Reality Is Not What It Seems: The Journey to Quantum Gravity

by Carlo Rovelli

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, or Maxwell’s or Dirac’s equations. Against expectations, however, all of its predictions have been confirmed.

The time has come to attempt some conclusions about what it is, precisely, that quantum mechanics tells us about the world. This 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.

Instead, quantum mechanics tells us that between five and six centimeters 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.* 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.

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.

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,”2 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.

The theory does not describe things as they “are”: it describes how things “occur,” and how they “interact with each other.”

Reality is reduced to interaction. Reality is reduced to relation.3

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 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.

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 to the notion of thing.

To summarize, quantum mechanics is the discovery of three features of the world: — Granularity (figure 4.7). The information in the state of a system is finite, and limited by Plank’s constant. Figure 4.7 Light is a wave on a field, but it has also a granular structure. — 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.

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 passed, 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 nobody really understands quantum mechanics.”

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?

I think that the obscurity of the theory is not the fault of quantum mechanics but is rather due to the limited capacity of our imagination.

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 spacetime curves and is described by Einstein’s equations.

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 space to a significant degree, and when we describe it we can forget the curvature of space. But there are situations where both curvature of space and quantum granularity matter, and for these we do not yet have an established physical theory that works.

In all these instances, today’s theories become confused and no longer tell us anything reasonable: quantum mechanics cannot deal with the curvature of spacetime, and general relativity cannot account for quanta. This is the problem of quantum gravity.

What, then, is quantum space? What is quantum time? This is the problem we call “quantum gravity.”

There is an intuitive way of understanding what happens. Suppose we want to observe a very, very, very small region of space. To do this, we need to place something in this area, to mark the point that we wish to consider. Say we place a particle there. Heisenberg had understood that you can’t locate a particle at a point in space for long: it soon escapes. The smaller the region where we try to locate a particle, the greater the velocity at which it escapes. (This is Heisenberg’s uncertainty principle.) If the particle escapes at great speed, it has a great deal of energy. Now let us take Einstein’s theory into account. Energy makes space curve. A lot of energy means that space will curve a great deal. A lot of energy in a small region results in curving space so much that it collapses into a black hole, like a collapsing star. But if a particle plummets into a black hole, I can no longer see it. I can no longer use it as a reference point for a region of space. I can’t manage to measure arbitrarily small regions of space, because if I try to do this, these regions disappear inside a black hole.

The result is general: quantum mechanics and general relativity, taken together, imply that there is a limit to the divisibility of space.

It is at this extremely minute scale that quantum gravity manifests itself.

Having absorbed Bronštejn’s lesson that quantum properties of the gravitational field imply a modification of the notion of space at a small scale, Wheeler looked for novel ideas to help conceive of this quantum space. He imagined it as a cloud of superimposed geometries, just as we can think of a quantum electron as a cloud of positions.

Imagine that you are looking at the sea from a great height: you perceive a vast expanse of it, a flat cerulean table. Now you descend and look at it more closely. You begin to make out the great waves swollen by the wind. You descend farther, and you see that the waves break up, and that the surface of the sea is a turbulent frothing. This is what space is like, as imagined by Wheeler.* On our scale space is smooth. If we move down to smaller and smaller regions of space, we reach a minute scale, called the “Planck scale,” where space shatters and foams.

The solutions had a curious peculiarity: they depended on closed lines in space. A closed line is a “loop.” Smolin and Jacobson could write a solution to the Wheeler-DeWitt equation for every loop: for every line closed on itself. What did this mean? The first works of what will later become known as “loop quantum gravity” emerge from these discussions, as the meaning of these solutions of the Wheeler-DeWitt equation gradually clarify. Upon these solutions, little by little, a coherent theory begins to be erected, inheriting the name “loop theory,” from the first solutions studied.

What is slowly being erected is the theory now known as “loop theory,” or “loop quantum gravity”: the theory to which the following chapters are devoted. It is not the only direction explored in the search for a quantum theory of gravity, but it is the one I consider the most promising.*

Remember Faraday’s lines? The lines that carry the electric force and that, in Faraday’s vision, fill space? The lines from which the concept of “field” originates? Well, the closed lines that appear in the solutions of the Wheeler-DeWitt equation are Faraday lines of the gravitational field. But two new ingredients are now added to Faraday’s ideas. The first is that we are dealing with quantum theory. In quantum theory, everything is discrete. This implies that the infinitely fine, continuous spiderweb of Faraday’s lines now becomes similar to a real spiderweb, with a finite number of distinct threads. Every single line determining a solution of the Wheeler-DeWitt equation describes one of the threads of this web. The second new aspect, the crucial one, is that we are speaking of gravity and therefore, as Einstein understood, we are not speaking of fields immersed in space, but of the very structure of space itself. Faraday’s lines of the quantum gravitational field are the threads with which space is woven.

The key technical result that clarifies the physical meaning of the graphs is the calculation of the spectra of volume and of area.

Take any region of space. For example, the room in which you are reading this, if you are in a room. How big is this room? The size of the space of the room is measured by its volume. Volume is a geometrical quantity that depends on the geometry of space, but the geometry of space—as Einstein understood, and I recounted in chapter 3—is the gravitational field. Volume is therefore a property of the gravitational field, expressing “how much gravitational field” there is between the walls of the room. But the gravitational field is a physical quantity and, like all physical quantities, is subject to the laws of quantum mechanics. In particular, like all physical quantities, volume may not assume arbitrary values, but only certain particular ones, as I described in chapter 4. The list of all possible values is called, recall, the “spectrum.” Hence, there should exist a “spectrum of the volume” (figure 6.2).

