Brunetto observes, is a surface where all the 'straight' lines get back to the starting point after traversing the same distance (the length of the equator). A 3-sphere is a three-dimensional s.p.a.ce with the same property.

Einstein's s.p.a.cetime is not curved in the sense that it curves 'in an external s.p.a.ce'. It is curved in the sense that its intrinsic geometry, that is to say, the web of distances between its points, which can be observed by staying within it, is not the geometry of a flat s.p.a.ce. It is a s.p.a.ce where Pythagoras's theorem is not valid, just as Pythagoras's theorem is not valid on the surface of the Earth.fn19 There is a way of understanding the curvature of s.p.a.ce from within it, and without looking at it from outside, which is important for what follows. Imagine you are at the North Pole and walk southwards until you reach the equator, carrying with you an arrow pointing ahead. Once you reach the equator, turn to the left without changing the direction of the arrow. The arrow still points south, which is now to your right. Advance a little towards the east along the equator and then turn again towards the north again without changing the direction of the arrow, which will now be pointing behind you. When you reach the North Pole again, you have executed a closed circuit a 'loop', as it is termed and the arrow does not point in the same direction as when you started out (figure 3.14). The angle through which the arrow has turned in the course of the loop measures the curvature.

I will return later to this method of measuring curvature by making a loop in s.p.a.ce. These will be the loops that give the name to the theory of loop quantum gravity.

Figure 3.14 An arrow carried parallel to itself along a circuit (a loop) in a curved s.p.a.ce arrives back rotated at the point of departure.

Dante leaves Florence in 1301, while the mosaics in the cupola of the Baptistery are being completed. The mosaic, representing h.e.l.l (the work of Coppo di Marcovaldo, the teacher of Cimabue), probably terrifying in the eyes of a medieval person, has often been indicated as a source of inspiration to Dante (figure 3.15).

Shortly before starting to write this book, I visited the Baptistery in the company of Emanuela Minnai, the friend who convinced me to write it. Entering the Baptistery and looking up, you see a shining point of light (the light source from the lantern at the summit of the cupola) surrounded by nine orders of angels, with the name of each order written: Angels, Archangels, Princ.i.p.alities, Powers, Virtues, Domains, Thrones, Cherubim and Seraphim. This corresponds exactly to the structure of the second sphere of Paradise. Imagine that you are an ant on the floor of the Baptistery and are able to walk in any direction; regardless of which direction you follow to climb the wall, you would reach the ceiling at the same point of light surrounded by angels: the point of light and its angels both 'surround' and 'are surrounded by' the rest of the decorated interior of the Baptistery (figure 3.16).

Figure 3.15 The mosaic depicting h.e.l.l, by Coppo di Marcovaldo, in the Baptistery of Florence.

Like every citizen of Florence at the end of the thirteenth century, Dante must have been profoundly awe-struck by the Baptistery, the grandiose architectural enterprise his city was completing. I believe that he may have been inspired by the Baptistery, not only by Coppo di Marcovaldo's Inferno, but also by its overall architecture, for his vision of the cosmos. The Paradiso reproduces its structure remarkably precisely, including the nine circles of angels and the point of light, just translating it from two to three dimensions. After describing the spherical universe of Aristotle, Brunetto had already written that beyond it lies the place of divinity and medieval iconography had already imagined Paradise as G.o.d surrounded by spheres of angels. In the end, Dante does no more than mount the pieces that already existed into a coherent architectural whole which follows the suggestive architecture of the Baptistery and resolves the ancient problem of the borders of the universe. In so doing, Dante antic.i.p.ates by six centuries Einstein's 3-sphere.

Figure 3.16 The interior of the Baptistery.

I don't know if the young Einstein had encountered the Paradiso during his intellectual wanderings in Italy, and whether or not the vivid imagination of the Italian poet may have had a direct influence on his intuition that the universe might be both finite and without boundary. Whether or not such influence occurred, I believe that this example demonstrates how great science and great poetry are both visionary, and may even arrive at the same intuitions. Our culture is foolish to keep science and poetry separated: they are two tools to open our eyes to the complexity and beauty of the world.

Dante's 3-sphere is only an intuition within a dream. Einstein's 3-sphere has mathematical form and follows from the theory's equations. The effect of each is different. Dante moves us deeply, touching the sources of our emotions. Einstein opens a road towards the unsolved mysteries of our universe. But both count among the most beautiful and significant flights that the mind can achieve.

