Reality Is Not What It Seems Part 8

Figure 9.4 Possible prediction of the spectrum of background radiation, of loop quantum gravity (shown by the solid line) compared with the current experimental errors (as represented by the points). Courtesy of A. Ashtekar, I. Agullo and W. Nelson.

Traces of the great primordial heat must also be in the gravitational field itself. The gravitational field, too, that is to say, s.p.a.ce itself, must be tremulous like the surface of the sea. Therefore, a cosmic gravitational background radiation must also exist older even than the electromagnetic one, because the gravitational waves are disturbed less by matter than the electromagnetic ones and were able to travel undisturbed even when the universe was too dense to let the electromagnetic waves pa.s.s.

We have now observed gravitational waves directly, with the LIGO detector, formed by two arms of a few miles in length, at a right angle to each other, in which laser beams measure the distance between three fixed points. When a gravitational wave pa.s.ses, the s.p.a.ce lengthens and shortens imperceptibly, and the lasers reveal this minuscule variation.fn46 The gravitational waves observed were generated by an astrophysical event: colliding black holes. These are phenomena described by general relativity which do not involve quantum gravity. But a more ambitious experiment called LISA is at the stage of being evaluated and is capable of doing the same thing but on a much larger scale: by putting into orbit three satellites, not around the Earth but around the Sun, as if they were miniature planets tracking the Earth in its...o...b..t. The three satellites are connected by laser beams measuring the distance between them or, better still, the variations in the distances when a gravitational wave pa.s.ses. If LISA is launched, it should be able to see not only the gravitational waves produced by stars and black holes but also the diffuse background of primordial gravitational waves generated at a time close to the Big Bang. These waves should tell us about the quantum bounce.

In the subtle irregularities of s.p.a.ce, we should be able to find traces of events which took place 14 billion years ago, at the origin of our universe, and confirm our deductions on the nature of s.p.a.ce and time.

10. Quantum Black Holes

Black holes populate our universe in great number. They are regions in which s.p.a.ce is so curved as to collapse in on itself, and where time comes to a standstill. As mentioned, they form, for instance, when a star has burned up all of the available hydrogen and collapses.

Frequently, the collapsed star formed part of a pair of neighbouring stars and, in this case, the black hole and the surviving counterpart circle one around the other; the black hole sucks matter from the other star continuously (as in figure 10.1).

Astronomers have found many black holes with a size (that is, ma.s.s) of the order of our Sun (a bit larger, in fact). But there are also gigantic back holes. There is one of these at the centre of almost all of the galaxies, including our own.

Figure 10.1 Representation of a couple star/black hole. The star loses matter, which is partly absorbed by the black hole, partly projected by it in jets in the direction of its poles.

The black hole at the centre of our own galaxy is currently being studied in detail. It has a ma.s.s a million times greater than our Sun. Every so often, a star gets too close to this monster, is disintegrated by the gravitational distortion and swallowed by the cyclopean black hole, like a small fish swallowed by a whale. Imagine a monster the size of a million Suns, which swallows in an instant our own Sun and its miniature planets ...

There is a wonderful ongoing project to construct a network of radio antennae distributed across the Earth from pole to pole, with which astronomers will be able to achieve a resolution sufficient to 'see' the galactic black hole. What we expect to see is a small black disc surrounded by the light produced by the radiation of the matter falling in.

What enters a black hole does not come out again, at least if we neglect quantum theory. The surface of a black hole is like the present: it can be crossed only in one direction. From the future, there is no return. For a black hole, the past is the outside; the future is the inside. Seen from outside, a black hole is like a sphere which can be entered but out of which nothing can come. A rocket could stay positioned at a fixed distance from this sphere, which is called the horizon of the black hole. To do so it needs to keep its engines firing intensely, to resist the gravitational pull of the hole. The powerful gravity of the hole implies that time slows down for this rocket. If the rocket stays near enough to the horizon for one hour, and then moves away, it would then find that, outside, in the meantime, centuries have pa.s.sed. The closer the rocket stays to the horizon, the slower with respect to the outside time runs for it. Thus, travelling to the past is difficult, but travelling to the future is easy: we need only to get close to a black hole with a s.p.a.ceship, keep within its vicinity for a while, and then move away.

