Einstein's Unfinished Revolution
Page 23
But, as I’ve already mentioned, pilot wave theory fails to satisfy another of our principles: Einstein’s principle of reciprocity. The pilot wave guides the particle, but the particle has no effect back on the wave. So we still have some distance to go.
The principle of sufficient reason advises us we can do better.
How shall we think of space and time in this new world of relations? Two chapters ago I drew a lesson from a survey of approaches to quantum foundations, which is that space and time cannot both be fundamental. Only one can be present at the deepest level of understanding; the other must be emergent and contingent. This seems ultimately to be forced on us by the nonlocality of entanglement, which leads to a tension between realist approaches to quantum mechanics and special relativity. The latter unifies space and time into spacetime, which the experimental tests of Bell’s restriction suggest is transcended in individual quantum processes. I would then like to suggest that the tension is resolved by making one of the pair space/time fundamental, while the other is an emergent and approximate description, ultimately a kind of illusion. For many reasons, some described here, some the subject of earlier books,1 I choose to focus on the hypothesis that time is fundamental, while space is emergent.
This is as far as principles take us. The next step is to frame hypotheses. I propose three hypotheses about what lies beyond spacetime and beyond the quantum:
Time, in the sense of causation, is fundamental. This means the process by which future events are produced from present events, called causation, is fundamental.
Time is irreversible. The process by which future events are created from present events can’t go backward. Once an event has happened, it can’t be made to un-happen.*
Space is emergent. There is no space, fundamentally. There are events and they cause other events, so there are causal relations. These events make up a network of relationships. Space arises as a coarse-grained and approximate description of the network of relationships between events.
This means that locality is emergent. Nonlocality must then also be emergent.
If locality is not absolute, if it is the contingent result of dynamics, it will have defects and exceptions. And indeed, this appears to be the case: how else are we to understand quantum nonlocality, particularly nonlocal entanglement? These, I would hypothesize, are remnants of the spaceless relations inherent in the primordial stage, before space emerges. Thus, by positing that space is emergent we gain a possibility of explaining quantum nonlocality as a consequence of defects which arise in that emergence.2
The combination of a fundamental time and an emergent space implies that there may be a fundamental simultaneity. At a deeper level, in which space disappears but time persists, a universal meaning can be given to the concept of now. If time is more fundamental than space, then during the primordial stage, in which space is dissolved into a network of relations, time is global and universal. Relationalism, in the form in which time is real and space is emergent, is the resolution of the conflict between realism and relativity.
Let’s give a name to this version of relationalism, which emphasizes the reality and irreversibility of time and the fundamentality of the flow of present moments. Let’s call it temporal relationalism. We can contrast it with eternalist relationalism, which investigates the hypothesis that space is fundamental, but time is emergent.
RELATIONAL HIDDEN VARIABLES
We thus seek a completion of quantum mechanics which is background independent and relational, and which is framed in a world where time is fundamental and space is emergent. If it involves hidden variables, these must express relations between particles. Thus, the hidden variables do not give us a more complete description of an individual electron; they must describe relations which hold between one electron and other electrons. We can call these relational hidden variables.
Indeed, what is more relational than the deepest and subtlest of the quantum mysteries, which is entanglement? A relational formulation of quantum physics will start by putting entanglement first. If, as we hypothesized, space is emergent, distance in space must be derivative of more fundamental relations. Perhaps this more fundamental relation, from which space emerges, is entanglement.*
The hidden variables in pilot wave theory are the trajectories of the particles. They are not relational; they do in fact just give us more information about each of the particles, individually. However, there is already a large dose of relationalism in pilot wave theory. This is inherent in the fact that for a system of more than one particle, the wave function lives not in ordinary space, but in the space of configurations of the total system, which consists of several particles. This is, as I explained in chapter 8, necessary to incorporate entanglement.
I first formulated the concept of a relational hidden variable theory, including the hypothesis that space is derivative of more fundamental relations, especially entanglement, early in my career. I wrote up3 a formulation of a relational hidden variable theory in 1983; this was the first of several such efforts.4
My 1983 theory was based on a simple idea. Suppose you have a system of particles in space. In an absolute description, you code in the location of each particle individually by giving coordinates in space. These coordinates are absolute; they refer to an observer outside the system—for Newton this was God himself. In a relational description you could use only the relative distances between each pair of particles. These no longer depend on reference to an observer outside the system.
There is a relative distance between every pair of particles. Hence, the relative distances can be represented as a table of numbers. The entry “10 down and 47 over” gives the distance between the 10th and 47th particles. Another name for such a table of numbers is a matrix. In my relational hidden variable theory, the hidden variables were such a matrix. My 1983 theory utilized a large matrix of complex numbers to describe a system of many particles living in a two-dimensional space. When the number of particles was large, the probabilities for the motion of the particles were approximately described by Schrödinger’s equation.
