by Lee Smolin
One worrying aspect of spontaneous collapse models is that the collapse of the wave function takes place all in one moment of time. As the wave function may be spread out over space, its collapse defines a moment of simultaneity over a whole region. This appears to contradict relativity theory, which asserts that there is no physically meaningful notion of simultaneity over regions of space. While this does seem to be a problem for the original dynamical collapse models, there have been proposals for making collapse models that are consistent with special relativity.4
But the most attractive feature of all the collapse models is that they predict new phenomena, which are subject to experimental testing. The random collapses introduce noise into a system. For some values of the parameters, the effect would be large enough to be seen. No need for such a noise source has been seen in several recent experiments, which rules out certain values of the parameters, if not the theory itself. This is real science, and the experiments continue. Nothing would be more wonderful than the discovery of an effect which contradicts quantum mechanics and confirms a prediction of one of its realist alternatives.
One weakness of some of these collapse models is that they make no reference to, or use of, other key questions of physics. It would be more compelling if the modifications we make to quantum mechanics are motivated by a problem besides the measurement problem, such as the problem of quantum gravity. This brings us to the work of Roger Penrose.
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IF THERE IS ONE living theorist whose achievement and depth of insight and influence match those of the sages of the early twentieth century, it is Roger Penrose. Simply put, he is the real thing.
Penrose follows his own compass and he has, as a result, novel and surprising things to say about most issues in fundamental physics, including quantum gravity and quantum foundations. Because everything he envisions is tied together by an often-hidden consistency, the best way to approach his proposal for quantum theory is by tracing his work back to his time as a young mathematician in the early 1960s, when he was fascinated by the foundations of our understanding of space, time, and the quantum.
It is easy, but inadequate, to describe Penrose as the most important contributor to general relativity since Einstein. In the early 1960s he invented revolutionary new mathematical tools to describe the geometry of spacetime, based on causality. Rather than talking about how far away two events are, or how much time elapses on a clock, he described spacetime in terms of which events were the causes of which events. This led him to posit and prove theorems that showed that, if general relativity is right, the gravitational field becomes infinitely strong within the core of black holes.5 Once that happens, the theory breaks down, because its equations stop working to predict the future. Such places, where time may start or stop, are called singularities. Afterward, working with Stephen Hawking, he extended his method to the expanding universe and proved that general relativity predicts that time began a finite time into our past, when the whole universe began its expansion in a state of infinite density.6
But his inventions exceed even these transformative contributions to general relativity. Like Einstein, Penrose cares more deeply than most for the coherence of our understanding of the world. And, just as it did in the cases of David Bohm and David Finkelstein, this passion has driven Penrose to develop a unique vision of fundamental physics, which is unmistakably his. Moreover, Penrose’s vision has, over the many years of his creative career, led him to invent mathematical structures that others later utilized.
After transforming the practice of general relativity, Penrose turned his attention to fundamental physics. He was struck by a sympathy between quantum entanglement and Mach’s principle—the idea, which had inspired Einstein’s invention of general relativity, that what is real in general relativity is relationships. Both ideas hint at a global harmony which ties the world together.
Penrose was the first to ask whether the relations which define space and time could emerge from quantum entanglement. Seeking insight into this question, he was inspired to invent a simple game based on drawing diagrams, the rules of which represented simultaneously quantum entanglement and aspects of physical geometry. This game, his first vision of a finite and discrete quantum geometry, Penrose called spin networks.
Most theoretical physicists work out their ideas by doing calculations in existing theories. Penrose works sometimes instead by inventing games. Their simplicity captures profound questions, which one investigates by playing the game. It is typical of Penrose that his main paper on spin networks was not only unpublished—it was never even typed up. Mimeographs (now they would be called photocopies) of his handwritten notes circulated among his students and from friend to friend. These notes were an exhilarating read, even though they ended in the middle of the main proof.*
For decades spin networks remained a kind of philosophical parlor trick, passed on by sketches on napkins during dessert at conference dinners. But they turned out, years later, to be the central structure in an approach to quantum gravity called loop quantum gravity. In that context, spin networks embody one way that the principles of quantum theory and general relativity can coexist.
Extending spin networks, Penrose discovered twistor theory, which is an extraordinarily elegant formulation of the geometry underlying the propagation of electrons, photons, and neutrinos. Intrinsic to twistors is a beautiful asymmetry of neutrino physics, which is called parity. We say that a system is parity symmetric if its mirror image exists in nature. We have two hands, which are each mirror images of the other, so our hands are parity symmetric. But overall humans are not parity symmetric, because our hearts and other internal organs are arranged asymmetrically, and we each tend to favor one hand. Neutrinos exist in states whose mirror images don’t exist, and hence are parity asymmetric. Penrose’s twistor theory expresses this feature of neutrinos, because it uses mathematical structures which are not the same when looked at in a mirror.