But let's return to 1917, when Einstein tries to insert the idea of the 3-sphere into his equations. Here he encounters a problem. He is convinced that the universe is fixed and immutable, but his equations tell him that this is not possible. It isn't difficult to understand why. Everything attracts, therefore the only way for a finite universe not to collapse on itself is for it to be expanding: just as the only way to prevent a football from falling to the ground is to kick it upwards. It either goes up, or falls down it can't stay still, suspended in the air.

But Einstein does not believe what his own equations are telling him. He even makes a silly physics mistake (he does not realize that the solution he considers is unstable) just to avoid accepting what his theory predicts: the universe is either contracting or expanding. He modifies his equations, trying to avoid the implication that it is expanding. It is for this reason that he adds the term gab in the equation written above. But it is a further mistake: the added term is correct, but it does not change the fact that the equation predicts that the universe must be expanding. For all his bravery, Einstein the genius lacks the courage to believe his own equations.

A few years later Einstein is forced to give up: it is his theory that is right, not his reservations about it. Astronomers realize that all galaxies are indeed moving away from us. The universe is expanding, exactly as the equations predicted. Fourteen billion years ago, the universe was concentrated almost to a single, furiously hot point. From there it expanded in a colossal 'cosmic' explosion and here the term 'cosmic' is not used in any rhetorical sense: it is, literally, a cosmic explosion. This is the 'Big Bang'.

Today we know the expansion is real. The definitive proof of the scenario foreseen by Einstein's equations arrives in 1964, when two American radio-astronomers, Arno Penzias and Robert Wilson, discover by accident a radiation diffused throughout the universe which turns out to be precisely what remains of the original immense heat of the early universe. Once again, the theory turns out to have been correct, up to its most amazing predictions.

Figure 3.17 Einstein's world: particles and fields which move on other fields.

Ever since we discovered that the Earth is round and turns like a mad spinning-top, we have understood that reality is not what it seems: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen. But the leap made by Einstein is unparalleled: s.p.a.cetime is a field; the world is made only of fields and particles; s.p.a.ce and time are not something else, something different from the rest of nature: they are just a field among the others (figure 3.17).

In 1953, a primary schoolchild writes to Albert Einstein, 'Our cla.s.s is studying the universe. I am very interested in s.p.a.ce. I would like to thank you for all that you have done so that we might understand it.'5 I feel the same way.

4. Quanta

The two pillars of twentieth-century physics general relativity and quantum mechanics could not be more different from each other. General relativity is a compact jewel: conceived by a single mind, based on combining previous theories, it is a simple and coherent vision of gravity, s.p.a.ce and time. Quantum mechanics, or quantum theory, on the other hand, emerges from experiments in the course of a long gestation over a quarter of a century, to which many have contributed; achieves unequalled experimental success and leads to applications which have transformed our everyday lives (the computer on which I write, for instance); but, more than a century after its birth, it remains shrouded in obscurity and incomprehensibility.

This chapter ill.u.s.trates the strange physics of this theory, relates how the theory came into being and the three aspects of reality it has unveiled: granularity, indeterminism and relationality.

Albert again

It's said that quantum mechanics was born precisely in 1900, virtually ushering in a century of intense thought. In 1900 the German physicist Max Planck tries to compute the amount of electromagnetic waves in equilibrium in a hot box. To obtain a formula reproducing the experimental results, he ends up using a trick which does not appear to make much sense: he a.s.sumes that the energy of the electric field is distributed in 'quanta', that is to say, in small packets, little bricks of energy. The size of the packets, he a.s.sumes, depends on the frequency (that is, the colour) of the electromagnetic waves. For waves of frequency , every quantum, or every packet, has energy

E = h

This formula is the first of quantum mechanics; h is a novel constant which today we call the Planck constant. It fixes how much energy there is in each packet of energy, for radiation of frequency (colour) . The constant h determines the scale of all quantum phenomena.

The idea that energy could be made up of finite packets is at odds with everything that was known at the time: energy was considered something that could vary in a continuous manner, and there was no reason to treat it as if it were made up of grains. For example, the energy of a pendulum measures the amplitude of the swing. There seems to be no reason for a pendulum to oscillate only with certain determined amplitudes and not others. For Max Planck, taking energy in finite-size packets was only a strange trick which happened to work for the calculation that is, to reproduce laboratory measurements but for utterly unclear reasons.

Five years later it is Albert Einstein him again who comes to understand that Planck's packets of energy are in fact real. This is the subject of the third of the three articles sent to the Annalen der Physik in 1905. And this is the true date of birth of quantum theory.