On the horizon itself, time stops: if we get extremely close to it and then move away after a few of our minutes, a million years might have elapsed in the rest of the universe.

The really surprising thing is that the properties of these strange objects, today commonly observed, were foreseen by Einstein's theory. Now, astronomers study these objects in s.p.a.ce, but until not long ago black holes were considered a scarcely credible and bizarre consequence of an outlandish theory. I remember my university professor introducing them as solutions to Einstein's equations, to which 'real objects were unlikely ever to correspond'. This is the stupendous capacity of theoretical physics to discover things before they are observed.

The black holes we observe are well described by Einstein's theory, and quantum mechanics is not needed to understand them. But there are two mysteries of black holes that do require quantum mechanics in order to be unravelled and, for each of these, loop theory offers a possible solution. One of these could also offer an opportunity to test the theory.

The first application of quantum gravity to black holes concerns a curious fact discovered by Stephen Hawking. Early in the 1970s he theoretically deduced that black holes are 'hot'. They behave like hot bodies: they emit heat. In doing so, they lose energy and hence ma.s.s (since energy and ma.s.s are the same thing), becoming progressively smaller. They 'evaporate'. This 'evaporation of black holes' is the most important discovery made by Hawking.

Objects are hot because their microscopic const.i.tuents move. A hot piece of iron, for example, is a piece of iron where the atoms vibrate very rapidly around their equilibrium position. Hot air is air in which molecules move faster than in cold air.

Figure 10.2 The surface of a black hole crossed by loops, that is to say, by links of the spin network that describe the state of the gravitational field. Each loop corresponds to a quantum area of the black hole's surface. John Baez.

What are the elementary 'atoms' that vibrate, making a black hole hot? Hawking left this problem unanswered. Loop theory provides a possible answer. The elementary atoms of a black hole that vibrate, and are thus responsible for its temperature, are the individual quanta of s.p.a.ce on its surface.

Thus, it is possible to understand the peculiar heat of black holes predicted by Hawking using loop theory: the heat is the result of the microscopic vibrations of the individual atoms of s.p.a.ce. These vibrate because in the world of quantum mechanics everything vibrates; nothing stays still. The impossibility of anything being entirely and continuously still in a place is at the heart of quantum mechanics. Black-hole heat is directly connected to loop quantum gravity's fluctuations of the atoms of s.p.a.ce. The precise position of the black hole's horizon is determined only in relation to these microscopic fluctuations of the gravitational field. Hence, in a certain sense, the horizon fluctuates like a hot body.

There is another way of understanding the origin of the heat of black holes. The quantum fluctuations generate a correlation between the interior and the exterior of a hole. (I will speak at length about correlations and temperature in Chapter 12). Quantum uncertainty across the horizon of the black hole generates fluctuations of the horizon's geometry. But fluctuations imply probability, and probability implies thermodynamics, and therefore temperature. Concealing from us a part of the universe, a black hole makes its quantum fluctuations detectable in the form of heat.

Figure 10.3 Stephen Hawking and Eugenio Bianchi. On the blackboard are the princ.i.p.al equations of loop quantum gravity which describe black holes.

It was a young Italian scientist, Eugenio Bianchi, today a professor in the United States, who completed an elegant calculation which shows how, starting from these ideas and from the basic equations of loop quantum gravity, it is possible to derive the formula for the heat of black holes foreseen by Hawking (figure 10.3).

The second application of loop quantum gravity to black-hole physics is more spectacular. Once collapsed, a star vanishes from external view: it is inside the black hole. But, inside the hole, what happens to it? What would you see if you let yourself fall into the hole?

At first, nothing in particular: you would cross the surface of the black hole without major injuries then you would plummet towards the centre, at ever greater speed. And then? General relativity predicts that everything is squashed at the centre into an infinitely small point of infinite density. But this is, once again, if we ignore quantum theory.

If we take quantum gravity into account, this prediction is no longer correct there is quantum repulsion the same repulsion that makes the universe bounce at the Big Bang. What we expect is that, on getting closer to the centre, the falling matter is slowed down by this quantum pressure, up to a very high but finite density. Matter gets squashed, but not all the way to an infinitely small point, because there is a limit to how small things can be. Quantum gravity generates a huge pressure that makes matter bounce out, precisely as a collapsing universe can bounce out into an expanding universe.

