Although quantum theory forms one of the two pillars of modern physics, along with general relativity, physicists disagree about what it says. Ask them. Their answers are apt to be inconsistent. Steven Weinberg recently declared himself unsatisfied with the approach to quantum mechanics upon which he had long relied.^{1} “Madness in great ones must not unwatched go,” Claudius remarks in *Hamlet*. The same is true for their doubts.

The predictions of quantum mechanics are statistical, but Schrödinger’s equation, which governs the evolution of its wave function, is deterministic. Given the initial state of a system, it implies that future states evolve in a unique way. How, then, are different outcomes physically possible? They are possible given the Born rule, which assigns probabilities to experimental outcomes. The rule has nothing to do with the equation, and the equation, nothing to do with the rule. Weinberg does not see how to reconcile one with the other. The measurement problem in quantum mechanics is of long standing; it may be found in Erwin Schrödinger’s three-part paper of 1935, “*Die gegenwärtige Situation in der Quantenmechanik* (The Present Situation in Quantum Mechanics).”^{2} It is there that his famous cat makes her first appearance, and, possibly, her last. Eighty years on, Weinberg is still struggling with the problem.

Peter Lewis’s *Quantum Ontology *provides a clear introduction to his subject. He begins with two-slit interference and Bell’s inequality, and develops the mathematical formalism used to explain them. For purposes of prediction, the formalism is enough, but Lewis is a philosopher; he wishes to know what is really going on.

Does quantum mechanics reflect a fundamental indeterminacy in the world? The probabilistic character of its predictions might suggest that this is so, but it might equally suggest that there are physical features of the world that the theory does not capture. It is here that Lewis urges his readers to step back. Quantum mechanics is not a physical theory at all, only a formalism. A physical theory must specify what exists in the world and how it behaves. In quantum mechanics, the same formalism allows different theories to make exactly the same predictions. Some theories are deterministic, and some not. Some contain a prosaic account of the universe; others assert that the universe is forever splitting into different branches. Some support a robust notion of causation; others do not.

Theories are undetermined by the evidence to which they appeal. This is an old philosophical issue, discussed for over a century, ever since Pierre Duhem published *The Aim and Structure of Physical Theory*.^{3} Given a theory, it is not hard to cook up a rival that makes the same predictions, but such rivals are often artificial. Scientific practice usually fails to produce competing theories. There is no alternative to the atomic theory of matter, or to the chemical account of photosynthesis. For all that, underdetermination haunts quantum mechanics. It is a specter. Various theories tell different stories about the physical world, and in some cases no experiment can decide between them.

Various theories? Lewis largely follows the lead of John Stewart Bell, who championed a number of unfamiliar theories for their clarity.^{4} They say what exists and how it behaves, and, their laws are expressed in precise mathematical terns. They make no appeal to such terms of art as measurement, system, apparatus, microscopic, or macroscopic.

## Ghirardi–Rimini–Weber

Suppose that electrons are fired from the right side of the laboratory toward a fluorescent screen on the left. The electron is associated with a wavefunction, one peaking on the right side of the laboratory. According to Schrödinger’s equation, the peak moves from right to left, spreading out as it moves. Nonetheless it is a small scintillation that is seen on the screen. How to connect the diffuse wave function and the point-like flash? The Born rule establishes that for any particular place on the screen, the square of the wavefunction yields the probability that the flash will occur there. After the flash occurs, the old, spread-out wave function is replaced by a new compact wave function centered on the location of the flash.

This is the legendary collapse of the wave function.

The collapse comes as a surprise, because it cannot be produced by Schrödinger’s equation. The wave function seems to undergo two completely different dynamics, one deterministic and the other stochastic. When does the wave function evolve by Schrödinger’s equation and when does it collapse? The standard answer is that there is a collapse when a measurement occurs. An answer like this is simply too vague to be used in the statement of a fundamental physical theory. Measurements are physical interactions. They ought to be analyzable using the same theoretical tools used for all other physical interactions.

The first mathematically precise version of a theory that included a wavefunction collapse was published by Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber in 1986. It is universally known as the GRW theory, or even as GRW.^{5} Lewis begins with it because it most closely follows the standard predictive formalism. GRW does not provide precise physical criteria for a measurement. The collapse that it encompasses is not related to measurements at all; it is not related to anything in the environment of a particle, or to any interaction between the particle and anything else. In GRW, a collapse occurs with fixed probability per particle per unit time. This yields a precise but indeterministic equation of motion. This theory dispenses with the vague idea of measurement in favor of an exact equation of motion. Bell approved.