For many years Penrose and a few students developed twistor theory, working in isolation in Oxford. In the late 1970s this caught the attention of Edward Witten, who many years later made twistors the keystone of a powerful reformulation of quantum field theory he invented with some younger theorists, which is still in progress.
What I find so remarkable about Penrose is that he has an inner narrative that connects everything he does into a coherent story. So it’s no surprise that his expansive vision of a new physics led him to a reinvention of quantum mechanics. This was part of a larger strategy to combine quantum theory with general relativity, to make a quantum theory of gravity.
Typically, Penrose started off his attack on quantum gravity by ignoring the obvious path taken by nearly everyone else. The standard path is to construct a quantum description of a system, a process called quantization. This starts with a description of the system given in the language of Newtonian physics. We “quantize” this by applying a certain algorithm. The details of this don’t concern us here, but suffice it to say the output is a quantum theory which is absolutely conventional and standard.
This technique works in many cases to give us successful quantum theories of atoms, elementary particles, and radiation. It can be applied to gravity; indeed, loop quantum gravity was made by “quantizing” general relativity.
Penrose took a different road. Quantum theory and general relativity clash on a few key points. The most crucial is that they have deeply different descriptions of time. Quantum mechanics has a single universal time. General relativity has many times—if by time we mean duration as measured by clocks. The beginning of Einstein’s theories of relativity is a discussion of synchronizing two clocks. You start off by synchronizing them, but they do not generally stay synchronized. They slip out of synchronicity at a rate that depends on their relative motions and relative positions in the gravitational field.
Another point on which the two theories clash is th
e superposition principle. As we discussed, given two states of a quantum system, we can make new states by adding them together. Something we haven’t needed to mention so far is that we can make a lot of different states from superposing the same two states. We do this by varying the contribution of each state to the superposition. Thus we can superpose CAT and DOG (from our earlier example) equally, as in
STATE = CAT + DOG
or we can choose instead
STATE = 3 CAT + DOG
or
STATE = CAT + 3 DOG
The number we multiply each state by is called an amplitude. Its square is related to the probability. Hence in the state CAT + DOG you are equally likely to find a cat lover or a dog lover, while someone in the state 3 CAT + DOG is nine times more likely to love cats than dogs.
General relativity does not have a superposition principle. You cannot add two solutions to the equations of the theory and get a new solution. Math-speak for this is to say that quantum mechanics is linear, while general relativity is nonlinear.
These two differences are related. The superposition principle is possible in quantum mechanics because there is a single universal time that we can use to clock how its states evolve in time. On the other hand, because distant clocks go out of sync, there is no simple way to add or combine two spacetimes to make a new spacetime.
Penrose embraces the multi-fingered nature of time in general relativity and the absence of superpositions as home truths. He suspects that the superposition principle must be violated once quantum phenomena are described in the language of general relativity. The simplicity and linearity of the superposition principle, he suspects, are only approximately true, and hold only to the extent to which the role of gravity can be ignored.
Thus, Penrose objects to quantizing gravity. Instead he suggests we should try to “relativize the quantum.” By this he means to introduce the multi-fingered notion of time into quantum theory by violating the superposition principle and making quantum states nonlinear.
Penrose is a realist, but he makes an unusual move for a realist on quantum theory. Rather than ascribing reality to both waves and particles, or inventing new “hidden variables,” Penrose takes reality to consist of the wave function alone. This leads him to take up the suggestion by Pearle and others that the collapse of the wave function during a measurement is a real physical process. The sudden change of the wave function is not, as some hold, due to an update in our knowledge of where the particle is; it is a genuine physical process.
Penrose, following the earlier work of Pearle and of GRW, proposed that collapse of the wave function is a physical process that occurs from time to time,7 interrupting the smooth changes mandated by Rule 1. And he took up a suggestion made by Diósi and Károlyházy: that the collapse process has something to do with gravity.8 When a wave function collapses, superpositions are wiped out. The rate at which a system’s wave function collapses depends on the size and mass of the system. As we discussed earlier, this rate can be specified so that atomic systems almost never collapse, while macroscopic systems collapse often, so that superpositions of large objects are impossible.
What is really exciting about the work of Diósi, Károlyházy, and Penrose is that they proposed a criterion for when collapses would take place that makes the collapse an effect of gravity. Roughly speaking, a superposition of an atom being here or there is collapsed to one location when the location of the atom would become measurable by the effect of its gravitational attraction.
This relates to the many-fingered time of general relativity. Imagine that the wave function is a superposition of an atom being in the living room with the atom being in the kitchen. Wherever the particle is, its mass has a gravitational field which affects clocks. One of the most striking predictions of general relativity is that clocks deeper in a gravitational field appear to slow down. This is well tested. Atoms on the surface of the sun have been observed to vibrate more slowly than the same atoms do on Earth. The effect is even seen by comparing the rates at which atomic clocks in the basement of a building tick compared to clocks on the roof.
