Einstein's Unfinished Revolution

by Lee Smolin

  It would be extremely interesting if there were ways to reproduce the successes of pilot wave theory that had just one class of beables and not two. Waves or particles, but not both. Or something else entirely.

  As a first try, we can ask what happens if we start with pilot wave theory and drop either the waves or the particles.

  If we drop the particles, we no longer solve the measurement problem—unless we radically alter the behavior of the wave by hypothesizing spontaneous collapse. So dropping the particles leads back to either collapse models or the Many Worlds Interpretation.

  We next try to drop the wave function, but keep the particles. What then is going to guide the particles? How are we to explain interference if we have only particles? Might we, for example, recover the wave function’s guidance by giving the particles strange new properties?

  Several physicists and mathematicians have tried to invent a theory of beables with just the particles, but none have succeeded. This is a long story, with some fascinating ins and outs, but the conclusion is simple: the wave function appears to capture an essential aspect of reality.13 The closest to success I know of is an approach by the mathematician Edward Nelson called stochastic quantum mechanics. For many years I thought this was the right way, but then I understood it requires a large amount of fine-tuning to avoid instabilities.

  This conclusion is upheld by a recent analysis by three specialists in quantum information theory, Matthew Pusey, Jonathan Barrett, and Terry Rudolph, who gave a new argument to the effect that the quantum state cannot be merely a representation of information an observer has about a system. It must be physically real, or represent something real.14 So we seem to have only two choices: keep the wave function itself as a beable, as it is in pilot wave theory and collapse models, or find another beable that captures, in some different form, the physical reality which the wave function represents.

  FOURTEEN

  First, Principles

  I went into physics hoping to contribute to solving the two great questions Einstein posed in his autobiographical notes: uniting quantum physics with spacetime and making sense of quantum physics.

  Despite the efforts of many brilliant people over the more than half a century since Einstein wrote his autobiographical notes, these two problems remain unsolved. It is worth taking some time to ask why.

  This question has been on my mind. Lately I find myself wondering if we have been going about completing Einstein’s twin revolutions all wrong. We invent theories, such as loop quantum gravity, string theory, pilot wave theory, and others, but these do not go deep enough. Theories like these are models, which embody our ideas about nature, but they are not the deepest or purest expressions of those ideas.

  Models exemplify ideas, but often in a simplified form, which allows the ideas’ essential features and implications to shine through. The game Monopoly is a model of capitalism. Nonscientists often fail to appreciate how useful models can be—exactly because they are incomplete and leave things out—when one is in the stage of exploring the implications of an idea.

  Ideas about nature are most purely expressed as either hypotheses or principles. A hypothesis is a simple assertion about nature, which is either true or false. “Matter is not infinitely divisible because it is made of atoms” is a hypothesis. So is “Light is a wave traveling through the electric and magnetic fields.” Both of these hypotheses turned out to be true, but the history of science is littered with those that proved false.

  A principle is a general requirement that restricts the form that a law of nature can take. “It is impossible to do any experiment that can determine an absolute sense of rest, or measure an absolute velocity” is a principle.

  Einstein knew what he was doing when he introduced special relativity: he began his 1905 paper with two principles and deduced consequences directly from them. It is worth noting that the idea of unifying space and time into a single entity called spacetime was not part of Einstein’s original conception of relativity. The idea of spacetime was introduced two years later by his teacher Hermann Minkowski as a model which exemplified Einstein’s principles.

  The problem with skipping the stage of principles and hypotheses and going right to models is that we can lose our way. It’s easy to get trapped in a microscopic focus while trying to work out the details of those models. As Feynman once told me, “Make every question you ask in research a question about nature. Otherwise you can waste your life in working out the minutiae of theories that most likely will never have anything to do with nature.” Even worse, we get caught up in petty competitions and academic turf battles between the adherents of different models.

  Einstein expressed this lesson by insisting that we distinguish two kinds of theories. Principle theories are those that embody general principles. They restrict what is possible, but they don’t suffice for the details. Those are supplied by the second kind of theory, which he called constitutive theories. These describe particular particles or specific forces that nature may or may not contain. Special relativity and thermodynamics are principle theories. Dirac’s theory of the electron and Maxwell’s theory of electromagnetism are constitutive theories.

  So, my conclusion is that we need to back off from our models, postpone conjectures about constituents, and begin again by talking about principles.

  Our strategy will then be to proceed to our goal of inventing a new fundamental theory in four steps: first, principles; second, hypotheses (which must satisfy the principles); third, models (which illustrate partial implications of the principles and hypotheses); then last, complete theories. Putting principles before theories raises an interesting question: Where do you find a language to state the principles, and a context to motivate and critique them? You don’t want to use the language of existing theories because the whole point of the exercise is to get beyond them. Einstein would never have invented general relativity had he restricted himself to reasoning within the language of Newtonian physics.

