The Order of Time

8 DYNAMICS AS RELATION

Sooner or later

the exact measurement of our time

will resume—

and we’ll be on the ship that’s bound

for the bitterest shore. (II, 9)

How does one describe a world in which everything occurs but there is no time variable? In which there is no common time and no privileged direction in which change occurs?

In the simplest way, the same way that we had thought about the world until Newton convinced us all that a variable time was indispensable.

To describe the world, the time variable is not required. What is required are variables that actually describe it: quantities that we can perceive, observe, and eventually measure. The length of a road, the height of a tree, the temperature of a forehead, the weight of a piece of bread, the color of the sky, the number of stars in the celestial vault, the elasticity of a piece of bamboo, the speed of a train, the pressure of a hand on a shoulder, the pain of a loss, the position of the hands on a clock, the height of the sun in the sky . . . These are the terms in which we describe the world. Quantities and properties that we see continuously changing. In these changes there are regularities: a stone falls faster than a feather. Sun and moon circle after each other in the sky, passing by each other once a month. . . . Among these quantities there are some that we see changing regularly with respect to others: the number of days, the phases of the moon, the height of the sun on the horizon, the position of the hands of a clock. It is useful to employ these as points of reference: let’s meet three days after the next full moon, when the sun is at its highest in the sky. I’ll see you tomorrow, when the clock shows 4:35. If we find a sufficient number of variables that remain synchronized enough in relation to each other, it is convenient to use them in order to speak of when.

There is no need in any of this to choose a privileged variable and call it “time.” What we need, if we want to do science, is a theory that tells us how the variables change with respect to each other. That is to say, how one changes when others change. The fundamental theory of the world must be constructed in this way; it does not need a time variable: it needs to tell us only how the things that we see in the world vary with respect to each other. That is to say, what the relations may be between these variables.⁶⁷

The fundamental equations of quantum gravity are effectively formulated like this: they do not have a time variable, and they describe the world by indicating the possible relations between variable quantities.⁶⁸

In 1967, an equation accounting for quantum gravity was written for the first time without any time variable. This equation was discovered by two American physicists—Bryce DeWitt and John Wheeler—and today it’s known as the Wheeler–DeWitt equation.⁶⁹

At first no one could understand the significance of an equation without a time variable, perhaps not even Wheeler and DeWitt themselves. (Wheeler: “Explain time? Not without explaining existence! Explain existence? Not without explaining time! To uncover the deep and hidden connection between time and existence . . . is a task for the future.”⁷⁰) The issue was discussed at great length; there were conferences, debates; rivers of ink flowed.⁷¹ I think that the dust has now settled and things have become much clearer. There is nothing mysterious about the absence of time in the fundamental equation of quantum gravity. It is only the consequence of the fact that, at the fundamental level, no special variable exists.

The theory does not describe how things evolve in time. The theory describes how things change one in respect to the others,⁷² how things happen in the world in relation to each other. That’s all there is to it.

Bryce and John left us some years ago. I knew them both and had great admiration and respect for them. In my study at the university in Marseille, I have hanging on the wall a letter that John Wheeler wrote to me when he became aware of my first work on quantum gravity. Every so often I reread it with a mixture of pride and nostalgia. I would like to have asked him more, during the handful of meetings that we had together. The last time I went to see him in Princeton, we took a long walk together. He spoke to me with the soft voice of an old man: I could not make out much of what he said but did not dare to ask him too frequently to repeat what he was saying. Now he is no longer with us. I can’t question him anymore, or tell him what I think. I can no longer tell him that it seems to me that his ideas are correct, and that they have guided me throughout a lifetime of research. I can no longer tell him I believe that he was the first to come close to the heart of the mystery of quantum gravity. Because he is no longer here—here and now. This is time for us. Memory and nostalgia. The pain of absence.

But it isn’t absence that causes sorrow. It is affection and love. Without affection, without love, such absences would cause us no pain. For this reason, even the pain caused by absence is, in the end, something good and even beautiful, because it feeds on that which gives meaning to life.

Bryce, I met in London, on the first occasion that I went to seek out a group working on quantum gravity. I was a young novice, fascinated by this arcane subject that no one in Italy was working on; he was its grand guru. I had gone to Imperial College to meet up with Chris Isham, and when I arrived I was told that he was on the top-floor terrace. Sitting together at a small table up there, I found Chris Isham, Karel Kuchar, and Bryce DeWitt—the three main authors whose ideas I had been studying in recent years. I remember vividly the intense impression I had on seeing them there through the glass, calmly discussing among themselves. I did not dare to interrupt. They seemed to me like three great Zen masters exchanging unfathomable truths between enigmatic smiles.

They were probably only deciding where to go for dinner. Revisiting and reflecting on the episode, I realize that at the time they were younger than I am now. This, too, is time: a strange shifting of perspective. Shortly before he died, Bryce gave a long interview in Italy, subsequently published in a small book.⁷³ And it was only then I became aware that he’d followed my work much more closely, and with more sympathy, than I had ever suspected from our conversations together, in which he was more prone to express criticism than encouragement.