## Pilot Waves

Having discussed GRW, Lewis goes on to examine Bohmian mechanics. Due to David Bohm, this theory had its origins in the work of Louis de Broglie during the 1920s. The discrepancy between a spread out wavefunction and its residue as a scintillating spot on a screen prompted Bell to remark that, “[e]ither the wavefunction as given by the Schrödinger equation is not everything, or it is not right.”^{6} The possibility that it is not right is addressed by GRW; the possibility that it is not everything, by Bohmian mechanics. Bohmian mechanics is a hidden variable theory. What is hidden are particles with a precise position. They follow exact trajectories, and their trajectories are governed by what are always called pilot wave equations. When considering two slits, they do not hesitate; they go through either one or the other, never both. Bohmian mechanics features two equations for the price of one. Schrödinger’s equation never collapses; the pilot wave equation never lapses. The ensuing theory is completely deterministic.

The theory is completely deterministic, but whether the world is deterministic or not is another question. GRW and Bohmian mechanics equally account for observed phenomena. The issue of determinism remains. Quantum theory, one might conclude, cannot answer questions about the fundamental nature of the world.

## Many Worlds

There is no end to underdetermination. It is possible to suppose that Schrödinger’s equation is both everything and right. Schrödinger’s cat is both alive and dead. But she is not alive or dead in the same world. Many worlds are required, the thesis first advanced by Hugh Everett III.

If one sends a water wave through a slit in a wall, an expanding circular wave forms on the other side. Two slits yield two such waves, side by side. As these waves propagate, they interact with each other. Where the crest of one wave meets the trough of another they cancel out; where crest meets crest or trough meets trough they reinforce, resulting in interference bands. Exactly the same thing happens with the wave function of an electron. The interaction of the electron with other systems can suppress such interference effects. If interference effects *are* suppressed, the wave function of the system has decohered. Never perfect, decoherence provides a rough way to divide the wave function of a large system into branches, with each branch evolving independently. These branches are the worlds in the many-worlds interpretation, and decoherence explains why we do not observe branches other than our own.

## The Big Three

Quantum mechanics admits, at least, these three interpretations: GRW, Bohmian mechanics, and many-worlds. None is better supported by the evidence than the other. At this point in the book, Lewis leaves off physics and takes up philosophy. In discussing indeterminism, he warms up with his coffee mug. We can assign the mug an approximate size, but trying to calculate its exact volume is profitless. At the microscopic scale, there is no precise point where the mug ends, nor is there a criterion for judging whether a particular atom belongs to the mug or not. Is the physics somehow indeterminate?

A long-standing idea connects the physical properties of a quantum-mechanical system with a mathematical operator on the wave function. These operators are called the observables of the system, and there is a standard way, given a wave function and an observable, to extract statistical predictions about what the outcome of a measurement of the observable might yield. When the prediction is certain, then the wave function is called an eigenstate of the observable. The system has a property if and only if it is in an eigenstate of the associated operator.

Certain questions now become purely mathematical. No system can simultaneously have values for all observables, since no wave function is an eigenstate of all operators. And the closer a wave function is to being an eigenstate of the position operator, the farther it is from being an eigenstate of the momentum operator. This is an illustration of the Heisenberg uncertainty principle. If the wave function is not an eigenstate of an observable, does it follow that the property is indeterminate? If so, then most physical properties of the system are indeterminate. Worse, when one system interacts with another so that their wave functions become entangled, neither system has any determinate properties at all. The world threatens to dissolve into a soup. A collapse in GRW, it is often said, localizes the wave function of a system, thereby giving it a location in space, but does not yield an eigenstate for any position operator. No GRW particle ever has a location in space.

How, then, can we use such particles to build a recognizable account of the world around us? Perhaps by loosening the property of being an eigenstate to being close to an eigenstate. Lewis shows that this laxness has some peculiar consequences. One can find a wave function for a collection of N marbles such that each of the marbles is in a box, but not all of them. Each of the individual wave functions is close enough to a position eigenstate for the marble to count as being in the box. When the collection with its single wave function is considered, the individual deviations add up. The collective wave function is not close enough to a position eigenstate for the collection to be in the box.