The bounce of a collapsing star can be very fast, if watched from down there. But remember time pa.s.ses much more slowly there than outside. Seen from the outside, the process of the bounce can take billions of years. After this time, we can see the black hole explode. In the end, basically, this is what a black hole is: a shortcut to the distant future.

Thus, quantum gravity might imply that black holes are not eternally stable objects, as cla.s.sical general relativity predicted, after all. They are, ultimately, unstable.

Seeing these black-hole explosions would be a spectacular confirmation for the theory. Very old black holes, such as those formed in the early universe, could be exploding today. Some recent calculations suggest that the signals of their explosion could be in the range of radio telescopes. It has even been suggested that certain mysterious radio pulses which radio astronomers have already measured, called Fast Radio Bursts, could be, precisely, signals generated by the explosion of primordial black holes. If this was confirmed, it would be fantastic: we would have a direct sign of a quantum gravitational phenomenon. Let's wait and see ...

11. The End of Infinity

When we take quantum gravity into account, the infinite compression of the universe into a single, infinitely small point predicted by general relativity at the Big Bang disappears. Quantum gravity is the discovery that no infinitely small point exists. There is a lower limit to the divisibility of s.p.a.ce. The universe cannot be smaller than the Planck scale, because nothing exists which is smaller than the Planck scale.

If we ignore quantum mechanics, we ignore the existence of this lower limit. The pathological situations predicted by general relativity, where the theory gives infinite quant.i.ties, are called singularities. Quantum gravity places a limit to infinity, and 'cures' the pathological singularities of general relativity.

The same happens at the centre of black holes: the singularity that cla.s.sic general relativity antic.i.p.ated disappears as soon as we take quantum gravity into account.

There is another case, of a different kind, in which quantum gravity places a limit to the infinite, and it regards forces such as electromagnetism. Quantum field theory, started by Dirac and completed in the 1950s by Feynman and his colleagues, describes these forces well but is full of mathematical absurdities. When we use it to compute physical processes, we often obtain results which are infinite, and mean nothing. They are called divergences. The divergences are then eliminated with calculations, using a baroque technical procedure which leads to finite final results. In practice, it works, and the numbers, in the end, come out right; they reproduce the experimental measurements. But why must the theory go via the infinite to arrive at reasonable numbers?

In the last years of his life, Dirac was very unhappy with the infinities in his theory and felt that, all things considered, his objective of truly understanding how things worked was not achieved. Dirac loved conceptual clarity, even if what was clarity to him was not always clarity to others. But infinities do not make for clarity.

But the infinities of quantum field theory follow from an a.s.sumption at the basis of the theory: the infinite divisibility of s.p.a.ce. For example, to calculate the probabilities of a process, we sum up as Feynman has taught us all of the ways in which the process could unfold, and these are infinite, because they can happen in any one of the infinite points of a spatial continuum. This is why the result can be infinite.

When quantum gravity is taken into account, these infinities also disappear. The reason is clear: s.p.a.ce is not infinitely divisible, there are no infinite points; there are no infinite things to add up. The granular discrete structure of s.p.a.ce resolves the difficulties of the quantum theory of fields, eliminating the infinities by which it is afflicted.

This is a tremendous result: on the one hand, taking quantum mechanics into account resolves the problems generated by the infinities of Einstein's theory of gravity, that is to say, the singularities. On the other, taking gravity into account solves the problems generated by quantum field theory, that is to say, the divergences. Far from being contradictory, as they at first seemed, the two theories each offer the solution to the problems posed by the other!

Putting a limit to infinity is a recurrent theme in modern physics. Special relativity may be summarized as the discovery that there exists a maximum velocity for all physical systems. Quantum mechanics can be summarized as the discovery that there exists a maximum of information for each physical system. The minimum length is the Planck length LP, the maximum velocity is the speed of light c, and the total information is determined by the Planck constant h. This is summarized in table 11.1.

The existence of these minimum and maximum values for length, velocity and action fixes a natural system of units. Instead of measuring speed in kilometres per hour, or in metres per second, we can measure it in fractions of the speed of light. We can fix the value 1 for the velocity c and write, for example, v = , for a body which is moving at half the speed of light. In the same way, we can posit by definition and measure length in multiples of Planck's length. And we can posit h = 1 and measure actions in multiples of Planck's constant. In this way, we have a natural system of fundamental unities from which the others follow. The unity of time is the time that light takes to cover the Planck length, and so on. The natural unities are commonly used in research on quantum gravity.