  Mathematics can sometimes provide new ideas and structures, and so is often a help. But new mathematics is usually not enough to invent new physics; otherwise Bernhard Riemann or William Kingdon Clifford would have invented general relativity. This is where a knowledge of philosophy can be the essential element, because a person with a philosophical education has in their toolbox a plethora of ideas and methods coming from the whole history of human beings’ attempting to think about the fundamentals of our description of the world. And when it comes to basic questions like the nature of space and time, that history is rich with useful arguments and strategies to be tried out. So Einstein was not alone when he faced the need for new notions of space and time. It was as if he carried Galileo, Newton, Leibniz, Kant, and Mach in his back pocket and was able to converse with them and benefit from their insights. Similarly, a good knowledge of Plato, Kant, and others gave Heisenberg a language to go beyond Newtonian particles.

  The twentieth century saw a flowering of philosophy in physics, which has further enriched the storehouse of useful ideas and arguments. Philosophy is indeed a living tradition, and if there was a time when philosophers of physics lagged behind in technical mastery of physics, that time is long over. So I will not apologize for going both to the sages of the past and to our contemporary philosophers to find language, contexts, and ideas to frame my search for new principles of physics.

  Starting with principles has an immediate consequence, which is that we realize that quantum gravity and quantum foundations are different sides of a single problem. When physicists try to solve quantum gravity without regard to the problems of quantum foundations, and vice versa, we are taking the wrong approach. These two problems are deeply related. One reason is that because of quantum nonlocality, going beyond quantum mechanics means going beyond spacetime.

  So I will proceed by putting forward principles which combine quantum phenomena with spacetime. After we have a good set of principl
es, the next step will be to frame hypotheses about how they are realized.

  Our aim is to combine quantum physics and spacetime at the level of fundamental principles. I believe the right principles to shape this unification are the following:

  PRINCIPLES FOR FUNDAMENTAL PHYSICS

  1. Background independence.

  A physical theory should not depend on structures which are fixed and which do not evolve dynamically in interaction with other quantities. This is a key concept, which takes some unpacking.

  All physical theories to date depend on structures which are fixed in time and have no prior justification; they are simply assumed and imposed. One example is the geometry of space, in all theories prior to general relativity. In Newtonian physics, the geometry of space is simply fixed to be Euclidean three-dimensional geometry. It’s arbitrary; it doesn’t change in time, it can’t be influenced by anything. Hence it is not subject to dynamical law.

  In Newton’s time, Euclid’s was the only geometry known, so he had no alternative and didn’t need to seek a justification for choosing it. But in the nineteenth century, Carl Friedrich Gauss, Nicholas Lobachevsky, and Riemann discovered an infinitude of alternate geometries. Any fundamental theory that comes after their work must justify the choice it makes for the geometry of space. The principle of background independence requires that the choice is made not by the theorist, but by the theory, dynamically, as a part of solving the laws of physics.

  Non-dynamical, fixed structures define a frozen background against which the system we are interested in evolves. I would maintain that these frozen structures represent objects outside the system we are modeling, which influence the system but do not themselves change. (Or whose changes are too slow to be noticed.) Hence these fixed background structures are evidence that the theory in question is incomplete.

  It follows that any theory with fixed external structures can be improved if the external elements can be unfrozen, made dynamical, and brought inside the circle of mutually interacting physical degrees of freedom. This was the strategy that led Einstein to general relativity. The geometry of space and time is frozen in Newtonian physics, and it is also frozen in special relativity. In these theories, the spacetime geometry provides an absolute and fixed background against which measurements are defined. General relativity unfreezes geometry, making it dynamical.

  This is turning out to be a multistage process, because our theories have layers of frozen elements, which were laid down, like layers of sedimentation, during the long and complex history of our subject. General relativity unfreezes some aspects of geometry, but deeper structures, such as dimension and structures needed to define the continuous numbers or define a rate of change, remain frozen. So general relativity, beautiful as it is, cannot be the end of our search, and will require further completion.

  Each step extends the range of the theory. It follows that the only complete theory of physics must be a cosmological theory, for the universe is the only system which has nothing outside of it. A theory of the whole universe must then be very different from theories of parts of the universe. It must have no fixed, frozen, timeless elements, as these refer to things outside the system described by the theory. It must be fully background independent.

