Shortly after completing the general formulation of quantum mechanics, Dirac realizes that the theory can be directly applied to fields such as electromagnetic ones, and can be made consistent with special relativity. (Making it consistent with general relativity will prove much harder, and is the main subject of this book). In doing this, Dirac discovers an ulterior, profound simplification of our description of nature: the convergence between the notion of particles used by Newton and the notion of fields introduced by Faraday.

The cloud of probability which accompanies electrons between one interaction and another does resemble a field. Faraday and Maxwell's fields, in turn, are made up of grains: photons. Not only are the particles in a certain sense diffused in s.p.a.ce like fields but the fields interact like particles. The notions of fields and particles, separated by Faraday and Maxwell, end up merging in quantum mechanics.

The way this happens in the theory is elegant: the equation of Dirac determines the values a variable can take. Applied to the energy of Faraday's lines, they tell us that this energy can only take on certain values, and not others. Since the energy of the electromagnetic field can take on only certain values, the field behaves like a set of packets of energy. These are precisely the quanta of energy introduced by Planck and Einstein thirty years earlier. The circle closes, and the story is complete. The equations of the theory, written by Dirac, account for the granular nature of light, which Planck and Einstein had intuited.

The electromagnetic waves are vibrations of Faraday's lines, but also, at a small scale, swarms of photons. When they interact with something else, as in the photoelectric effect, they manifest themselves as particles: to our eye, light rains in separate droplets, in single photons. Photons are the quanta of the electromagnetic field.

But the electrons and all the other particles of which the world is made are equally quanta of a field a 'quantum field' similar to Faraday and Maxwell's, subject to granularity and to quantum probability. Dirac writes the equations for the field of the electrons and of the other elementary particles.fn23 The sharp distinction between fields and particles introduced by Faraday vanishes.

The general form of quantum theory compatible with special relativity is thus called quantum field theory, and it forms the basis of today's particle physics. Particles are quanta of a field, just as photons are quanta of light. All fields display a granular structure in their interactions.

During the course of the twentieth century the list of fundamental fields was repeatedly updated, and today we have a theory called the standard model of elementary particles which describes almost all we see, with the exception of gravity,fn24 in the context of quantum field theory. The development of this model occupied physicists for a good part of the last century, and represents in itself a wonderful adventure of discovery. I don't present this side of the story here: it is quantum gravity that I would like to get on to. The standard model is completed by the 1970s. There are approximately fifteen fields, whose quanta are the elementary particles (electrons, quarks, muons, neutrinos, Higgs, and little else), plus a few fields similar to the electromagnetic one, which describe electromagnetic force and the other forces operating at a nuclear scale, whose quanta are similar to the photons.

The standard model was not taken very seriously at first, due to its somewhat cobbled-together aspect, so different from the airy simplicity of general relativity and Maxwell's or Dirac's equations. Against expectations, however, all of its predictions have been confirmed. For more than thirty years, every single experiment of particle physics has done nothing but repeatedly reconfirm the standard model. A recent confirmation was the discovery of the Higgs particle, which caused a sensation in 2013. Introduced to render the theory coherent, the Higgs field seemed a bit artificial until the Higgs particle, the quantum of this field, was actually observed and found to have precisely the properties predicted by the standard model.fn25 (The fact that it has been called 'the G.o.d particle' is so stupid as to be unworthy of comment.) In short, despite its unjustly modest name, the standard model has been a triumph.

Figure 4.6 What is the world made of?

Quantum mechanics, with its fields/particles, offers today a spectacularly effective description of nature. The world is not made up of fields and particles but of a single type of ent.i.ty: the quantum field. There are no longer particles which move in s.p.a.ce with the pa.s.sage of time, but quantum fields whose elementary events happen in s.p.a.cetime. The world is strange, but simple (figure 4.6).

Quanta 1: Information is finite

The time has come to attempt some conclusions about what it is, precisely, that quantum mechanics tells us about the world. It isn't an easy task, because quantum mechanics is not conceptually clear and its true meaning remains controversial; but it's a necessary exercise, to gain clarity and go forward. I think that quantum mechanics has revealed three aspects of the nature of things: granularity, indeterminacy and the relational structure of the world. Let's look at each of these more closely.