The identification of these three fundamental constants places a limit to what seemed to be the infinite possibilities of nature. It suggests that what we call infinite often is nothing more than something which we have not yet counted, or understood. I think this is true in general. 'Infinite', ultimately, is the name that we give to what we do not yet know. Nature appears to be telling us that there is nothing truly infinite.

Table 11.1 Fundamental limitations discovered by theoretical physics.

Physical Quant.i.ty Fundamental constant Theory Discovery Velocity c Special relativity A maximum velocity exists Information (actions) Quantum mechanics A minimum of information exists Length Lp Quantum gravity A minimum length exists There is another infinity which disorientates our thinking: the infinite spatial extension of the cosmos. But as I ill.u.s.trated in Chapter 3, Einstein has found the way of thinking of a finite cosmos without borders. Current measurements indicate that the size of the cosmos must be larger than 100 billion light years. This is the order of magnitude of the universe we have indirect access to. It is around 10120 times greater than the Planck length, a number of times which is given by a 1 followed by 120 zeroes. Between the Planck scale and the cosmological one, then, there is the mind-blowing separation of 120 orders of magnitude. Huge. Extraordinarily huge. But finite.

In this s.p.a.ce between the size of the minute quanta of s.p.a.ce, up to quarks, protons, atoms, chemical structures, mountains, stars, galaxies (each formed by one hundred billion stars), cl.u.s.ters of galaxies, and right up until the seemingly boundless visible universe of more than 100 billion galaxies unfolds the swarming complexity of our universe; a universe we know only in a few aspects. Immense. Finite.

The cosmological scale is reflected in the value of the cosmological constant , which enters into the basic equations of our theories. The fundamental theory contains, therefore, a very large number: the ratio between the cosmological constant and the Planck length. It is this large number that opens the way to the vast complexity of the world. But what we see and understand of the universe is not an infinity to drown in. It is a wide sea, but a finite one.

The book of Ecclesiasticus, or Sirach,fn47 opens with a stupendous question: Who can number the sand of the sea, and the drops of rain, and the days of eternity? Who can find out the height of heaven, and the breadth of the earth, and the deep, and wisdom?

Not much longer after these lines were composed, another great text was written, with an opening which still resounds: Some think, O King Hiero, that the grains of sand cannot be counted.

This is the opening of Psammites (The Sand Reckoner) by Archimedes, in which the greatest scientist of antiquity ... counts the grains of sand in the universe!

He does so in order to demonstrate that their number is large but finite, and can be determined. The numerical system of antiquity did not allow for dealing with very large numbers easily. In The Sand Reckoner, Archimedes develops a new system of numbering, similar to our exponentials, that makes it possible to deal with very large numbers, and shows its power by counting (certainly playfully) how many grains of sand there are, not just on the seash.o.r.es but in the entire universe.

The Sand Reckoner is playful, but profound. With a flight of fancy that seems to antic.i.p.ate the Enlightenment by millennia, Archimedes rebels against the form of knowledge that insists on there being mysteries which are intrinsically inaccessible to human thought. He does not claim to know the exact dimensions of the universe, or the precise number of grains of sand. It isn't the completeness of his knowledge that he is a.s.serting. On the contrary, he is explicit about the approximate and provisional nature of his estimates. He speaks about possible alternatives regarding the true size of the universe between which he does not make a definite choice. The point at stake here is not the presumption of knowing everything. It is the opposite: an awareness that yesterday's ignorance may have light shed on it today, and that today's might be illuminated tomorrow.

The central point is rebellion against the renunciation of the desire to know: a declaration of faith in the comprehensibility of the world, a proud retaliation to those who remain satisfied with their own ignorance, who call infinite that which we don't understand and delegate knowledge to elsewhere.

