The Order of Time

9 TIME IS IGNORANCE

Do not ask

about the outcome of my days, or of yours,

Leuconoe—

it’s a secret, beyond us.

And don’t attempt abstruse calculations. (I, 11)

There is a time to be born and a time to die, a time to weep and a time to dance, a time to kill and a time to heal. A time to destroy and a time to build.⁷⁷ Up to this point, it has been a time to destroy time. Now it is time to rebuild the time that we experience: to look for its sources, to understand where it comes from.

If, in the elementary dynamic of the world, all the variables are equivalent, what is this thing that we humans call “time”? What is it that my watch measures? What is it that always runs forward, and never backward—and why? It may not be part of the elementary grammar of the world, but what is it?

There are so many things that are not part of the elementary grammar of the world and that simply “emerge” in some way. For example:

A cat is not part of the elementary ingredients of the universe. It is something complex that emerges, and repeats itself, in various parts of our planet.

A group of boys on a field decide to have a match. They form teams. This is how we used to do it: the two most enterprising would take turns choosing the players they wanted, having tossed a coin to see who would have first pick. At the end of this solemn procedure, there were two teams. Where were the teams before they were chosen? Nowhere. They emerged from the procedure.

Where do “high” and “low” come from—terms that are so familiar and yet are not in the elementary equations of the world? From the Earth that is so close to us and that attracts. “High” and “low” emerge in certain circumstances in the universe, as when there is a large mass nearby.

In the mountains, we see a valley covered by a sea of white clouds. The surface of the clouds gleams, immaculate. We start to walk toward the valley. The air becomes more humid, then less clear; the sky is no longer blue. We find ourselves in a fog. Where did the well-defined surface of the clouds go? It vanished. Its disappearance is gradual; there is no surface that separates the fog from the sparse air of the heights. Was it an illusion? No, it was a view from afar. Come to think of it, it’s like this with all surfaces. This dense marble table would look like a fog if I were shrunk to a small enough, atomic scale. Everything in the world becomes blurred when seen close up. Where exactly does the mountain end and where do the plains begin? Where does the savannah begin and the desert end? We cut the world into large slices. We think of it in terms of concepts that are meaningful for us, that emerge at a certain scale.

We see the sky turning around us every day, but we are the ones who are turning. Is the daily spectacle of a revolving universe “illusory”? No, it is real, but it doesn’t involve the cosmos alone. It involves our relation with the sun and the stars. We understand it by asking ourselves how we move. Cosmic movement emerges from the relation between the cosmos and ourselves.

In these examples, something that is real—a cat, a football team, high and low, the surface of clouds, the rotation of the cosmos—emerges from a world that at a much simpler level has no cats, teams, up or down, no surfaces of clouds, no revolving cosmos. . . . Time emerges from a world without time, in a way that has something in common with each of these examples.

The reconstruction of time begins here, in two little chapters—this one and the next—that are brief and technical. If you find them heavy going, skip them and go directly to chapter 11. From there, step by step, we will gradually reach more human things.

THERMAL TIME

In the frenzy of thermal molecular mingling, all the variables that can possibly vary do so continuously.

One, however, does not vary: the total amount of energy in any isolated system. Between energy and time there is a close bond. They form one of those characteristic couples of quantities that physicists call “conjugate,” such as position and momentum, or orientation and angular momentum. The two terms of these couples are tied to each other. On the one hand, knowing what the energy of a system may be⁷⁸—how it is linked to the other variables—is the same as knowing how time flows, because the equations of evolution in time follow from the form of its energy.⁷⁹ On the other, energy is conserved in time, hence it cannot vary, even when everything else varies. In its thermal agitation, a system⁸⁰ passes through all the configurations that have the same energy, but only these. The set of these configurations—which our blurred macroscopic vision does not distinguish—is the “(macroscopic) state of equilibrium”: a placid glass of hot water.

The usual way of interpreting the relation between time and state of equilibrium is to think that time is something absolute and objective; energy governs the time-evolution of a system; and the system in equilibrium mixes all configurations of equal energy. The conventional logic for interpreting this relation is therefore:

time → energy → macroscopic state⁸¹

That is: to define the macroscopic state, we first need to know the energy, and to define energy we first need to know what is time. In this logic, time comes first and is independent from the rest.

But there is another way of thinking about this same relationship: by reading it in reverse. That is, to observe that a macroscopic state, which is to say a blurred vision of the world, may be interpreted as a mingling that preserves an energy, and this in its turn generates a time. That is:

macroscopic state → energy → time⁸²

This observation opens up a new perspective: in an elementary physical system without any privileged variable that acts like “time”—where, in effect, all the variables are on the same level but we can have only a blurred vision of them described by macroscopic states. A generic macroscopic state determines a time.

I’ll repeat this point, because it is a key one: a macroscopic state (which ignores the details) chooses a particular variable that has some of the characteristics of time.

In other words, a time becomes determined simply as an effect of blurring. Boltzmann understood that the behavior of heat involves blurring, from the fact that inside a glass of water there is a myriad of microscopic variables that we do not see. The number of possible microscopic configurations for water is its entropy. But something further is also true: the blurring itself determines a particular variable—time.