The first is the existence of a fundamental granularity in nature. The granularity of matter and light is at the heart of quantum theory. It isn't the same granularity intuited by Democritus, however. For Democritus, atoms were like little pebbles, whereas in quantum mechanics particles vanish and reappear. But the root of the idea of the substantive granularity of the world is still to be found in ancient atomism, and quantum mechanics strengthened by centuries of experiments, by powerful mathematics, and by its extraordinary capacity for making correct predictions is a genuine recognition of the profound insights on the nature of things reached by the great philosopher of Abdera.

Say we make measurements on a physical system and find that the system is in a particular state. For instance, we measure the amplitude of the oscillations of a pendulum and find that it has a certain value say, somewhere between five centimetres and six centimetres (no measurement is exact in physics). Before quantum mechanics we would have said that, since there are an infinite number of possible values between five and six centimetres (for instance 5.1 or 5.101 or 5.101001 ...), then there are infinite possible states of motion in which the pendulum could find itself: the amount of our ignorance about the pendulum state is still infinite.

Instead, quantum mechanics tells us that between five and six centimetres there is a finite number of possible values of the amplitude, hence our missing information about the pendulum is finite.

This goes for everything in general.fn26 Therefore, the first meaning of quantum mechanics is the existence of a limit to the information that can exist within a system: a limit to the number of distinguishable states in which a system can be. This limitation upon infinity this granularity of nature glimpsed by Democritus is the first central aspect of the theory. Planck's constant h measures the elementary scale of this granularity.

Quanta 2: Indeterminacy

The world is a sequence of granular quantum events. These are discrete, granular and individual; they are individual interactions of one physical system with another. An electron, a quantum of a field or a photon does not follow a trajectory in s.p.a.ce but appears in a given place and at a given time when colliding with something else. When and where will it appear? There is no way of knowing with certainty. Quantum mechanics introduces an elementary indeterminacy to the heart of the world. The future is genuinely unpredictable. This is the second fundamental lesson learned with quantum mechanics.

Due to this indeterminacy, in the world described by quantum mechanics, things are constantly subject to random change. All the variables 'fluctuate' continually, as if, at the smallest scale, everything is constantly vibrating. We do not see these omnipresent fluctuations only because of their small scale; they cannot be observed at a large scale, as when we observe macroscopic bodies. If we look at a stone, it stays still. But if we could see its atoms, we would observe them constantly spread here and there, and in ceaseless vibration. Quantum mechanics reveals to us that, the more we look at the detail of the world, the less constant it is. The world is not made up of tiny pebbles. It is a world of vibrations, a continuous fluctuation, a microscopic swarming of fleeting micro-events.

The atomism of antiquity had antic.i.p.ated also this aspect of modern physics: the appearance of laws of probability at a deep level. Democritus a.s.sumed (just like Newton) that the movement of atoms was rigorously determined by their collisions. But his successor, Epicurus, corrects the determinism of the master and introduces into atomism the notion of indeterminacy in the same way in which Heisenberg introduces indeterminacy into Newton's determinism. For Epicurus, atoms can on occasion deviate by chance from their course. Lucretius says this in beautiful words: this deviation occurs 'incerto tempore ... incertisque loci':3 at an uncertain place, at an uncertain time. The same randomness, the same appearance of probability at an elementary level, is the second key discovery about the world that quantum mechanics expresses.

So, how do we compute the probability that an electron in a certain initial position A will reappear, after a given time, in one or another final position B?

In the 1950s, Richard Feynman, who I've already mentioned, found a suggestive method of making this calculation: consider all possible trajectories from A to B, that is to say, all possible trajectories the electron can follow (straight, curved, zigzagging ...). Each trajectory determines a number. The probability is obtained from the sum of all these numbers. The details of this calculation are not important: what matters is the fact that all trajectories from A to B contribute: it is as if the electron, in order to go from A to B, pa.s.sed 'through all possible trajectories', or, in other words, unfurled into a cloud in order then to converge mysteriously on point B, where it collides again with something else (figure 4.7).

Figure 4.7 In order to move from A to B an electron behaves as if pa.s.sing through all possible trajectories.

This technique for computing the probability of a quantum event is called Feynman's sum over paths,fn27 and we shall see that it plays a role in quantum gravity.