Centuries have pa.s.sed, and the text of Ecclesiasticus, along with the rest of the Bible can be found in countless homes, while Archimedes' text is read only by the few. Archimedes was slaughtered by the Romans during the sacking of Syracuse, the last proud remnant of Magna Grecia to fall under the Roman yoke, during the expansion of that future empire which would soon adopt Ecclesiasticus as one of the foundational texts of its official religion, a position which it was to occupy there for more than a thousand years. During that millennium, the calculations made by Archimedes languished in a state of incomprehensibility: no one was able to use, or even to understand them.

Near Archimedes' Syracuse there is one of the most beautiful sites in Italy, the theatre of Taormina, which looks out at the Mediterranean and upon Mount Etna, the smoking volcano. In Archimedes' time, the theatre was used to stage plays by Sophocles and Euripides. The Romans adapted it for gladiatorial combat, for the pleasure of watching gladiators die.

The sophisticated playfulness of The Sand Reckoner is perhaps not only about an audacious mathematical construction, or the virtuosity of one of the most extraordinary minds of antiquity. It is also a defiant cry of reason, which recognizes its own ignorance but refuses to delegate to others the source of knowledge. It is a small, reserved and powerfully intelligent manifesto against infinity against obscurantism.

Quantum gravity is one of the many lines that continue the quest of The Sand Reckoner. We are counting the grains of s.p.a.ce of which the cosmos is made. A vast cosmos, but a finite one.

The only truly infinite thing is our ignorance.

12. Information

We are approaching the conclusion of our journey. In the previous few chapters, I spoke about the concrete applications of quantum gravity: the description of what happened to the universe around the time of the Big Bang; the description of the properties of the heat of black holes and the suppression of infinity.

Before concluding, I would like to return to the theory, but looking at its future, and to speak about information: a spectre that is haunting theoretical physics, arousing enthusiasm and confusion.

This chapter is different from the preceding ones, where I spoke of ideas and theories not yet tested but well defined; here, I'm speaking of ideas still confused, badly in need of organization. If, dear reader, you have found the journey so far a little rough, then hold on tighter, because we're now flying between voids of air. If this chapter seems particularly opaque, it's not because your ideas are confused. It's because the one with the confused ideas is me.

Many scientists suspect today that the concept of 'information' may turn out to be a key for new advances in physics. Information is mentioned in the foundations of thermodynamics, the science of heat, the foundation of quantum mechanics and in other areas besides, with the word quite often used very imprecisely. I believe there is something important in this idea. I'll try to explain why, and to show what information has to do with quantum gravity.

Before anything else, what is information? The word 'information' is used in common parlance to mean a variety of different things, and this imprecision is a source of confusion in science as well. The scientific notion of information, however, was defined with clarity in 1948, by the American mathematician and engineer Claude Shannon, and is something very simple: information is the measure of the number of possible alternatives for something. For example, if I throw a die, it can land on one of six faces. When we've seen it fall on a particular one of these, we have an amount of information N = 6, because the possible alternatives are six in number. If I don't know which day of the year is your birthday; there are 365 distinct possibilities. If you tell me the date, I have the information N = 365. And so on.

Instead of the number of alternatives N, scientists measure information in terms of a quant.i.ty called S, for 'Shannon information'. S is defined as the logarithm in base 2 of N: S = log2 N. The advantage of using the logarithm is that the unit of measurement S = 1 corresponds to N = 2 (because 1 = log2 2), making the unit of information the minimum number of alternatives: the choice between two possibilities. This unit of measurement is called 'bit'. When I know at roulette that a red number has come up rather than a black, I have one bit of information; when I know that a red, even number has won, I have two bits of information; when an even red number 'manque' (eighteen or less, in roulette parlance) wins, I have three bits. Two bits of information correspond to four alternatives (red even, red uneven, black even, black uneven). Three bits of information correspond to eight alternatives. And so on.fn48 A key point is that information can be located somewhere. Imagine, for instance, that you have in your hand a ball which can be either black or white. Imagine that I also have a ball which can be either black or white. There are two possibilities on my part, and two on yours. The total number of possibilities is four (2 x 2): white-white; white-black; black-white and black-black. Now, suppose that for some reason we are certain that the two b.a.l.l.s are opposite in colour (for instance, because we have taken the b.a.l.l.s from a box that contained only one white and one black ball). The total number of alternatives is then only 2 (white-black or black-white), even if the alternatives are still two on my part and two on yours. Note that, in this situation, something peculiar happens: if you look at your ball, then you know the colour of mine. In this case, we say that the colours of the two b.a.l.l.s are correlated, that is to say, linked to one another. We say that my ball 'has information' about yours (as well as vice versa).

