by Lee Smolin
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LET’S COME BACK TO the Schrödinger’s cat experiment and see how pilot wave theory resolves it.
Pilot wave theory asserts that quantum mechanics applies universally. There is only Rule 1, and it applies to all cases. This means that measurements are no different from other processes.
Everything—atoms, photons, Geiger counters, cats, and people—has a dual existence, as a wave and a particle. Both sides of this double existence are complex for large complicated objects such as Geiger counters or cats, which are made of many particles working together. We need a word to talk about all the ways that the particles making up a cat may be arranged in space, and we have one: the configuration of the atoms. If you speak of where all the atoms making up the cat are located with respect to each other, you are describing the configuration of the cat. Because there are many atoms, it takes a great deal of information to describe the cat’s configuration.
All this information must be coded into a list of numbers. How many numbers does it take to describe a cat? For just one atom it takes three numbers. These locate the atom in three-dimensional space. For two atoms it takes six numbers, three for each atom. So to locate the atoms in a cat takes three numbers per atom. There are very roughly 1025 atoms in a cat, so it takes 1025 multiplied by three to describe the cat’s configuration.
The important thing about pilot wave theory is that the atoms are all real, and they are each located somewhere definite in space. Each atom has a location, which is a point in space. Each cat has a configuration, which amounts to saying that each of its atoms is located somewhere definite in space.
An atom also has a wave, which is located in three-dimensional space. Each cat also has a wave associated with it. The strange thing is where that wave is located. It isn’t a wave in three-dimensional space. Instead, it’s a wave in a very high-dimensional space, called the configuration space (see figure 9). Each point of this space corresponds to a configuration of the cat.
It is difficult, if not impossible, to visualize a space with many dimensions. I once watched in awe as Roger Penrose did a calculation on the blackboard which required him to slide a two-dimensional surface around a six-dimensional obstacle in an eight-dimensional space, and I did have the thrilling experience of following step by step, but that’s the limit of my experience. Most mathematicians are not as visually gifted, but we can reason our way around in a high-dimensional space. When I draw a three-dimensional object, I am really drawing a two-dimensional projection of it. Likewise, what I see in my mind when I imagine a configuration space like that of a cat, with perhaps 3 × 1025 dimensions, is a three-dimensional projection, together with a silent admonishment to be careful and not draw false conclusions from this totally inadequate visualization.
FIGURE 9. CONFIGURATION SPACE Two atoms live on a line, in one dimension. Their configuration is measured by two numbers, so their combined configuration space is a point on a plane, in two dimensions. We treat the two atoms as identical, so atom 2 is always the rightmost atom.
A wave on the configuration space carries a vast amount of information. Recall, for example, the state CONTRARY, which describes correlations between the answers to questions asked simultaneously of two particles, while telling nothing at all about each particle separately. To code quantum states like this, in total generality, we need more than a three-dimensional wave for each atom in the cat. We need a wave flowing on the space of all possible configurations of the cat.
Once one accepts the existence of a wave on the space of all the configurations of a cat, the resolutions of the quantum puzzles follow directly.
There is only one cat, which all the time is in some configuration. The configuration of the cat may be one in which the cat is alive, or it may be one in which the cat is dead. It must be one or the other, but it cannot be both. So the cat is at every moment either alive or dead.
The wave function of the cat can be the sum of two waves, because you can always add waves. That is what they do: waves superpose, which means they add. The wave guides the configuration, just as it does for a single electron. The wave function may flow simultaneously through configurations of a live cat and configurations of a dead one. Just as a river can branch, and take both branches, a wave function may branch and take both the branch over living configurations and the branch over dead configurations.
The wave function ends up related to the probability of finding different configurations. When the wave function is large over some configuration, so is the probability. So the probability of finding the cat in a live configuration or a dead configuration may be each roughly one half. But there is only one cat, and just as an electron can be in only one place at a time, the one cat is either dead or alive.
Is it weird that the wave function will spawn branches that flow to where the particles or their configurations are not? A bit, but this must be, because the particle can follow only one branch. But an empty branch may have consequences in the future. The different branches may flow back together in the future, making interference patterns that influence where the particles go.
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EIGHTEEN YEARS AGO I had a difficult decision to make. Two futures beckoned to me, each of which seemed, from all the information I could gather, attractive. Of course one never has enough information to make a decision like this, for everything about my future was at stake. The question was, Which country and which city would I call home? Whom I might marry, who my children would be, what languages they might speak, and how long I might live would all be influenced by this decision.
Unable to decide, I consulted friends with a quantum lab and let a radioactive atom decide for me. If the atom decayed within its half-life, I would take the new opportunity in a new city and country; if it didn’t, I would stay with the familiar. Well, it decayed, and here I am in Toronto, with my family, friends, and neighbors, none of whom I would ever have met had that atom held off decaying a bit longer.
