by Lee Smolin
We should be mindful of two lessons from the history of the original atomic hypothesis. The first is that progress on the first challenge—that of discovering the laws of atomic physics—didn’t begin to be made until we had experiments that could verify that atoms really existed and reveal to us some of their properties.
History also teaches us that the second challenge—deriving the bulk properties of the various phases of matter—may be easier to address than the first. Half a century before we began to make real progress on uncovering the laws of atomic physics, a few pioneers had already made substantial progress on the second challenge. The reason is that the behavior of matter in bulk turns out not to depend very much on the details of atomic physics. One needed to know only that there are atoms and that they interact by forces that are short-range (i.e., could only act over a short distance).
This lesson is taken to heart by some quantum gravity theorists who seek to derive the law that governs spacetime on a macroscopic scale, namely general relativity, from simple hypotheses about the atoms of spacetime. This direction was pioneered by Ted Jacobson,3 and it has succeeded to a significant degree. This makes it likely that the known laws of physics, which operate on scales we can observe—much larger than the fundamental Planck scale—don’t depend very much on the laws that govern the atoms of spacetime.
This is bad because it means that the known laws hold few clues which might help reveal the truly fundamental laws. Indeed, there are, it would seem, just two clues. The first has to do with how information flows through spacetime, and is the following: To derive general relativity from the properties of hypothetical atoms of spacetime, one must posit that there is a maximum rate at which information may flow through a surface in space. This rate of information flow cannot be greater than the area of that surface, when counted up in fundamental Planck units.* This is called the (weak4) holographic hypothesis.*
If this holographic hypothesis is fundamental, then it has to make sense to speak of a flow of information all the way down at the tiny scales where quantum gravity operates. But information is influence, as is expressed by defining it as the distinction that makes a difference. So a flow of information defines (or depends on) a causal structure. Thus the holographic hypothesis requires that we must have a causal structure to guide, or express, the flows of information. This is one reason to believe that causal structure is fundamental.
The second clue is that to derive general relativity following Jacobson’s argument, we have to keep track of the flows of energy through the same surfaces. This suggests that energy is a fundamental quantity that makes sense all the way down to the level of the fundamental events. The great insight of Jacobson is then to have realized that, most fundamentally, the equations of general relativity encode a relationship between flows of energy and flows of information, both flows being guided by the causal structure.
Because of the first clue, I favor the hypothesis that the history of the universe contains a set of events and their causal relations, i.e., that the universe is a causal set. But, because of the inverse problem, I do not believe the radical hypothesis that the only properties events enjoy are causal relations. I am willing to believe that the causal relations are the only relational properties needed, but I believe there must be further properties, which are intrinsic to the events. The second clue leads me to posit that among these intrinsic properties, events are endowed with energy, which flows between them, following the causal relations.
I then would propose that each event has a certain quantity of energy, and that energy is transmitted from past events to future events along the causal relations. An event’s energy is the sum of the energies received from the events in its immediate causal past. That energy is divided up and transmitted to the events in its immediate causal future. In this way the law of conservation of energy, according to which energy is never created or destroyed, is respected.
Special relativity tells us that energy is unified with momentum, so I would have momentum propagated from past events to future events as well. In collaboration with Marina Cortês, I invented a causal set model which incorporates flows of energy and momentum, which we call an energetic causal set.5
The history of the universe, according to an energetic causal set model, consists of events which are each the causes of future events, to which they transfer some energy and momentum. But there is no spacetime, fundamentally; there is just the discrete set of events connected by causal relations, with the events and the relations endowed with energy and momentum.
One early success of this approach was a solution to the inverse problem. At least in simple cases in which space and time each have one dimension, we were able to derive the emergence of spacetime directly from the energetic causal set models.
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IT IS TIME we talked about energy.
Each of the major physical theories, from Newtonian physics down through general relativity and on to quantum field theory, has equations of motion that tell how some entity changes in time. For Newton, that entity is the position of a particle, while for quantum field theory it is the value of a field at every point in space. It is highly significant that all these equations of motion share a common structure. There is a configuration variable—the positions of the particles or the values of the fields. Then there are certain additional, dynamical quantities, which are called so because they come into the laws that tell how the particles move around or how the fields oscillate. The most important of these are momentum and energy.
Each particle carries a certain quantity of energy and momentum. When two particles interact, they exchange some of their energy and momentum. One may gain a bit, while the other loses, so long as the total energy and total momentum are conserved.