  This recognition that a cosmological theory cannot be achieved by just scaling up our current theories, but must be a radically new kind of theory, is the most important lesson learned so far in the search for a completion of Einstein’s twin revolutions.*

  It follows that quantum mechanics cannot be a theory of the whole universe because it too has fixed elements. These include the observables of the system and various relations they have, as well as the structure that gives rise to probabilities.*

  This implies that there is no wave function of the universe, because there is no observer outside the universe who could measure it. The quantum state is, and must remain, a description of part of the universe.

  We then seek to complete quantum theory by eliminating background structures. We do this by exposing and then unfreezing the background and giving it dynamics. In other words, rather than quantizing gravity we seek to gravitize the quantum. We mean by that the process of identifying and unfreezing those aspects of quantum theory which are arbitrary and fixed, making them subject to dynamical laws. Turning this around, we hope to understand the challenging features of quantum physics as consequences of separating the universe into two parts: the system we observe, and the rest, containing the observer and their measuring instruments.

  Closely related to background independence is another key idea: that the observables of physical theories should describe relationships.

  Leibniz, Mach, and Einstein taught us to distinguish absolute notions of space and time from relational notions. We say that location in space is absolute when there is a fixed meaning to where something is. A relative location is defined with reference to something else. Three blocks south of the supermarket is a relative location. Similarly, an absolute time is meaningful without reference to anything else, while relational time is always defined by its relation to another event or set of events.

  This leads to our second principle:

  2. Space and time are relational.

  A relational observable, or property, is one that describes a relationship between two entities. In a theory without background structures, all properties that refer to location in space or time should be relational. Background-independent theories speak to us about nature through relational observables.

  The third principle tells us nothing is left out.

  3. Principle of causal completeness.

  If a theory is complete, everything that happens in the universe has a cause, which is one or more prior events. It is never the case that the chain of causes traces back to something outside the universe.

  Our next principle was introduced by Einstein, in his papers on general relativity.

  4. Principle of reciprocity.

  This principle states that if an object, A, acts on a second object, B, then B must also act back on A.

  There is one more of these principles, and it is both subtle and powerful.

  5. Principle of the identity of indiscernibles.

  This states that any two objects that have exactly the same properties are in fact the same object.

  Putting them in order, we have five closely related principles:

  The principle of background independence

  The principle that space and time are relational

  The principle of causal completeness

  The principle of reciprocity

  The principle of the identity of indiscernibles

  These are all aspects of a single principle, which Leibniz called the principle of sufficient reason. This states that every time we identify some aspect of the universe which seemingly might be different, we will discover, on further examination, a rational reason why it is so and not otherwise.

  For example, given present knowledge, it seems that space might have more or less than three dimensions. (By this I mean the three large dimensions that we see around us; this doesn’t count hypothetical tiny, “rolled-up” dimensions perceivable only on a subatomic scale.) This is because all our current theories, including general relativity and quantum mechanics, would also make sense in a world with a different number of spatial dimensions. Leibniz’s principle of sufficient reason advises us that this must be because our current theories are incomplete. We must seek to complete our theories, and one sign of success will be when we find out why the number of large spatial dimensions is three.*

  Leibniz believed we could uncover a rational explanation for every apparent choice God might seem to have made in the creation of the universe. He spoke of the state in which this understanding would be achieved as one of having “sufficient reason.” His principle of sufficient reason states that the universe can be completely
understood.

  Each of the principles I’ve stated expresses this idea. For example, we could ask why the universe came into being where it was and not ten meters to the left. But everything would have happened just the same way, so this can’t be a meaningful question. Therefore, absolute position is meaningless; only relative position is meaningful. A scientist who aspires to be rational must be a relationalist.

  Our theories express these principles incompletely, but over time there has been a clear trend toward theories that explain more. Each time we explain a feature of the world in a way that limits the choice a creator might have had, we eliminate some of the arbitrariness we formerly perceived in the design of the world. As we understand the world better, it appears to us to be more rational. This happens each time we discover a hidden unity. A good example of this was Maxwell’s discovery that light, electricity, and magnetism are not separate phenomena, but are different aspects of a single force. This discovery shows us that a world could not exist that has magnetism but no electric forces. And we understand that any world with electricity and magnetism must also have light.

  I do not know if a complete understanding of nature will ever be attained. But I do believe that our goal should be to always progress toward ever more complete understanding, which means we seek always less arbitrariness and more rationality. Hence I would propose we seek ever more sufficient reason.

  I believe the progress of science is measured by such increases in our understanding of nature.

  Special relativity is an improvement over Newtonian physics, and general relativity, by embracing a purely relational spacetime geometry, is an improvement on both. We can also say that quantum mechanics satisfies the principle of reciprocity better than Newtonian mechanics, but that pilot wave theory comes still closer to sufficient reason because it explains things quantum mechanics leaves unexplained, such as why individual events take place where and when they do.