Does quantum theory commit us to this sort of indeterminacy? Bohmian mechanics gives a different result. Here there is no attempt to use operators on wave functions to define the locations of things; the particles always have precise locations. There can be no problem about marbles in boxes. Either the particles that constitute a particular marble are enclosed by the particles that constitute the box, or they are not. The wave function determines how the particles move and so underwrites some of their properties, but not by virtue of being an eigenstate of any operator.

Bohmian mechanics rests on an ontology of what Bell called local beables. These are pieces of physical ontology that always have well-defined locations in space. The distribution of such local beables corresponds to the world we see around us. Empirical predictions are determined by the implications of the theory for the motion of local beables. The methodological advantage provided by local beables has not been lost on the advocates of GRW, which has evolved into variants also making use of local beables. In one variant, there is, in addition to the wave function, a field of matter defined on all of space; in another, the wave function is supplemented by a collection of point events in space-time. In both variants, the distribution of matter in space is always perfectly sharp and mathematically well defined. Points in space no longer have the laxness that might be associated with marbles in a box in classical quantum mechanics.

## Causation

It was Bertrand Russell who remarked that the concept of cause has no place in physics.^{7} It is a concept that does not figure in GRW, Bohmian mechanics, or many-worlds, but there is no getting away from causes in everyday life and one would like to see how the concept is integrated with these theories. Although Bohmian mechanics makes no direct use of the concept of cause, the fact that it is a deterministic theory makes possible certain counterfactual conclusions. Ask what the world would have been like had something been different and the theory will tell you. What is more, Bohmian mechanics, Lewis’s analysis reveals, is causally nonlocal. Bell’s theorem established that certain correlations between experiments cannot be recovered from a local theory. Quantum mechanics predicts the correlations but gives up the locality. It has been widely verified by experiment.

Nonlocality appears in Bohmian mechanics in a particularly egregious form. Let a pair of particles in a particular entangled state separate, and consider all the different experiments that can be carried out on each of the particles individually. Quantum formalism makes statistical predictions for the outcomes of the individual experiments and for correlations between them. In order to recover these predictions, there must be some connection or influence between the particles, no matter how far apart they are. In Bohmian mechanics, this influence is causal. For certain initial conditions, the theory implies that the outcome of an experiment on one particle would have been different had a different experiment been performed on the other, or even if the same experiment had been performed at a different time. This counterfactual dependency of one result on the other is undeniable.

The many-worlds theory is also deterministic: there is only the wave function, constantly evolving in accordance with Schrödinger’s equation. But the theory does not lend itself to straightforward causal explanations of everyday events. If the wave function of a particle is in a definite state of spin in the *x* direction, and a measurement of spin is made in an orthogonal direction *z*, the predictive formalism assigns a fifty percent chance of an up outcome and a fifty percent chance of a down outcome.

If we ask in Bohmian mechanics why one results rather than the other, there is a clear answer: the initial position of the particle determines the outcome. But if we ask the same question in many-worlds, there is no clear answer. The theory predicts that both outcomes will occur. It also predicts that the observer will split into two descendants, the first seeing one result and the second seeing the other. But it is unclear what it would mean for there to be a cause of being one descendant rather than the other.

GRW is a stochastic theory. Outcomes of experiments are determined by where and when collapses occur, and the collapses are random, having no causes themselves. Lewis, however, argues that the collapses can be causes. Since a collapse changes, all at once, the global structure of the wave function, it is not a local cause.

The final chapters of the book focus on the wave function. As a mathematical object, it is central to the predictive apparatus, but what sort of physical entity does it represent? It is a function defined on the configuration space of a system. If there are N particles in the system there are 3N dimensions in configuration space; the configuration space for the visible universe is on the order of 10^{80} dimensions. Some philosophers have argued that if the wave function is real, then so is a high-dimensional space. This position calls into question the status of the three-dimensional space of everyday experience. If three-space is illusory, why should a higher dimensional space represent it so well? If it is the higher dimensional space that is illusory, why do physical theories point so insistently in its direction?

*Quantum Ontology* is written for philosophers, particularly metaphysicians who want their work to be informed by current physics. Since many in the intended audience have no advanced training in the subject, the theories must be presented from scratch. The result is an excellent book for anyone interested in the foundations of quantum theory. It offers accessible descriptions of GRW, Bohmian mechanics, and many-worlds, but still presents their foundations in a technically rigorous way.