FIGURE 11. A matrix is a table of numbers, made up of rows of columns.
There are by now several proposals that go beyond the quantum by starting with pilot wave theory and trading in the wave function for a deeper structure described in terms of matrices. Relational hidden variables theories based on matrices have also been proposed by Stephen Adler5 and Artem Starodubtsev.6
A matrix assigns a number to every pair of particles. Another structure that does so is a graph, which is a simple structure built of points, connected by lines. Each pair of points is either connected by a line or not. We can assign a one to the pair if they are connected and a zero if they are not, and then we have a matrix representing the same structure.
Graphs and matrices are thus both ways to express the hypothesis that the fundamental beables underlying physics are a network of relations. These relations may express quantum entanglement and nonlocality.
There is no purer model of a system of relations than a graph or network. Interestingly enough, networks are ubiquitous in those approaches to quantum gravity which are in accord with the principle of background independence. These include loop quantum gravity, causal sets, and causal dynamical relations. This suggests two exciting deepenings of our hypotheses: First, space emerges from the fundamental network. Second, quantum physics arises from nonlocal interactions left over when space emerges.
However, networks fit uneasily into space, if “nearby” in the emergent space is to correspond with “nearby” in the network. The reason is simple: consider two points in the graph, each corresponding to a point in the emergent space. Suppose they are far away from each other in space and also far away on the graph. But now add a link to the graph directly connecting those two points. All of a sudden they are neighbors on the graph, but still far away from each other when considered in terms of the emergen
t space.
In our work with Fotini Markopoulou, we called such connections defects of locality. They look like narrow wormholes. We showed that they will be common in loop quantum gravity.7 This led us to another paper where we derived quantum mechanics from averaging over the nonlocal interactions which might arise from such defects of locality.8 A bit tongue in cheek, we called this “Quantum Theory from Quantum Gravity.”*
FIGURE 12. DISORDERED LOCALITY (A) A lattice of points, embedded in space, which we call local because points which are far away in terms of steps on the lattice are far away in the space it is embedded in. (B) By adding a new link that connects far away points, we disrupt locality because the connected points are still far away in space, but are only one step apart on the lattice.
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I MET RICHARD FEYNMAN ONLY a few times, but on two occasions he was kind enough to ask about my work. Each time he responded the same way. He listened carefully and then suggested that the idea I described to him wasn’t crazy enough to have a chance to work. What I believe he meant by that was that my idea didn’t go deep enough. In any case, that is how I feel about my earlier attempts to make a relational hidden variable theory based on matrices and networks. They solve the problem of giving a completion of quantum mechanics at a technical level, but in other aspects they come up short. One way to tell is that the Schrödinger equation only comes out as a prediction of the theory if you hammer out the imperfections and fine-tune the equations.
To go deeper into the relational idea, we can go back to Leibniz for inspiration. Leibniz sketched a purely relational view of the universe in a short book, The Monadology,9 written in 1714. Since we are interested only in getting inspiration from Leibniz, we don’t care to accurately reproduce his vision. We are free to creatively misinterpret his book. Here is one such loose reading of The Monadology.
We shall call the elements of a relational model of the universe nads because they are only partially in accord with Leibniz’s elements, which he called monads. Nads have two kinds of properties: intrinsic properties, which belong to each individual nad, and relational properties, which depend on several of the nads. A nadic universe may be pictured as a graph, with the relational properties represented by labels on links that connect pairs of nads.
It is not a coincidence that so far this picture accords with the description of the world given in loop quantum gravity. There, a state of the world is described by a graph with labels on it.
Each nad has a view of the universe, which summarizes its relations with the rest. One way to talk about the views is in terms of neighborhoods (or zones) of the graph. Let’s talk about the view of a nad called Sam. Consider the nads one step away in the graph from Sam: they are the first, or nearest, neighbors. The first neighborhood consists of Sam and her nearest neighbors, together with the relations they share, which are indicated on the links between them.
To construct Sam’s second neighborhood, add in the nads two steps away from her, and all their relations with each other and with their neighbors who are one step away (who are also included). And so on. These neighborhoods constitute Sam’s views of her universe.
We can compare Sam’s views to the views of another nad—let’s call him Sue. Sam and Sue have identical first and second neighborhoods, which is to say, we couldn’t tell them apart if we can only see that far.
But let us posit that our relational, nadic universe obeys Leibniz’s principle of the identity of indiscernibles. Then Sam’s and Sue’s neighborhoods must differ at some point; otherwise they would have identical views, which is forbidden by that principle. This implies there must be some number of steps at which the two neighborhoods differ. We call that number the distinction of Sue and Sam.