The calculation took time, first to formulate and then to complete, and made us suffer.

It was completed in the mid-1990s, and the answer, as expected (Feynman used to say that we should never do a calculation without first knowing the result) is that the spectrum of the volume is discrete.

A link is an individual quantum of a Faraday line. Now we can understand what it represents: if you imagine two nodes as two small “regions of space,” these two regions will be separated by a small surface. The size of this surface is its area. The second quantity, after the volume, which characterizes the quantum webs of space, is the area associated with each line.*

The area, just as in the case of the volume, is a physical variable and has a spectrum that may be calculated using Dirac’s equation.* Area is not continuous; it is granular. There is no such thing as an arbitrarily small area.

When we say that the volume of a room is, for example, one hundred cubic meters, we are in effect counting the grains of space—the “quanta of the gravitational field”—that it contains.

The idea that measuring length, area, and volume is a question of counting individual elements had been proposed in the nineteenth century by Riemann himself. The mathematician who developed the theory of continuous curved mathematical spaces was already aware that a discrete physical space is ultimately more reasonable than a continuous one.

To summarize, the theory of loop quantum gravity, or “loop theory,” combines general relativity with quantum mechanics in a rather conservative way, because it does not employ any other hypotheses apart from those of the two theories themselves, suitably rewritten to render them compatible. But the consequences are radical.

General relativity taught us that space is something dynamic, like the electromagnetic field: an immense, mobile mollusk in which we are immersed, which stretches and bends. Quantum mechanics teaches us that every field of this sort is “made of quanta,” has a fine, granular structure. It follows that physical space, being a field, is “made of quanta” as well. The same granular structure characterizing the other quantum fields also characterizes the quantum gravitational field, and therefore space. We expect space to be granular. We expect “qu...

The central prediction of loop theory is therefore that space is not a continuum, it is not divisible ad infinitum, it is formed of “atoms of space,” a billion billion times smaller than the smallest of atomic nuclei.

The calculation of the quantum spectra of geometric quantities indicates that the answer is negative: arbitrarily small chunks of space do not exist. There is a lower limit to the divisibility of space. It is at a very small scale indeed, but it is there.

The crucial difference between photons, the quanta of the electromagnetic field, and the nodes of the graph, the “quanta of gravity,” is that photons exist in space, whereas the quanta of gravity constitute space itself.

The quanta of gravity, that is, are not in space; they are themselves space.

Physical space is the fabric resulting from the ceaseless swarming of this web of relations. The lines themselves are nowhere—they are not in a place but rather create places through their interactions. Space is created by the interaction of individual quanta of gravity.

Space as an amorphous container of things disappears from physics with quantum gravity. Things (the quanta) do not inhabit space; they dwell one over the other, and space is the fabric of their neighboring relations. As we abandon the idea of space as an inert container, similarly we must abandon the idea of time as an inert flow, along which reality unfurls. Just as the idea of the space continuum containing things disappears, so too does the idea of a flowing continuum “time,” during the course of which phenomena happen.

In the same sense, time no longer exists in the fundamental theory: the quanta of gravity do not evolve in time. Time just counts their interactions.

The watches that we usually wear on our wrists, or have on our mobile phones, are not sufficiently precise to allow us to verify this fact, but in physics laboratories all over the world, there are timepieces precise enough to demonstrate the discrepancy that occurs: the watch left on the floor is slow when compared to the one that has been raised above it.

Because time does not pass in the same way everywhere in the world. In some places it flows more quickly, in others more slowly. The closer you get to Earth, where gravity is more intense, the slower time passes.*

We must not think of time as if there were a great cosmic clock that marks the life of the universe. We have known for more than a century that we must think of time instead as a localized phenomenon: every object in the universe has its own time, running at a pace determined by the local gravitational field.

But even this notion of a localized time no longer works when we take the quantum nature of the gravitational field into account. Quantum events are no longer ordered by the passage of time, at the Planck scale. Time, in a sense, ceases to exist. What does it mean to say that time does not exist?

It’s a fine story, but on reflection, it leaves us perplexed—and this perplexity goes to the heart of the problem of time. How could Galileo know that his own individual pulse-beats all lasted for the same amount of time?* Not many years after Galileo, doctors began to measure their patients’ pulses by using a watch—which is nothing, after all, but a pendulum. So we use the beats to assure ourselves that the pendulum is regular, and then the pendulum to ascertain the regularity of the pulse-beats. Is this not somewhat circular? What does it mean? It means that we, in reality, never measure time itself; we always measure the physical variables A, B, C . . . (oscillations, beats, and many other things) and compare one variable with another, that is to say, we measure the functions A(B), B(C), C(A), and so on. We can count how many beats for each oscillation; how many oscillations for every tick of my stopwatch; how many ticks of my stopwatch between intervals of the clock on the bell tower. . . .

In other words, the existence of the variable time is a useful assumption, not the result of an observation.

If we want to understand the world widely, if we want to understand how it functions in the less familiar situations where quantum gravity matters, we need to abandon this schema. The idea of a time t that flows by itself, and in relation to which all things evolve, is no longer a useful one. The world is not described by equations of evolution in time t. What we must do is simply enumerate the variables A, B, C . . . that we actually observe, and write equations expressing relations between these variables, and nothing else: that is, equations for the relations A(B), B(C), C(A) . . . that we observe, and not for the functions A(t), B(t), C(t) . . . , which we do not observe.