There is nothing special about me. All of us are made of particles that have been guided to the present by a wave function on our vast space of possible configurations.
The wave function surrounds where I am now, but it also has other branches where I might be, but am not. Some of them branch from that experiment and develop the empty history that I did not take, but would have had the atom not decayed. The empty wave function of the particle flows on from there, to this day. This empty branch of my wave function continues to inhabit London.
Have we not all felt a bit wistful contemplating lives we might have had, had a decision turned out just a bit differently? If the pilot wave theory is right, then these lives not taken are traced by an empty wave function, ready to guide my atoms, which, however, are elsewhere.
A few years before that, my wave function faced another fork, from which two very different branches flowed. I took one branch, but had I taken another, I would have faced a very different fate.
I was booked on a Swissair flight from New York to a conference in Vienna. The night before my trip, I heard from the organizers that my talk wasn’t scheduled till the end of the meeting, and so, on a total whim, for a reason I no longer remember, I called the travel agent and rebooked for a day later. Before going to sleep the next evening I turned on the radio and heard that the flight I would have been on had crashed off Halifax. So, if pilot wave theory is correct, it is really true that a branch of the wave function of the atoms that then constituted me is to this day bunched up at the bottom of St. Margarets Bay, off the village of Peggy’s Cove, Nova Scotia.
That branch is empty, as are myriad others. But if pilot wave theory is right, they are real. The only difference between them and the one branch that guides me now is that only one branch coincides with, and guides, the atoms that make me up. The myriad other branches flow on, empty.
Do I care about these other branches? Should I? T
here is always the chance that at some time in the future an empty branch recombines with my branch, causing interference, which changes my life abruptly.
The chances for this to happen are extraordinarily small. They are in a category of possible events that would be permissible under the laws of physics but which essentially never happen. All the atoms in the air in the room where I’m typing this might by chance line up together and fly out the window, asphyxiating me. But this would be extraordinarily unlikely, given that the atoms spend their day bouncing around randomly.
So there is basically no chance that the empty branches representing the lives we didn’t live and the choices we didn’t make will have any effect on our futures. But were we mere atoms, interference between full and empty branches of the wave function would be happening all the time.
So for all practical and moral purposes, if pilot wave theory is right, we can ignore the empty branches. We are real only once, and live out that life on that one occupied branch. We need care about, and be responsible for, only what the one real version of each of us does.
NINE
Physical Collapse of the Quantum State
Experiment and common sense suggest that there are no superpositions of macroscopic objects, because every large body is always somewhere particular. Rule 2 was invented to accommodate this, at least as regards the behavior of measurement instruments and the systems that come into contact with them. To avoid superpositions of the states of a measuring instrument, Rule 2 dictates that just after a measurement of a particle’s position, its wave function immediately collapses to a state corresponding to the position that was measured.
Just before the measurement, a certain atom’s wave function might have been spread all around the Earth, giving it an equal probability of being found anywhere on the globe. But when a measurement is done of its position, and if that measurement reports the atom’s location to be somewhere in New York City, then, at the moment that report is made, the atom’s wave function collapses down to the extent of the five boroughs.
In standard quantum mechanics this collapse of the wave function happens only as the result of a measurement. This raises a problem for realism, because it is only our use and interpretation of the result that determines whether an interaction with a large body is a measurement or not.
According to a realist perspective, a measuring instrument is a physical system, which happens to be large, and which has a special capacity to amplify tiny differences in an atom’s behavior to make a record of what was n that can be etched in a macroscopic change. But because it is a physical system it should obey the same laws as the atoms which compose it. If the atoms can be in superpositions, the same should be true of the vast collections of atoms that make up the measuring instrument. In the last chapter, we saw that in pilot wave theory, part of the price we pay for realism is a world full of empty branches of wave functions, which have long since disconnected from the objects they might guide.
But what if the collapse were a real physical process that occurs whenever a large body is involved in an interaction? The collapse would be triggered by the size of the object, measured in mass or in the number of atoms that make it up, irrespective of its use as a measuring instrument. The wave functions of all large bodies would collapse, wiping out their superpositions. The measurement systems, made of myriad atoms, would collapse too. This suggests a strategy for a realist version of quantum physics.
The idea would be to modify quantum mechanics by combining Rule 1 and Rule 2 into a single rule, which specifies how wave functions evolve in time. When the system it is applied to is microscopic, the old Rule 1 is a good approximation. Collapses of the wave functions of atoms may happen, but only rarely. But when the system is large, collapse happens frequently, so that it appears that the body is always somewhere definite.