FIGURE 13. The first and second neighborhoods of Sam and Sue, defined by the connectivity of the graph they inhabit, are identical, but the third and higher neighborhoods distinguish them.
Leibniz posited that the actual universe is distinguished from many possible universes by “having as much perfection as possible.” If we strip this of its poetic or allegorical meaning, what Leibniz is doing is positing that there is some observable quantity which is larger in the real universe than in all the other possible universes. This is shockingly modern, as it anticipates a method for formulating laws of nature that was developed later and only came into fruition during the twentieth century. The quantity that is maximized, which Leibniz called “perfection,” we call an action.
Feynman liked to emphasize that a beautiful feature the laws of physics enjoy is that they can be formulated in several different ways. These seem, at first, to be very different, but when you know them better you come to understand that they are all equivalent to each other. I can illustrate this with Newton’s laws of motion and gravity. These describe the motion of the planets, moons, and other bodies in the solar system. One way to describe the laws is by specifying how the positions of these bodies change in time. This is usually done by setting their accelerations equal to the sum of the gravitational forces from the other bodies, divided by the masses.
But another way to specify the same laws is to delineate a set of quantities that are fixed, and don’t change as the planets move, such as their total energy. A third way, equivalent to the first two, is to say that the planets move in such a way that a certain quantity is made as large as possible. We call this the action;* Leibniz called it perfection.
Leibniz tells us what goes into the “perfection.” He defined a world with “as much perfection as possible” to be one having “as much variety as possible, but with the greatest order possible.”
What does Leibniz mean here by “variety”? I believe that Leibniz meant that the views of the different monads should differ as much as possible. So by maximizing perfection, Leibniz means we should maximize the variety of different views.
Inspired by this picture, Julian Barbour and I constructed numerical measures of the variety inherent in a system of relations.10 We noticed that as variety increases, less information is needed to pick out and distinguish each view from the others. That is, everything else being equal, we prefer worlds where any pair of nads has neighborhoods which differ at a small number of steps.
For Leibniz, sufficient reason had to be founded on a notion of maximal perfection.
And this [sufficient] reason can be found only in the fitness, or in the degrees of perfection, that these worlds possess. . . . This interconnection (or accommodation) of all created things to each other, and each to all the others, brings it about that each simple substance has relations that express all the others, and consequently, that each simple substance is a perpetual, living mirror of the universe.
He then reaches for a metaphor to describe this, and comes up with the different views of a city.
Just as the same city viewed from different directions appears entirely different, and, as it were, multiplied perspectively, in just the same way it happens that, because of the infinite multitude of simple substances, there are, as it were, just as many different universes, which are, nevertheless, only perspectives on a single one.11
This is indeed a metaphor that Jane Jacobs would have appreciated, as it captures a notion of urban diversity championed by her and embraced by philosophers of the city, such as Richard Florida, since.
This urban metaphor inspires a hypothesis about how space and locality break down. If you stand next to me and we both look out, by virtue of our proximity we have similar views of the rest of the universe. Our views cannot be identical, because we cannot coincide, by virtue of both the Pauli exclusion principle and the identity of indiscernibles. But the closer to each other we stand, the more similar are our views.
Because we are close to each other, we can interact easily, and indeed, the closer we stand the higher is the probability that we interact through an interchange of quanta such as photons. This is basically what we
mean when we say physical interactions are local.
But suppose we have this backward. What if we interact with high probability exactly because our views are similar? Suppose that the probability for us to interact increases with the increasing similarity of our views, and decreases if our views begin to differ.
If this is right, then the fundamental relation determining how often we interact is how similar our views are—and distance in space is derivative from that.
Now, for big, clunky things like ourselves, made up of vast numbers of atoms, this is as far as it goes. But consider what it takes for atoms to have similar views. Atoms have many fewer degrees of freedom, hence fewer relational properties. So atoms which are far away from each other in space may still have similar neighborhoods, just because there are vastly fewer configurations their local neighborhoods could take. This suggests that perhaps similar atoms, with the same constituents and similar surroundings, interact with each other just because they have similar views.
These interactions would be highly, highly nonlocal. But in my recent work, I have showed that this could be the basis of quantum physics.12
Consider a hydrogen atom in a water molecule dancing in the air in front of me. This has a first neighborhood consisting of an oxygen atom, and a second neighborhood consisting of the whole molecule. The same is true of every hydrogen atom in a water molecule everywhere in the universe. So I am going to trust my relational instincts and take the crazy step of positing that all these atoms are interacting with each other, just because their views are similar. More specifically, I will posit that the interactions act to increase the differences between these atoms’ views. This will go on until the system has maximized the variety of views the atoms have of the universe.