Quanta 3: Reality is relational

The third discovery about the world articulated by quantum mechanics is the most profound and difficult and one which was not antic.i.p.ated by the atomism of antiquity.

The theory does not describe things as they are: it describes how things occur and how they interact with each other. It doesn't describe where there is a particle but how the particle shows itself to others. The world of existent things is reduced to a realm of possible interactions. Reality is reduced to interaction. Reality is reduced to relation.4 In a certain sense, this is just an extension of relativity, albeit a radical one. Aristotle was first to emphasize that we only perceive relative speed. On a ship, for example, we talk of our speed relative to the ship; on land, relative to the Earth. Galileo understood that this is the reason why the Earth can move with respect to the Sun without us feeling the movement. Speed is not a property of an object on its own: it is the property of the motion of an object with respect to another object. Einstein extended the notion of relativity to time: we can say that two events are simultaneous only relatively to a given motion (see here). Quantum mechanics extends this relativity in a radical way: all variable aspects of an object exist only in relation to other objects. It is only in interactions that nature draws the world.

In the world described by quantum mechanics there is no reality except in the relations between physical systems. It isn't things that enter into relations but, rather, relations that ground the notion of 'thing'. The world of quantum mechanics is not a world of objects: it is a world of events. Things are built by the happening of elementary events: as the philosopher Nelson Goodman wrote in the 1950s, in a beautiful phrase, 'An object is a monotonous process.' A stone is a vibration of quanta that maintains its structure for a while, just as a marine wave maintains its ident.i.ty for a while before melting again into the sea.

What is a wave, which moves on water without carrying with it any drop of water? A wave is not an object, in the sense that it is not made of matter that travels with it. The atoms of our body, as well, flow in and away from us. We, like waves and like all objects, are a flux of events; we are processes, for a brief time monotonous ...

Quantum mechanics does not describe objects: it describes processes and events which are junction points between processes.

To summarize, quantum mechanics is the discovery of three features of the world: Granularity (figure 4.8). The information in the state of a system is finite, and limited by Plank's constant.

Indeterminacy. The future is not determined unequivocally by the past. Even the more rigid regularities we see are, ultimately, statistical.

Relationality. The events of nature are always interactions. All events of a system occur in relation to another system.

Quantum mechanics teaches us not to think about the world in terms of 'things' which are in this or that state but in terms of 'processes' instead. A process is the pa.s.sage from one interaction to another. The properties of 'things' manifest themselves in a granular manner only in the moment of interaction, that is to say, at the edges of the processes, and are such only in relation to other things. They cannot be predicted in an unequivocal way but only in a probabilistic one.

Figure 4.8 The 'light box' in Einstein's mental experiment, as drawn by Bohr.

This is the vertiginous dive taken by Bohr, Heisenberg and Dirac into the depth of the nature of things.

But do we really understand?

Certainly, quantum mechanics is a triumph of efficacy. And yet ... are you sure, dear reader, that you have fully understood what quantum mechanics reveals to us? An electron is nowhere when it is not interacting ... mmm ... things only exist by jumping from one interaction to another ... well ... Does it all seem a little absurd?

It seemed absurd to Einstein.

On the one hand, Einstein proposed Werner Heisenberg and Paul Dirac for the n.o.bel Prize, recognizing that they had understood something fundamental about the world. On the other, he took every opportunity to grumble that, however, none of this made much sense The young lions of the Copenhagen group were dismayed: how could this come from Einstein himself? Their spiritual father, the man who had the courage to think the unthinkable, now pulled back and feared this new leap into the unknown the very leap which he had himself triggered. How could it be that the same Einstein, who had taught us that time is not universal and that s.p.a.ce bends, was now saying that the world could not be this strange?

Niels Bohr patiently explained the new ideas to Einstein. Einstein objected. Bohr, in the end, always managed to find answers to the objections. The dialogue continued for years, by way of lectures, letters, articles ... Einstein devised mental experiments to show that the new ideas were contradictory: 'Imagine a box filled with light, from which is let escape for a brief instant a single photon ...': thus one of the most famous examples of these begins (figure 4.8).fn28 During the course of the exchange, both great men had to give way, to alter their ideas. Einstein was obliged to recognize that there was actually no contradiction within the new ideas. But Bohr had to recognize that things were not as simple and as clear as he thought. Einstein did not want to relent on what for him was the key point: the notion that there is an objective reality, independent of whatever interacted with what. He refused to accept the relational aspect of the theory, the fact that things manifest themselves only through interactions. Bohr did not want to concede on the validity of the profoundly new way in which the real was conceptualized by the theory. Ultimately, Einstein accepts that the theory represents a gigantic leap forward in our understanding of the world, and that it is coherent. But he remains convinced that things could not be as strange as this theory proposed and that, 'behind' it, there must be a further, more reasonable explanation.