If you think about it, this is precisely what happens in life when we communicate: for example, when I phone you, I know that the phone causes the sounds on your side to be dependent on the sounds on mine. The sounds on both sides are linked, like the colour of the b.a.l.l.s.

The example is not chosen at random: Shannon, who invented the theory of information, worked for a telephone company, and was looking for a way to measure accurately how much a telephone line could 'carry'. But what does a telephone line carry? It carries information. It carries the capacity to distinguish between alternatives. For this reason, Shannon defined information.

Why is the notion of information useful, perhaps even fundamental, to understanding the world? For a subtle reason: because it measures the ability of one physical system to communicate with another physical system.

Let's return for a final time to the atoms of Democritus. Let's imagine a world formed of an interminable sea of atoms which bounce, attract and cling together, and of nothing else. Aren't we missing something?

Plato and Aristotle insisted on the fact that something was indeed missing; they thought that the form of things was this something extra that had to be added to the substance of which things were made in order to understand the world. For Plato, forms exist by themselves, in an ethereal ideal world of forms, a world of 'ideas'. The idea of a horse exists prior to and independently of any actual horse. For Plato, a real horse is nothing but a pale reflection of the idea of a horse. The atoms which make up the horse count for little: what counts is the 'horseness', the abstract form. Aristotle is a bit more realistic, but for him, too, the form cannot be reduced to the substance. In a statue, there is more than the stone of which it is made. This more, for Aristotle, is the form. This is the basis of the critique of Democritus's materialism in antiquity. It still remains a common critique of materialism.

But was Democritus really proposing that everything can be reduced to atoms? Let's look at it more closely. Democritus says that when atoms combine what counts is their form, their arrangement in the structure, as well as the way in which they combine. He gives the example of the letters of the alphabet: there are only twenty or so letters but, as he puts it, 'It is possible for them to combine in diverse modes, in order to produce comedies or tragedies, ridiculous stories or epic poems.'

There are more than just atoms in this idea: what counts is the way in which they are combined, one in relation to another. But what relevance can the way in which they are combined have, in a world in which there is nothing but other atoms?

If the atoms are also an alphabet, who is able to read the phrases written with this alphabet?

The answer is subtle: the way in which the atoms arrange themselves is correlated with the way other atoms arrange themselves. Therefore, a set of atoms can have information, in the technical, precise sense described above, about another set of atoms.

This, in the physical world, happens continuously and throughout, in every moment and in every place: the light which arrives at our eyes carries information about the objects which it has played across; the colour of the sea has information on the colour of the sky above it; a cell has information about the virus that is attacking it; a new living being has plenty of information because it is correlated with its parents, and with its species; and you, dear reader, when reading these lines, receive information about what I am thinking while writing them, that is to say, about what is happening in my mind at the moment in which I write this text. What occurs in the atoms of your brain is not any more independent from what is happening in the atoms of mine: we communicate.

The world isn't, then, just a network of colliding atoms: it is also a network of correlations between sets of atoms, a network of real reciprocal information between physical systems.

In all of this, there is nothing idealistic or spiritual; it's nothing but an application of Shannon's idea that alternatives can be counted. All this is as much a part of the world as the stones of the Dolomites, the buzzing of bees and the waves of the sea.

Once we have understood that this network of reciprocal information exists in the universe, it is natural to seek to use this treasure to describe the world. Let's start with an aspect of nature well understood since the end of the nineteenth century: heat. What is heat? What does it mean to say that something is hot? Why does a cup of scalding-hot tea cool itself down, rather than heating itself up further?

It was the Austrian scientist Ludwig Boltzmann, the founder of statistical mechanics, who first understood why.fn49 Heat is the random microscopic movement of molecules: when the tea is hotter, the movement of the molecules is more agitated. Why does it cool down? Boltzmann hazarded a splendid hypothesis: because the number of possible states of the molecules in hot tea and cold air is smaller than the number in cool tea and slightly warmer air. The combined state evolves from a situation where there are less possible states to a situation where there are more possible states. The tea can't warm itself up, because information cannot increase by itself.