The structure of these theories is always the same: there are two fundamental equations. The first tells how the positions of the particles change in time, in a way that depends on the particles’ momentum.* The second equation tells how the momentum changes in time, and this depends on the particles’ positions. So the two quantities, position and momentum, are intertwined; the change of one depends on the other. We say that two quantities, related in this way, are dual. Position and momentum are dual. So are the electric and magnetic fields.
I believe that the fact that this pattern of dual equations is universal in physics is a deep property of nature. It is also restricted to physics. Other sciences describe systems that change in time, such as computers or ecosystems or markets or organisms. They each have their equations. But in none of these cases do the equations have this dual structure involving configuration variables, momenta, and energy, the latter two of which are conserved in total. This is one reason I don’t think it’s very helpful to imagine that the physical universe is a computer.
The conservation of momentum is important for another reason. It explains the principle of inertia, which is the deepest principle of physics so far posited.
Why is there this duality, involving configuration and momentum variables? Why is the world such that energy and momentum are conserved? There is an old answer to these questions, which is based on a deep theorem of Emmy Noether, which she proved in 1915. It involves the notion of symmetry, which is a transformation that changes a system in some way that doesn’t change the laws of motion. Rotations are symmetries, as are translations in space and time; so long as the entire system is rotated or translated together, then the laws of motion are unaffected. Noether’s theorem states that for every symmetry in nature that is based on a transformation that varies continuously, there is a conserved quantity. Symmetry in space implies that momentum is conserved. Symmetry in time explains the conservation of energy.*
This suggests that space is fundamental, while energy and momentum are emergent properties of space, reflecting its symmetries. This is a standard view, but I believe the reverse is closer to the truth.
While No
ether’s theorem reflects a true insight, it cannot apply to a fundamental theory. This is because we require that the fundamental theory satisfy the principle of the identity of indiscernibles. But that principle implies that there are no symmetries in nature. Think of a body that is invariant under a rotation, such as a sphere or a cylinder. The fact that it is symmetrical means that it is unchanged by a rotation. That is, an observer cannot tell the difference between the body before and after it is rotated. But this is true because there are on the body circles of points, which are all identical to each other. Similarly, an infinite straight line is invariant under a translation along its length because under such a translation, each point is taken to another point with identical properties. In each case we see that the existence of a symmetry means there are distinct points with identical properties, which violates the principle.
Symmetries are properties of fixed backgrounds, and the occurrence of a symmetry in a theory is a clear sign that that theory is background dependent. A symmetry is an operation that translates or rotates the system we are studying, with respect to the background, which is left unchanged. Symmetries characterize a system that has been isolated from a larger universe, and arise from what is ignored in that isolation.
We have posited that the fundamental theory is background independent, which means there are no symmetries. This in turn means that we cannot regard energy and momentum, and their conservation, as emergent from the properties of space. But we still have to explain why energy and momentum play the ubiquitous role they do in the structure of the equations of physics.
Further, we have hypothesized that space is not present at the fundamental level in nature, but is emergent. So if we want energy and momentum to play a role in physics, there seems to be no alternative but to put them in at the beginning.
What we want is an inverse of Noether’s theorem, which assumes that energy and momentum and their conservation are fundamental, and tells us the conditions under which space may emerge as an approximate description of subsystems of the whole.
So we are left with a picture in which causal relations, energy, and momentum are fundamental. Energetic causal sets are a working out of this picture.
The energetic causal set models realize the principles and hypotheses of temporal relationalism that I introduced in the previous chapter within a concrete framework. In these principles, time, in the sense of the continual becoming of the present moment, is fundamental to nature. Indeed, our experience of time’s passage is the one thing we directly perceive about the world which is truly fundamental. All the rest, including the impression that there are unchanging laws, is approximate and emergent. This view, and the case for it, had been developed during a long collaboration with Roberto Mangabeira Unger. An important consequence is that the laws of nature, rather than being timeless, evolve in time. This reverses the belief, common among physicists, that time is not present in the most fundamental laws, but rather emerges from those laws. Instead, we argue that time, in the sense of the present moment and its passage, is fundamental, while the laws are emergent and subject to change.
Marina Cortês insisted that the laws at the most fundamental level must be irreversible, in two senses. First, the laws are not the same if you reverse the direction of time. If you take a video of a lawful process, you do not get another lawful process by playing it backward. This directly contradicts a widely held belief that the laws of nature are unchanged if you reverse the direction of time.