A century has pa.s.sed, and we are at the same point. Richard Feynman, who more than anyone has known how to juggle with the theory, has written, 'I think I can state that n.o.body really understands quantum mechanics.'

The equations of the theory and their consequences are used daily in a wide variety of fields: by physicists, engineers, chemists and biologists. But they remain mysterious: they do not describe physical systems but only how physical systems interact with and affect one another. What does this mean?

Physicists and philosophers continue to ask themselves what the real meaning of the theory might be and, in recent years, articles and conferences on the issue have proliferated. What is quantum theory, a century after its birth? An extraordinary dive deep into the nature of reality? A blunder that works, by chance? Part of an incomplete puzzle? Or a clue to something profound regarding the structure of the world, which we have yet to fully decipher?

The interpretation of quantum mechanics which I have presented here is the one which seems least unreasonable to me. It is called the 'relational interpretation', and it has been discussed by serious philosophers such as Bas van Fraa.s.sen, Michel Bitbol and Mauro Dorato.5 But there is no consensus on how to think about quantum mechanics: there are other ways of thinking about it, discussed by other physicists and other philosophers. We are on the brink of that which we don't know, and opinions diverge.

Quantum mechanics is only a physics theory: perhaps tomorrow it will be corrected by an understanding of the world which is different and even more profound. Some scientists today try to iron it out a bit, to render it more in keeping with our intuition. In my opinion, its dramatic empirical success should compel us to take it seriously, and to ask ourselves not what there is to change in the theory but rather what is limited about our intuition that makes it seem so strange to us.

I think that the obscurity of the theory is not the fault of quantum mechanics but, rather, is due to the limited capacity of our imagination. When we try to 'see' the quantum world, we are rather like moles used to living underground to whom someone is trying to describe the Himalayas. Or like the men imprisoned at the back of Plato's cave.

When Einstein died, his greatest rival, Bohr, found for him words of moving admiration. When, a few years later, Bohr in turn died, someone took a photograph of the blackboard in his study. There's a drawing on it. It represents the 'box of light' of Einstein's thought experiment. To the very last, the desire to debate, to understand more. To the very last, doubt.

This permanent doubt, the deep source of science.

Part Three.

QUANTUM s.p.a.cE AND RELATIONAL TIME.

If you have followed me this far, you now have all the elements with which to understand the current image of the world suggested by fundamental physics its power, its weaknesses, its limits.

There is a curved s.p.a.cetime born 14 billion years ago n.o.body knows how and still expanding. This s.p.a.ce is a real object, a physical field with its dynamics described by Einstein's equations. s.p.a.ce bends and curves under the weight of matter and plunges into black holes when matter is too concentrated.

Matter is distributed in 100 billion galaxies, each containing 100 billion stars, and is made up of quantum fields which manifest themselves in the form of particles, such as electrons and photons, or as waves, such as the electromagnetic ones that bring us television images and the light of the Sun and the stars.

These quantum fields make up atoms, light and the full contents of the universe. They are strange objects: their quanta are particles that appear when they interact with something else; left alone, they unfurl into a 'cloud of probability'. The world is a swarming of elementary events, immersed in the sea of a vast dynamical s.p.a.ce which sways like the water of an ocean.

With this image of the world, and the few equations that make it concrete, we can describe almost everything that we see.

Almost. Something is missing. And it is this something that we are seeking. The rest of the book talks about this missing part.

Turning the page, you pa.s.s from what, for good or ill, we credibly know about the world, to what we don't yet know but are trying to glimpse.

Turning the page is like leaving the security of our small s.p.a.cecraft of near-certainties and stepping into the unknown.

5. s.p.a.cetime is Quantum.