John and Bryce were my spiritual fathers. Thirsting, I found in their ideas fresh, clear water to drink. So, thank you, John; thanks, Bryce. As human beings, we live by emotions and thoughts. We exchange them when we are in the same place at the same time, talking to each other, looking into each other’s eyes, brushing against each other’s skin. We are nourished by this network of encounters and exchanges. But, in reality, we do not need to be in the same place and time to have such exchanges. Thoughts and emotions that create bonds of attachment between us have no difficulty in crossing seas and decades, sometimes even centuries, tied to thin sheets of paper or dancing between the microchips of a computer. We are part of a network that goes far beyond the few days of our lives and the few square meters that we tread. This book is also a part of that weave. . . .

But I have digressed and lost my thread. Nostalgia for John and Bryce has caused me to deviate from my path. All I had intended to say in this chapter is that they had discovered the extremely simple structure of the equation that describes the dynamics of the world. It describes possible events and the correlations between them, and nothing else.

It’s the elementary form of the mechanics of the world, and it does not need to mention “time.” The world without a time variable is not a complicated one. It’s a net of interconnected events, where the variables in play adhere to probabilistic rules that, incredibly, we know for a good part how to write. And it’s a clear world, windswept and full of beauty as the crests of mountains; aridly beautiful as the cracked lips of the adolescent you loved.

ELEMENTARY QUANTUM EVENTS AND SPIN NETWORKS

The equations of loop quantum gravity⁷⁴ on which I work are a modern version of the theory of Wheeler and DeWitt. There is no time variable in these equations.

The variables of the theory describe the fields that form matter, photons, electrons, other components of atoms and the gravitational field—all on the same level. Loop theory is not a “unified theory of everything.” It doesn’t even begin to claim that it’s the ultimate theory of science. It’s a theory made up of coherent but distinct parts. It seeks to be “only” a coherent description of the world as we understand it so far.

The fields manifest themselves in granular form: elementary particles, photons, and quanta of gravity—or rather “quanta of space.” These elementary grains do not exist immersed in space; rather, they themselves form that space. The spatiality of the world consists of the web of their interactions. They do not dwell in time: they interact incessantly with each other, and indeed exist only in terms of these incessant interactions. And this interaction is the happening of the world: it is the minimum elementary form of time that is neither directional nor linear. Nor does it have the curved and smooth geometry studied by Einstein. It is a reciprocal interaction in which quanta manifest themselves in the interaction, in relation to what they interact with.

The dynamic of these interactions is probabilistic. The probabilities that something will happen—given the occurrence of something else—can in principle be calculated with the equations of the theory.

We cannot draw a complete map, a complete geometry, of everything that happens in the world, because such happenings—including among them the passage of time—are always triggered only by an interaction with, and with respect to, a physical system involved in the interaction. The world is like a collection of interrelated points of view. To speak of the world “seen from outside” makes no sense, because there is no “outside” to the world.

Representation of the web of elementary grains of space (or spin network).

The elementary quanta of the gravitational field exist at the Planck scale. They are the elementary grains that weave the mobile fabric with which Einstein reinterpreted Newton’s absolute space and time. It is these, and their interactions, that determine the extension of space and the duration of time.

The relations of spatial adjacency tie the grains of space into webs. We call these “spin networks.” The name “spin” comes from the mathematics that describe the grains of space.⁷⁵ A ring in the spin network is called a “loop,” and these are the loops that give “loop theory” its name.

The webs, in turn, transform into each other in discrete leaps, described in the theory as structures called “spinfoam.”⁷⁶

The occurrence of these leaps draws the patterns that on a large scale appear to us like the smooth structure of spacetime. On a small scale, the theory describes a “quantum spacetime” that is fluctuating, probabilistic, and discrete. At this scale, there is only the frenzied swarming of quanta that appear and vanish.

Representation of spinfoam.

This is the world with which I seek daily to come to terms. An unusual world, but not a meaningless one.

In my research group in Marseille, for example, we are attempting to calculate the time needed for a black hole to explode when it passes through a quantum phase.

During the course of this phase, inside the black hole and within its immediate vicinity there is no longer a single and determinate spacetime. There is a quantum superposition of spin webs. Just as an electron can unfurl into a cloud of probabilities between the moment it is emitted and the moment it arrives on a screen, passing through more than one place, so the spacetime of the quantum collapse of a black hole passes through a phase in which time fluctuates violently, there is a quantum superposition of different times, and then, later, a return to a determined state after the explosion.

For this intermediate phase, where time is wholly indeterminate, we still have equations that tell us what happens. Equations without time.

This is the world described by loop theory.

Am I certain that this is the correct description of the world? I am not, but it is today the only coherent and complete way that I know of to think about the structure of spacetime without neglecting its quantum properties. Loop quantum gravity shows that it is possible to write a coherent theory without fundamental space and time—and that it can be used to make qualitative predictions.

In a theory of this kind, time and space are no longer containers or general forms of the world. They are approximations of a quantum dynamic that in itself knows neither space nor time. There are only events and relations. It is the world without time of elementary physics.