The result is that clocks in the room where the particle is run slower than clocks in the other room. But what of a state which is a superposition of the atom being in the living room and in the kitchen? This seems to imply that the gravitational field must be in a superposition of states such that each clock runs slow.
But there is no such state, because one cannot add spacetime geometries to get new spacetime geometries. Hence the wave function must collapse.
Penrose gives a prediction for when the wave function will collapse, and work is underway to build an experiment to test Penrose’s prediction. Very recently two experimental teams9 have proposed that they may be able to construct superpositions of different gravitational fields, contrary to Penrose’s hypothesis. This is fabulous, but what is worrying is that Penrose has not put forward a detailed theory unifying gravity and quantum theory from which his heuristic model can be derived.
Penrose has at least proposed a model for how this might all work, which combines the usual evolution of the quantum state, given by Rule 1, with collapse of the wave function, given by Rule 2. They go together into a single evolution rule.
Penrose’s theory is not quantum mechanics; it is a new theory, which contains quantum mechanics within a realistic framework, based on a new evolution law, called the Schrödinger-Newton law. This unifies Rule 1 and Rule 2 into a single dynamical law.
If we focus on the behavior of atoms and radiation, this single evolution law mimics standard quantum mechanics. The superposition principle is satisfied to a good approximation. The wave function behaves like a wave, and Rule 1 is satisfied. Schrödinger’s equation for the wave function is then recovered for atomic systems.
But if we pull back to describe the macroscopic world, Penrose’s model describes a wave function which is collapsed and concentrated on single configurations. These concentrated wave functions behave like particles. So on the macroscopic level, Newton’s laws for the motion of particles are recovered.
Thus, in the microscopic regime, this theory reproduces quantum mechanics, while in the opposite situation, it predicts that macroscopic objects behave like particles and obey Newton’s laws.
Physical collapse models continue to be developed. Recently Pearle has made progress constructing a collapse model consistent with special relativity.10 The idea that gravity is responsible for causing the quantum state to lose coherence, and hence collapse, has also been developed by Rodolfo Gambini and Jorge Pullin, who call their proposal the Montevideo interpretation of quantum mechanics.11 And Steve Adler has found a role for spontaneous collapse in a hidden variables model he has been developing.12
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PILOT WAVE THEORY and the collapse models have given us options for quantum physicists who want to be realists. The differences are striking, but so are the similarities.
One option is to believe there are both waves and particles; this leads to pilot wave theory. This easily resolves the measurement problem, but at a cost. The pilot wave theory is doubly extravagant. It has a doubled ontology, but an asymmetric dynamic by which the wave function guides the particles without there being any reciprocal action by which the particles influence the wave. And we have to live with a vast world in which the wave function has many empty, ghostlike branches.
The collapse models avoid all these objections. There are only waves, so there is no doubled ontology and no issue with reciprocation, and there are no empty branches because they are eliminated by the collapses. This also solves the measurement problem, but here, too, there is a price, which is that the theory comes with new adjustable parameters that must be tuned to keep the theory out of harm’s way.
Both approaches agree on two key lessons: the wave function is an aspect of reality, and there is a tension with relativity theory. These a
re vital clues for the future of physics.
TEN
Magical Realism
Every quantum transition taking place on every star, in every galaxy, in every remote corner of the universe is splitting our local world on earth into myriads of copies of itself.
—BRYCE DEWITT
We saw in the last few chapters that there are options for realists, but notice that they all require changing the theory. The spontaneous collapse models make the sudden collapse of the wave function part of the dynamics of the theory. The collapse occurs whether or not measurements take place, and without regard to what we know. The resulting theories disagree with quantum mechanics generally, but preserve a domain of agreement sufficient for them not to contradict the results of experiments done so far.
Pilot wave theory is another option for realists. Rule 2 is suspended, so the wave function evolves always according to Rule 1. But a new element is added: particles, whose travel is guided by the wave function. So this theory is also different from quantum mechanics. When the particles are in quantum equilibrium, the predictions of the two theories overlap, but out of quantum equilibrium, the predictions of pilot wave theory differ from those of quantum mechanics.
It would be wonderful if someday experiments confirm that nature favors one of these realist theories over quantum mechanics. But suppose it turns out that after many years, or indeed centuries, we have no experimental results which require a modification or completion of quantum mechanics. In particular, what if no limit is found to how large or complex a system can be and still be put in a superposition? Suppose, in other words, that quantum mechanics in its original form appears to be completely correct. Would there be any options for realists?
The reason it is hard to be a realist and believe in quantum mechanics is Rule 2, which gives a special role to measurement. The suddenness of the collapse of the wave function on measurement dictated by Rule 2 means quantum states change in time in a way that pays no heed to locality or energy and instead seems to depend on what we know or believe. Since it makes the quantum state depend on our knowledge, this cannot be part of a realist theory.