There is a paradox at the heart of our understanding of the physical world. General relativity and quantum mechanics, the two jewels that the twentieth century has left us, have been prolific in gifts for comprehending the world and for today's technology. From the first of these, cosmology has developed, as well as astrophysics, the study of gravitational waves and of black holes. The second has provided the foundation for atomic physics, nuclear physics, the physics of elementary particles and of condensed matter, and of much else besides.

And yet between the two theories there is something that grates. They cannot both be true, at least not in their present forms, because they appear to contradict each other. The gravitational field is described without taking quantum mechanics into account, without accounting for the fact that fields are quantum fields and quantum mechanics is formulated without taking into account the fact that s.p.a.cetime curves and is described by Einstein's equations.

A university student attending lectures on general relativity in the morning, and others on quantum mechanics in the afternoon, might be forgiven for concluding that his professors are fools, or that they haven't talked to each other for at least a century. In the morning, the world is a curved s.p.a.cetime where everything is continuous; in the afternoon, the world is a flat one where discrete quanta of energy leap and interact.

The paradox resides in the fact that both theories work remarkably well.

With every experiment and every test, nature continues to say 'you are right' to general relativity, and continues to say 'you are right' to quantum mechanics as well, despite the seemingly opposite a.s.sumptions on which the two theories are founded. It is clear that something still eludes us.

In most situations we can neglect quantum mechanics or general relativity (or both). The Moon is too large to be sensitive to minute quantum granularity, so we can forget the quanta when describing its movements. On the other hand, an atom is too light to curve s.p.a.ce to a significant degree, and when we describe it we can forget the curvature of s.p.a.ce. But there are situations where both curvature of s.p.a.ce and quantum granularity matter, and for these we do not yet have an established physical theory that works.

An example is the interior of black holes. Another is what happened to the universe during the Big Bang. In more general terms, we do not know how time and s.p.a.ce behave at very small scale. In all these instances, today's theories become confused and no longer tell us anything reasonable: quantum mechanics cannot deal with the curvature of s.p.a.cetime, and general relativity cannot account for quanta. This is the problem of quantum gravity.

The problem goes even deeper. Einstein understood that s.p.a.ce and time are manifestations of a physical field: the gravitational field. Bohr, Heisenberg and Dirac understood that physical fields have a quantum character: granular, probabilistic, manifesting through interactions. It follows that s.p.a.ce and time must also be quantum ent.i.ties possessing these strange properties.

What, then, is quantum s.p.a.ce? What is quantum time? This is the problem we call quantum gravity. A band of theoretical physicists scattered across five continents is laboriously seeking to solve the problem. Their objective is to find a theory, that is to say, a set of equations but, above all, a coherent vision of the world with which to resolve the current schizophrenia between quanta and gravity.

It isn't the first time that physics has found itself faced with two highly successful but apparently contradictory theories. The effort to synthesize has in the past been rewarded with great strides forward in our understanding of the world. Newton discovered universal gravity precisely by combining Galileo's physics of how things move on Earth with Kepler's physics of the heavens. Maxwell and Faraday found the equations of electromagnetism by bringing together what was known about electricity and what was known about magnetism. Einstein found special relativity in order to resolve the apparent conflict between Newton's mechanics and Maxwell's electromagnetism and then general relativity in order to resolve the resulting conflict between Newton's mechanics and his own special relativity.

Theoretical physicists are thus only too happy when they discover a conflict of this type: it is an extraordinary opportunity. The question to ask is: can we construct a conceptual structure compatible with what we have learned about the world with both theories?

To comprehend what quantum s.p.a.ce and quantum time are, we need once more to revise in depth the way we conceive things. We need to rethink the grammar of our understanding of the world. Just as happened with Anaximander, who understood that Earth flies in s.p.a.ce, and that 'up' and 'down' do not exist in the cosmos; or with Copernicus, who understood that we are moving across the heavens at great speed; or with Einstein, who understood that s.p.a.cetime squashes like a mollusc, and time pa.s.ses differently in different places ... once again, in seeking a coherent vision of the world in keeping with what we have learned about it, our ideas about the nature of reality have to change.

Figure 5.1 Matvei Brontejn.

The first to realize that our conceptual basis must change in order to understand quantum gravity was a romantic and legendary character: Matvei Brontejn, a young Russian who lived during the Stalin era and died tragically.