In the article, Einstein argues that light truly is made up of small grains, particles of light. He considers a phenomenon that had been recently observed: the photoelectric effect. There are substances that generate a weak electric current when struck by light. That is to say, they emit electrons when light shines on them. Today we use them, for example, in the photoelectric cells which open doors when we approach them by detecting if light arrives, or not, in a sensor. That this happens is not strange, because light carries energy (it warms us, for example), and its energy makes the electrons 'jump out' of their atoms; it gives them a push.

But something is strange: it seems reasonable to expect that if the energy of light is scarce namely, if the light is dim the phenomenon would not take place; and that it would take place when the energy is sufficient namely, when the light is bright. But it isn't like this: what is observed is that the phenomenon happens only if the frequency of light is high and does not happen if the frequency is low. That is to say, it happens or doesn't happen depending on the colour of light (the frequency) rather than its intensity (energy). There is no way of making sense of this with standard physics.

Einstein uses Planck's idea of the packets of energy, with a size that depends upon frequency, and realizes that if these packets are real, the phenomenon can be explained. It isn't difficult to understand why. Imagine that the light arrives in the form of grains of energy. An electron will be swept out of its atom if the individual grain hitting it has a great deal of energy. What matters is the energy of each grain, not the number of grains. If, as in Planck's hypothesis, the energy of each grain is determined by frequency, the phenomenon will occur only if frequency is sufficiently high, that is to say, if the individual grains of energy are sufficiently large, independently from the total amount of energy that's around.

It is like when it hails: what determines whether your car will be dented is not the total quant.i.ty of hail that falls but the size of the individual hailstones. There could be an enormous amount of hail, but it will do no damage if all the stones are small. In the same way, even if light is intense which amounts to saying that there are a great deal of light packets the electrons would not be extracted from their atoms if the individual grains of light are too small, that is, if the frequency of light is too low. This explains why it is the colour and not the intensity which determines whether the photoelectric effect occurs or not. For this simple reasoning Einstein was awarded the n.o.bel Prize. It is easy to understand things once someone has thought them through. The difficulty lies in thinking them through in the first place.

Today we call these packets of energy 'photons', from the Greek word for light: . Photons are the grains of light, its 'quanta'. In the article Einstein writes: It seems to me that the observations a.s.sociated with blackbody radiation, fluorescence, the production of cathode rays by ultraviolet light, and other related phenomena connected with the emission or transformation of light are more readily understood if one a.s.sumes that the energy of light is discontinuously distributed in s.p.a.ce. In accordance with the a.s.sumption to be considered here, the energy of a light ray spreading out from a point source is not continuously distributed over an increasing s.p.a.ce but consists of a finite number of 'energy quanta' which are localized at points in s.p.a.ce, which move without dividing, and which can only be produced and absorbed as complete units.1 These simple and clear lines are the real birth certificate of quantum theory. Note the wonderful initial 'It seems to me ...', which recalls the hesitations of Faraday, or those of Newton; or the uncertainty of Darwin in the first pages of On the Origin of Species. True genius is aware of the momentousness of the steps it is taking, and is always hesitant ...

There is a clear relation between Einstein's work on Brownian motion (discussed in Chapter 1) and his work on the quanta of light, both completed in 1905. In the first, Einstein had managed to find a demonstration of the atomic hypothesis, that is to say, of the granular structure of matter. In the second he extends this same hypothesis to light: light must have a granular structure as well.

At first, Einstein's idea that light could be made up of photons is regarded by his colleagues as no more than youthful waywardness. Everyone commends him for his theory of relativity, but everybody judges the notion of photons to be outlandish. Scientists had only recently been persuaded that light was a wave in the electromagnetic field: how could it be made up of grains? In a letter addressed to the German Ministry, recommending that Einstein should have a professorship inaugurated for him in Berlin, the most distinguished physicists of the day write that the young man is so brilliant that he 'may be excused' certain excesses, such as the idea of photons. Not many years later, the very same colleagues award him the n.o.bel Prize, precisely for having understood that photons exist. Light falls on a surface like a gentle hail shower.

To comprehend how light may be simultaneously an electromagnetic wave and a swarm of photons will require the entire construction of quantum mechanics. But the first building block of this theory has been established: there exists a fundamental granularity in all things, including light.

Niels, Werner and Paul

If Planck is the biological father of the theory, Einstein is the parent who gave birth to and nurtured it. But as is often the case with children, the theory then went its own way, barely recognized by Einstein as his own.