In fundamental relativistic physics, where no variable plays a priori the role of time, we can reverse the relation between macroscopic state and evolution of time: it is not the evolution of time that determines the state, it is the state—the blurring—that determines a time.

Time that is determined in this way by a macroscopic state is called “thermal time.” In what sense may it be said to be a time? From a microscopic point of view, there is nothing special about it—it is a variable like any other. But from a macroscopic one, it has a crucial characteristic: among so many variables all at the same level, thermal time is the one with behavior that most closely resembles the variable we usually call “time,” because its relations with the macroscopic states are exactly those that we know from thermodynamics.

But it is not a universal time. It is determined by a macroscopic state, that is, by a blurring, by the incompleteness of a description. In the next chapter, I will discuss the origin of this blurring, but before I do, let’s take another step by bringing quantum mechanics into consideration.

QUANTUM TIME

Roger Penrose is among the most lucid of those scientists who have focused on space and time.⁸³ He reached the conclusion that the physics of relativity is not incompatible with our experience of the flowing of time but that it does not seem sufficient to account for it. He has suggested that what’s missing might be what happens in a quantum interaction.⁸⁴ Alain Connes, the great French mathematician, has pointed out the deep role of quantum interaction at the root of time.

When an interaction renders the position of a molecule concrete, the state of the molecule is altered. The same applies for its speed. If what materializes first is the speed and then the position, the state of the molecule changes in a different way than if the order of the two events were reversed. The order matters. If I measure the position of an electron first and then its speed, its state changes differently than if I were to measure its velocity first and then its position.

This is called the “noncommutativity” of the quantum variables, because position and speed “do not commute,” that is to say, they cannot exchange order with impunity. This noncommutativity is one of the characteristic phenomena of quantum mechanics. Noncommutativity determines an order and, consequently, a germ of temporality in the determination of two physical variables. To determine a physical variable is not an isolated act; it involves interaction. The effect of such interactions depends on their order, and this order is a primitive form of the temporal order.

Perhaps it is the very fact that the effect of these interactions depends on the order in which they take place that is at the root of the temporal order of the world. This is the fascinating idea suggested by Connes: the first germ of temporality in elementary quantum transitions lies in the fact that these interactions are naturally (partially) ordered.

Connes has provided a refined mathematical version of this idea: he has shown that a kind of temporal flow is implicitly defined by the noncommutativity of the physical variables. Due to this noncommutativity, the set of physical variables in a system defines a mathematical structure called “noncommutative von Neumann algebra,” and Connes has shown that these structures have within themselves an implicitly defined flow.⁸⁵

Surprisingly, there is an extremely close relation between Alain Connes’s flow for quantum systems and the thermal time that I have discussed above. Connes has shown that, in a quantum system, the thermal flows determined by different macroscopic states are equivalent, up to certain internal symmetries,⁸⁶ and that, together, they form precisely the Connes flow.⁸⁷ Put more simply: the time determined by macroscopic states and the time determined by quantum noncommutativity are aspects of the same phenomenon.

And it is this thermal and quantum time, I believe,⁸⁸ that is the variable that we call “time” in our real universe, where a time variable does not exist at the fundamental level.

The intrinsic quantum indeterminacy of things produces a blurring, like Boltzmann’s blurring, which ensures—contrary to what classic physics seemed to indicate—that the unpredictability of the world is maintained even if it were possible to measure everything that is measurable.

Both the sources of blurring—quantum indeterminacy, and the fact that physical systems are composed of zillions of molecules—are at the heart of time. Temporality is profoundly linked to blurring. The blurring is due to the fact that we are ignorant of the microscopic details of the world. The time of physics is, ultimately, the expression of our ignorance of the world. Time is ignorance.

Alain Connes has coauthored with two friends a short science fiction novel. Charlotte, the protagonist, manages to have for a moment a totality of information about the world, without blurring. She manages to “see” the world directly, beyond time:

I have had the unheard-of good fortune of experiencing a global vision of my being—not of a particular moment, but of my existence “as a whole.” I was able to compare its finite nature in space, against which no one protests, with its finite nature in time, which is instead the source of so much outrage.

And then returning to time:

I had the impression of losing all the infinite information generated by the quantum scene, and this loss was sufficient to drag me irresistibly into the river of time.

The emotion that results from this is an emotion of time:

This re-emergence of time seemed to me like an intrusion, a source of mental confusion, anguish, fear and alienation.⁸⁹

Our blurred and indeterminate image of reality determines a variable, thermal time that turns out to have certain peculiar properties which begin to resemble what we call “time”: it is in the correct relation with equilibrium states.

Thermal time is tied to thermodynamics, and hence to heat, but does not yet resemble time as we experience it, because it does not distinguish between the past and the future, has no direction, and lacks what we mean when we speak of its flow. We have not yet reached the time of our own experience.

The difference between the past and the future that is so important to us:

Where does that come from?