But all the known fundamental laws, including quantum mechanics, general relativity, and the standard model, are invariant under such a time reversal.* There must be more-fundamental laws, which are not reversible. This raises two challenges: First, can we invent candidates for an irreversible fundamental law? Second, might it happen that reversible laws emerge as an approximation to more fundamental irreversible laws? These were the questions which energetic causal set models were invented to address.
Cortês also insisted on a deeper sense in which a theory that takes events as primary is irreversible. An event is something that happens. As we stated above, once something happens it cannot un-happen. However, the effect of an event can be reversed. If an event changes A to B, it can be followed by an event that changes B back to A. But that makes a history with two events. Once an event has happened it is in the past, and that fact cannot be erased by a future event, even if that future event reverses the effect of the original event.
This thought led us to view the passage of time as a process by which new events are steadily created from present events. While we may give diverse meanings to the word “time,” we posited that the passage of time expresses an active process of creation and that this “activity of time” is the creation of novel events, each one after the other.
More specifically, we invented several models, for the purposes of making a concrete realization of our principles and hypotheses. In one model we studied, each event is created from two “parent” events, and then, in turn, becomes the parent of two “child” events.
At each stage in the process there is a vanguard of events, which have been created but have yet to have had all their children. These events make up what we call “the present,” as they are the events that will still influence the future.
This process of the continual becoming of events creates a history.
Once an event has had its full allotment of children, it may no longer play a direct role in creating the future, so we say it is in the past. Each past event has a causal past, consisting of those prior events that have directly or indirectly influenced it. Its causal future is the continually growing set of events it directly or indirectly influences. Thus, the past has the structure of a causal set.
We next added energy and momentum, making our model of a growing future an energetic causal set. Each event has a total energy and a total momentum, which are the sums of those of their parents. These are divided up and passed down to their children.
To complete this model, we must answer two questions. How does the process that creates new events out of present events, which we called the activity of time, choose which pair of present events will be the next parents of a novel event? Second, how do events distribute energy and momentum to their children? To answer these questions, we need to prescribe a rule for the creation of new events.
In choosing this rule we were guided by two of the principles I enunciated earlier. The theory should be background independent, which in this context means that the different events should be named, or labeled or distinguished, only by dynamically created structures. Moreover, these structures should not refer to the order in which the events were created. These requirements are satisfied if events are labeled or described only by the structure of their causal pasts.
This makes it natural to invoke the identity of indiscernibles as our second principle. If events are distinguished by their causal pasts, then the causal past of each event must be unique. The event creation rule should then ensure that each event it creates has a causal past different from all the others so far created.
In the models I studied with Cortês, we found two very interesting results. The first, already mentioned, is that the inverse problem appears to be solved, in that there emerges a spacetime into which the events and their causal relations can be mapped. We also found that the systems begin in a very time-asymmetric and disordered phase, which evolves to a phase that is ordered and approximately time symmetric.*
We thus learned an important lesson from the energetic causal set models, which is that time-reversible laws can emerge from more fundamental, irreversible laws. This contradicts the way most physicists think about irreversibility.
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WE BEGAN in the last chapter with five principles, which are all ways of expressing Leibniz’s principle of sufficient reason, and three hypotheses, which express the fundamental and irreversible character of time
and the contrasting, emergent, and contingent nature of space. The theory we seek, which would complete Einstein’s twin revolutions, I believe may be the consistent expression of all of these. But, before going all the way there, we introduced several models, which were not meant to be the complete theory, but rather explorations of some aspects it may have by applying only a subset of the principles.
The real ensemble formulation is a relational hidden variable theory. It is not a full application of the principles, as it is situated in a fixed background of time and space, but otherwise it takes the principle of the identity of indiscernibles extremely seriously when it postulates that two events, which have the same view of the universe, are to be identified. I then postulated that the reason two bodies interact more strongly when they are nearer in space is in fact that their views of the rest of the universe are similar. That is, I propose to explain the principle of locality as arising from a deeper principle of similarity of views. To ensure that the identity of indiscernibles is realized, we introduce a force between subsystems that seeks to increase their distinctiveness, or maximize the overall variety. This, as I described earlier, leads to a derivation of quantum mechanics.
Energetic causal sets are models of discrete or quantum universes that explore the hypotheses we made about space and time. In particular, they embody the idea that there is no background space or spacetime. Instead, they take to be fundamental an active, irreversible notion of time and causation, as well as energy and momentum. Spacetime, and space, are emergent and contingent.