During the first two decades of the twentieth century, it is the Dane Niels Bohr who is responsible for guiding its development. Bohr studies the structure of atoms, which was beginning to be explored at the turn of the century. Experiments had shown that an atom is like a small solar system: the ma.s.s is concentrated in a heavy central nucleus, around which light electrons revolve, more or less like the planets around the Sun. This picture, however, did not account for a simple fact: matter is coloured.

Salt is white, pepper is black, chilli is red. Why? Studying the light emitted by atoms, it is apparent that substances have specific colours. Since colour is the frequency of light, light is emitted by substances at certain fixed frequencies. The set of the frequencies that characterizes a given substance is known as the 'spectrum' of this substance. A spectrum is a collection of fine lines of different hues, in which the light emitted by a given substance is decomposed (for instance, by a prism). The spectra of a few elements are shown in figure 4.2.

Figure 4.1 Niels Bohr.

Spectra of numerous substances had been studied and catalogued in many laboratories at the turn of the century, and n.o.body knew how to explain why each substance had this or that spectrum. What determines the colour of those lines?

Figure 4.2 The spectra of some elements: sodium, mercury, lithium and hydrogen.

Colour is the speed at which Faraday's lines vibrate, and this is determined by the vibrations of the electric charges which emit light. These charges are the electrons that move inside the atoms. Therefore, studying spectra, we can understand how electrons move around nuclei. The other way around, we could predict the spectrum of each atom by computing the frequencies of the electrons circling their nucleus. Easy to say, but in practice n.o.body was able to do so. In fact, the whole thing seemed impossible, because in Newton's mechanics an electron can revolve around its nucleus at any speed, and hence emit light at any frequency. But then why does the light emitted by an atom not contain all colours, rather than just a few particular ones? Why are atomic spectra not a continuum of colours, instead of just a few separate lines? Why, in technical parlance, are they 'discrete' instead of continuous? For decades, physicists seemed incapable of finding an answer.

Bohr finds a tentative solution, by way of a strange hypothesis. He realizes that everything could be explained if the energy of electrons in atoms could only a.s.sume certain 'quantized' values certain specific values, just as was hypothesized by Planck and by Einstein for the energy of the quanta of light. Once again, the key is a granularity, but not now for the energy of light but rather for the energy of the electrons in the atom. It begins to become clear that granularity is something widespread in nature.

Bohr makes the hypothesis that electrons can exist only at certain 'special' distances from the nucleus, that is, only on certain particular orbits, the scale of which is determined by Planck's constant h. And that electrons can 'leap' between one orbit with the permitted energy to another. These are the famous 'quantum leaps'. The frequency at which the electron moves on these orbits determines the frequency of the emitted light and, since only certain orbits are allowed, it follows that only certain frequencies are emitted.

These hypotheses define Bohr's 'atomic model', whose centenary was commemorated in 2013. With these a.s.sumptions (outlandish, but simple) Bohr manages to compute the spectra of all atoms, and even to predict accurately spectra not yet observed. The experimental success of this simple model is astonishing.

Clearly, there must be some truth in these a.s.sumptions, even if they run contrary to all contemporary notions of matter and dynamics. But why are there always only just certain orbits? And what does it mean to say that electrons 'leap'?

In Bohr's inst.i.tute in Copenhagen, the most brilliant young minds of the century gather to try to give order to this jumble of incomprehensible behaviours in the atomic world, and to construct a coherent theory. The research is arduous and protracted, until a young German finds the key to unlock the door of the mystery of the quantum world.

Werner Heisenberg is twenty-five years old when he writes the equations of quantum mechanics, the same age as Einstein was when he wrote his three major articles. He does so on the basis of dizzying ideas.

Figure 4.3 Werner Heisenberg.

The intuition comes to him one night in the park behind the Copenhagen Inst.i.tute of Physics. The young Werner walks about pensively in the park. It is really dark there; we are in 1925. There is only an occasional streetlamp, casting dim islands of light here and there. The pools of light are separated by large expanses of darkness. Suddenly, Heisenberg sees a figure pa.s.s by. Actually, he does not see him pa.s.s: he sees him appear beneath a lamp, then disappear into the dark before reappearing beneath another lamp, and then vanishing back into the dark again. And so on, from pool of light to pool of light, until he eventually disappears altogether into the night. Heisenberg thinks that, 'evidently', the man does not actually vanish and reappear: in his mind, he can easily reconstruct the man's trajectory between one streetlamp and another. After all, a man is a substantial object, big and heavy and big, heavy objects do not simply appear and vanish ...