Theories of this kind have been constructed since the 1960s; they are called physical collapse models.
The first physical collapse model was invented in 1966 by Jeffrey Bub, a student of David Bohm, and developed by the two of them.1 In the same year, F. Károlyházy published a paper arguing that noisy fluctuations in the geometry of spacetime could cause the wave function to collapse. As with pilot wave theory and Bell’s work of the same period, the response to these pioneering papers was slow. The first person to develop a completely precise version of a theory of this kind was Philip Pearle, an American theorist who has done very important work in spite of spending his career at a small undergraduate college. He struggled for almost a decade to invent a consistent theory for physical wave-function collapse, and his first theory incorporating a physical collapse of the wave function was published in 1976.2
Pearle’s version of a collapse model adds a random element, so that there is something akin to a roll of the dice that decides when and where a wave function collapses. The rolls are infrequent for wave functions of atoms, and so small systems consisting of a few atoms collapse infrequently. But collapse occurs often for macroscopic systems containing many atoms. Pearle called his theory continuous spontaneous localization, or CSL.
For several years Pearle was nearly the only one working on this approach to realism. Then in 1986 three Italians working in Trieste proposed a rather elegant version of the idea, which has been known since as the GRW theory after their names, Ghirardi, Rimini, and Weber.3 Other people joined in to develop these dynamical collapse models, including Lajos Diósi, Lane Hughston, and Nicolas Gisin.
These theories differ from each other at the level of details, but they share the key feature that the behavior of any quantum system is a mixture of Rule 1 and Rule 2. Most of the time the wave function of an atomic system changes slowly and smoothly, following Rule 1. But from time to time it jumps abruptly into a definite state, following a form of Rule 2.
One defect of these spontaneous collapse models is that the rate of the spontaneous collapses has to be carefully specified so that the collapses are rare enough not to corrupt interference patterns built by delicate superpositions in atomic systems. This guarantees the successes of quantum mechanics by preserving the coherence of superpositions of microscopic systems, where it is needed. But the wave function of a large body will get hit with a collapse far more often, because it consists of many atoms. Events that are rare for one atom will happen frequently to some atom or another in a large collection of them. But when one atom collapses, so must the others making up the same body. As a result, the model can be tuned so that the wave functions describing macroscopic systems collapse far more frequently, explaining why large-scale objects are always somewhere. This solves the measurement problem.
These theories have no need of particles, in the sense of pilot wave theory. There are only waves, but the result of a spontaneous collapse will be a wave highly concentrated around one location. Such a concentrated wave is hard to distinguish from a particle.
Because there are no particles, the mysteries of the wave-particle duality evaporate. One just has to understand why waves evolve under two very different processes.
These collapse theories are entirely realist. The wave function is the system, and there are no mysteries as to how to interpret it. By collapsing the wave function down to only what is physically relevant, collapse theory avoids the extravagant proliferation of branches that burden the pilot wave theory. There is no measurement problem because big objects, including measuring devices, are always in collapsed states. There is no special role for consciousness, information, or measurement. What you see is what you get.
To define one of these theories you have to decide which of the incompatible questions the collapsed wave function is to answer. The usual answer is position in space. The collapsed wave functions are peaked somewhere in space, which makes them like particles.
One consequence is that energy is no longer precisely conserved. A metal block should slowly heat up as a result of all the collapses the wave functions of its atoms u
ndergo. This, for me, is the least attractive feature of spontaneous collapse models. On the plus side, there are experiments planned to look for this heating.
As is often the case with new theories, there is a lot of freedom. One is free to adjust how often the collapses take place. One can make this rate depend on the mass or the energy of the atoms. If the hypothesis of spontaneous collapse is to be viable, there must be a way to set the rates so that wave functions of atoms and elementary particles rarely collapse, while big things collapse often enough that they are always in some definite place. And one has to make sure all unintended consequences, such as heating up of matter, are undetectable. Remarkably, these conditions can all be met, so these theories are viable.
In some of these models the spontaneous collapses are random processes. The theory specifies only a probability for collapse to happen. This leads to uncertainties and probabilities, which are built in from the beginning. The probabilities are coded into the fundamental laws, rather than being a consequence of ignorance or belief. The intrinsic randomness of the collapse process then explains the uncertainties in quantum physics, and it does so in a way that does not single out measurements. Thus, the probabilities are explained in a way that is perfectly compatible with realism. That is a great advantage. (Of course, if one wants a deterministic theory, this is a disadvantage.) Related to this is the fact that the fundamental laws are irreversible, so that the arrow or direction of time is coded in at the bottom level. Some may see these as defects, but my view is that they are very positive features of the collapse models.