I'll elaborate. The molecules of tea are extremely numerous and extremely small, and we don't know their precise movements. Therefore, we lack information. This lack of information or missing information can be computed. (Boltzmann did it: he computed the number of distinct states the molecules can be in. This number depends on the temperature.) If the tea cools, a little of its energy pa.s.ses into the surrounding air; therefore, the molecules of tea move more slowly and the molecules of air move more quickly. If you compute your missing information, you discover that it has increased. If, instead, tea absorbed heat from the colder air, then the missing information would be decreased. That is, we would know more. But information cannot fall from the sky. It cannot increase by itself, because what we don't know, we just don't know. Therefore, the tea cannot warm up by itself in contact with cold air. It sounds a bit magical, but it works: we can predict how heat behaves just on the basis of the observation that our information cannot increase for free!

Boltzmann was not taken seriously. At the age of fifty-six, in Duino, near Trieste, he committed suicide. Today, he is considered one of the geniuses of physics. His tomb is incised with his formula S = k log W which expresses (missing) information as the logarithm of the number of alternatives, Shannon's key idea. Boltzmann pointed out that this quant.i.ty coincides with the entropy used in thermodynamics. Entropy is 'missing information', that is, information with a minus sign. The total amount of entropy can only increase, because information can only diminish.fn50 Today, physicists commonly accept the idea that information can be used as a conceptual tool to throw light on the nature of heat. More audacious, but defended today by an increasing number of theorists, is the idea that the concept of information can be useful also to the mysterious aspects of quantum mechanics ill.u.s.trated in Chapter 5.

Remember that a key result of quantum mechanics is precisely the fact that information is finite. The number of alternative results that we can obtain measuring a physical systemfn51 is infinite in cla.s.sical mechanics; but, thanks to quantum theory, we have understood that, in reality, it is finite. Quantum mechanics can be understood as the discovery that information in nature is always finite.

In fact, the entire structure of quantum mechanics can be read and understood in terms of information, as follows. A physical system manifests itself only by interacting with another. The description of a physical system, then, is always given in relation to another physical system, the one with which it interacts. Any description of a system is therefore always a description of the information which a system has about another system, that is to say, the correlation between the two systems. The mysteries of quantum mechanics become less dense if interpreted in this way, as the description of the information that physical systems have about one another.

The description of a system, in the end, is nothing other than a way of summarizing all the past interactions with it, and using them to predict the effect of future interactions.

The entire formal structure of quantum mechanics can in large measure be expressed in two simple postulates:1 The relevant information in any physical system is finite.

You can always obtain new information on a physical system.

Here, the 'relevant information' is the information that we have about a given system as a consequence of our past interactions with it: information allowing us to predict what will be the result for us of future interactions with this system. The first postulate characterizes the granularity of quantum mechanics: the fact that a finite number of possibilities exists. The second characterizes its indeterminacy: the fact that there is always something unpredictable which allows us to obtain new information. When we acquire new information about a system, the total relevant information cannot grow indefinitely (because of the first postulate), and part of the previous information becomes irrelevant, that is to say, it no longer has any effect upon predictions of the future. In quantum mechanics when we interact with a system, we don't only learn something, we also 'cancel' a part of the relevant information about the system.fn52 The entire formal structure of quantum mechanics follows in large measure from these two simple postulates. Therefore, the theory lends itself in a surprising way to being expressed in terms of information.

The first to realize that the notion of information was fundamental to the understanding of quantum reality was John Wheeler, the father of quantum gravity. Wheeler coined the phrase 'It from bit' to express this idea, meaning that 'everything is information'.

Information reappears, then, in the context of quantum gravity. Remember: the area of any surface is determined by the spins of the loop which intersect this surface. These spins are discrete quant.i.ties, and each one contributes to the area.

A surface with a fixed area may be formed from these elementary quanta of area in many different ways, say, in a number of ways N. If you know the area of a surface but don't know exactly how its quanta of area are distributed, you have missing information about the surface. This is one of the ways of computing the heat of black holes: the quanta of area of a black hole enclosed in a surface of a certain area can be in N different possible distributions. It is like for the cup of tea, in which the molecules can move in N different possible ways. Thus we can a.s.sociate a quant.i.ty of missing information, that is to say, entropy, with a black hole.