Ah! These objects, which are substantial, large and heavy, don't vanish and reappear ... but what do we know about electrons? A light flashes on in his mind. Why should small objects such as electrons do the same? What if, effectively, electrons could vanish and reappear? What if these were the mysterious quantum leaps which appeared to underlie the structure of the atomic spectra? What if, between one interaction with something, and another with something else, the electron could literally be nowhere.

What if the electron could be something that manifests itself only when it interacts, when it collides with something else; and that between one interaction and another it had no precise position? What if always having a precise position is something which is acquired only if one is substantial enough large and heavy like the man that pa.s.sed by a little while ago, like a ghost in the dark, and then disappeared into the night ...?

Only someone in his twenties can take such delirious propositions seriously. You have to be a twenty-something to believe that they can be turned into a theory of the world. And perhaps you have to be this young to understand better than anyone else, for the first time, the deep structure of nature. Just as Einstein was in his twenties when he realized that time does not pa.s.s in the same way for everyone, so, too, was Heisenberg on that Copenhagen night. Perhaps, it is no longer a good idea to trust your intuitions after the age of thirty ...

Heisenberg returns home gripped by feverish emotion, and plunges into calculations. He emerges, some time later, with a disconcerting theory: a fundamental description of the movement of particles, in which they are described not by their position at every moment but only by their position at particular instants: the instants in which they interact with something else.

This is the second cornerstone of quantum mechanics, its hardest key: the relational aspect of things. Electrons don't always exist. They exist when they interact. They materialize in a place when they collide with something else. The quantum leaps from one orbit to another const.i.tute their way of being real: an electron is a combination of leaps from one interaction to another. When nothing disturbs it, an electron does not exist in any place. Instead of writing the position and velocity of the electron, Heisenberg writes tables of numbers (matrices). He multiplies and divides tables of numbers representing possible interactions of the electron. And, as if from the magical abacus of a magus, the results correspond exactly with what was observed. These are the first fundamental equations of quantum mechanics. From here on, these equations will do nothing but work, work, work. Up until now, incredible as it may seem, they have never failed.

In the end, it is another twenty-five-year-old who picks up the work initiated by Heisenberg, takes the new theory in his hands and constructs its entire formal and mathematical scaffolding: the Englishman Paul Adrien Maurice Dirac, considered by many to be the greatest physicist of the twentieth century after Einstein.

Despite his scientific stature, Dirac is much less well-known than Einstein. This is due, in part, to the rarefied abstraction of his science, and partly due to his disconcerting character. Silent in company, extremely reserved, incapable of expressing emotions, frequently unable to recognize the faces of acquaintances incapable even of conducting an ordinary conversation, or of, apparently, understanding simple questions he seemed virtually autistic, and perhaps fell within the spectrum of this condition.

Figure 4.4 Paul Dirac.

During one of his lectures, a colleague said to him, 'I don't understand that formula.' After a short, silent pause, Dirac continued on regardless. The moderator interrupted him, asking if he would like to reply to the question. Dirac, sincerely astonished, replied, 'Question? What question? My colleague has made an a.s.sertion.' And so, in a very pedantic sense, he had. It wasn't arrogance: the man who could discover secrets of nature which had eluded everyone else could not understand the implicit meaning of language, could not grasp its non-literal usage, and took every phrase at face value.2 And yet, in his hands, quantum mechanics is transformed from a jumble of intuitions, half-baked calculations, misty metaphysical discussions and equations that work well, but inexplicably, into a perfect architecture: airy, simple and extremely beautiful. Beautiful, but stratospherically abstract.

The venerable Bohr said of him, 'Of all physicists, Dirac has the purest soul.' And don't his eyes, in figure 4.4 show so? His physics has the pristine clarity of a song. For him, the world is not made of things, it's const.i.tuted of an abstract mathematical structure which shows us how things appear and how they behave when manifesting themselves. It's a magical encounter between logic and intuition. Deeply impressed, Einstein remarked, 'Dirac poses problems for me. To maintain an equilibrium along this vertiginous course, between genius and madness, is a daunting enterprise.'

Dirac's quantum mechanics is the mathematical theory used today by any engineer, chemist or molecular biologist. In it, every object is defined by an abstract s.p.a.cefn20 and has no property in itself, apart from those that are unchanging, such as ma.s.s. Its position and velocity, its angular momentum and its electrical potential, and so on, acquire reality only when it collides 'interacts' with another object. It is not just its position which is undefined, as Heisenberg had recognized: no variable of the object is defined between one interaction and the next. The relational aspect of the theory becomes universal.

When it suddenly appears, in the course of an interaction with another object, a physical variable (velocity, energy, momentum, angular momentum) does not a.s.sume just any value. Dirac provides the general recipe to compute the set of values that a physical variable can take.fn21 These values are a.n.a.logous to the spectra of the light emitted by atoms. Today we call the set of the particular values which a variable may a.s.sume the 'spectrum' of that variable, by a.n.a.logy with the spectra into which the light of elements decomposes the first manifestation of this phenomenon. For example, the radius of the orbitals of an electron around a nucleus can acquire only specific values, those that Bohr had hypothesized, which form the 'spectrum of the radius'.

The theory also gives information on which value of the spectrum will manifest itself in the next interaction, but only in the form of probabilities. We do not know with certainty where the electron will appear, but we can compute the probability that it will appear here or there. This is a radical change from Newton's theory, where it is possible, in principle, to predict the future with certainty. Quantum mechanics brings probability to the heart of the evolution of things. This indeterminacy is the third cornerstone of quantum mechanics: the discovery that chance operates at the atomic level. While Newton's physics allows for the prediction of the future with exact.i.tude, if we have sufficient information about the initial data and if we can make the calculations, quantum mechanics allows us to calculate only the probability of an event. This absence of determinism at a small scale is intrinsic to nature. An electron is not obliged by nature to move towards the right or the left; it does so by chance. The apparent determinism of the macroscopic world is due only to the fact that the microscopic randomness cancels out on average, leaving only fluctuations too minute for us to perceive in everyday life.

Dirac's quantum mechanics thus allows us to do two things. The first is to calculate which values a physical variable may a.s.sume. This is called 'calculation of the spectrum of a variable'; it captures the granular nature of things. When an object (atom, electromagnetic field, molecule, pendulum, stone, star, and so on) interacts with something else, the values computed are those which its variables can a.s.sume in the interaction (relationism). The second thing that Dirac's quantum mechanics allows us to do is to compute the probability that this or that value of a variable appears at the next interaction. This is called 'calculation of an amplitude of transition'. Probability expresses the third feature of the theory: indeterminacy the fact that it does not give unique predictions, only probabilistic ones.

This is Dirac's quantum mechanics: a recipe for calculating the spectra of the variables and a recipe for calculating the probability that one or another value in the spectrum will appear during an interaction. That's it. What happens between one interaction and the next is not mentioned in the theory. It does not exist.

The probability of finding an electron or any other particle at one point or another can be imagined as a diffuse cloud, denser where the probability of seeing the particle is stronger. Sometimes it is useful to visualize this cloud as if it were a real thing. For instance, the cloud that represents an electron around its nucleus indicates where it is more likely that the electron appears if we look at it. Perhaps you encountered them at school: these are the atomic 'orbitals'.fn22 The efficacy of the theory soon proves extraordinary. If today we build computers, have advanced molecular chemistry and biology, lasers and semiconductors, it is thanks to quantum mechanics. For a certain number of decades it was as if it were Christmas every day for physicists: for every new problem, there was an answer which followed from the equations of quantum mechanics, and it was always the correct answer. One example of this will suffice.

The matter surrounding us is made up of a thousand different substances. During the nineteenth and twentieth centuries chemists understood that all these different substances are just combinations of a relatively small number (less than a hundred) of simple elements: hydrogen, helium, oxygen, and so on to uranium. Mendeleev put these elements in order (according to weight) in the famous periodic table which is pinned to the walls of so many cla.s.srooms and which summarizes the properties of the elements of which the world is made not only on Earth but all over the universe in all galaxies. Why these specific elements? What explains the periodic structure of the table? Why does each element have certain properties and not others? Why, for instance, do some elements combine easily, whereas others do not? What is the secret of the curious structure of Mendeleev's table?

Figure 4.5 Light is a wave on a field, but it has also a granular structure.

Well, take the equation of quantum mechanics that determines the form of the orbitals of an electron. This equation has a certain number of solutions, and these solutions correspond exactly to hydrogen, helium, oxygen ... and the other elements! Mendeleev's periodic table is structured exactly like the set of these solutions. The properties of the elements, with everything else, follows from the solution of this equation. Quantum mechanics deciphers perfectly the secret of the structure of the periodic table of elements.

Pythagoras and Plato's ancient dream is realized: to describe all of the world's substances with a single formula. The infinite complexity of chemistry, captured by the solutions of a single equation! And this is just one of the applications of quantum mechanics.

Fields and particles are the same thing