4 All the Worlds Encoded in the Quantum State of the Cosmos

1. What Is Matter? From Lumps in the Void to Fields

So far, our rapid review of physics has consisted of (i) the rise of mechanics, especially Newton’s theory of gravity (sections 3 to 6 of chapter 2), and the idea of a state space (section 3 of chapter 3).

To understand the Everettian multiverse, we need to understand how, in quantum physics, the notion of state is very different from the notion in classical physics.

To prepare for that, the clearest and most vivid route is to review how our conception of matter developed historically, from ancient times to the end of the nineteenth century.

In this development, the main theme is that the idea of matter as a lump of stuff surrounded by void (a vacuum) gave way to the idea of a field that pervades all of space. (Here, I hasten to explain that field has nothing to do with fields of wheat, etc., in the countryside.) A field is, rather, there being a physical quantity associated with each place in space; and a state of the field is therefore an assignment of a value to the quantity at each place. An elementary example is the temperature of the air throughout a room: hotter here, cooler there. Strictly speaking, temperature does not make sense at an extensionless point of space; it is an average property of the air in a small volume, say, a cubic millimetre, around the point. But let us idealize and speak of a temperature at every point of space in the room. Then an assignment of temperature values to all points is a state of the temperature field.

So let us begin with lumps in the void. We discussed this conception of matter in sections 2 and 3 of chapter 2. We saw that it was advocated not just by ancient atomists like Democritus and Lucretius, but by many seventeenth-century mechanical philosophers, including Newton himself. In particular, we discussed how non-obvious, indeed unclear, it is, as regards both how it might explain the very varied phenomena we see around us and how such lumps might interact (whether by contact action, or by action at a distance).

Then, in section 3 of chapter 3, we noted how complicated a collision between two such lumps really is. This led to the idea of the point particle: mass concentrated at an extensionless spatial point. So, on this conception, ordinary objects are clouds, more or less dense, of such point particles. This idea was introduced as an idealization by Euler (1707–1783) and then advocated as physically real, i.e., the true nature of matter, by Boscovich (1711–1787).

Again, we should pause over how non-obvious, even problematic, the idea is. For any point particle, its density, i.e., the ratio of mass to volume, is infinite. So, if one advocates point particles with different masses, one must accept different sizes of infinity in order to describe their mass densities. Besides, how do point particles exert force on each other? And what happens if they ever collide? Boscovich himself—writing in an era when Newton’s theory of gravity with its action at a distance was accepted—suggested that at very short distances, a repulsive force, that is ever stronger at shorter distances, comes into play and overcomes the particles’ gravitational attraction, so that collisions never occur. But whatever you say about collisions, the question arises: Can you give a good account of the contact and interaction of ordinary objects?

Difficult questions like these suggest a rival conception of matter as continuous. On this view, there is no void anywhere, not even on the tiniest length scales: matter fills space completely. As we mentioned in chapter 2, Descartes endorsed this conception. He explicitly identified matter with extendedness (while in his metaphysics, mind was essentially unextended). During the eighteenth century, this conception continued to develop. Indeed, the exact mathematical description of how continuous matter moves, and how one part of it exerts forces on, and responds to forces from, various other parts, is a very subtle affair. It requires a lot of advanced calculus, as well as physical insight. Unsurprisingly, there was, from the time of Euler onwards, a century-long struggle to achieve this description.

The result, in brief, is to describe matter as a field, in the above sense. For think of a continuous piece of matter. To begin, let us suppose for simplicity that the matter is utterly rigid. That is, the distance between any two of its material parts, no matter how tiny, remains constant over time. So, think of a metal bar, and set aside your knowledge that it has layers of microscopic structure, i.e., crystals, atoms, etc. Although it is rigid, properties such as mass density and temperature may well vary across its expanse, and so these properties call for a field description. Besides, for continuous matter that is not rigid—that can be deformed (like an elastic solid, such as a pencil eraser) and/or compressed (like a liquid or gas)—the positions and velocities of its material parts are not “locked in step” together. So, the parts’ positions and velocities, as well as their density and temperature, also call for a field description. No wonder that some advanced calculus is required.

In the nineteenth century, electricity and magnetism “went the way” of continuous matter, such as fluids. That is, it turned out that, whatever the ultimate microstructure of matter was (point particles or continuous), electric and magnetic forces take time to propagate across space, between, say, positively and negatively charged matter. As discussed in chapter 2, section 3, this is unlike gravitational force, as it had been described by Newton. According to him, gravitational force propagates instantaneously and unmediated, i.e., without the need of an intervening medium.

Besides, these propagating electric and magnetic forces call for a field description. That is, one needs to attribute to each point of physical space a vector, i.e., a line segment in physical space (given by three real numbers relative to a coordinate system at the point), which is the (value of the) electric field at that point. This vector represents the electric force that would accelerate a stationary point-like electric charge, if the charge were at that point. And similarly for the magnetic field, though with the difference that it represents the force felt by a moving electric charge. (Here, the phrases “if it were at,” and “would accelerate” signal a counterfactual conditional. This is another example of science being up to its neck in modality: cf. sections 3 and 7 of chapter 3.)

Besides, this field description of the electric (or the magnetic) field is not “just” a very convenient way of stating an infinite conjunction of counterfactual conditionals: namely, as a mathematical function from spatial points to vectors located there. For Maxwell (1831–1879), in his stupendously successful theory of the electric and magnetic fields, showed that there is much more to these fields than their describing how a charge would accelerate.

Maxwell’s theory unified electricity and magnetism as two aspects of a single field: the electromagnetic field. It also showed light (and later, radio waves, etc.) to be waves in this field. That is, light is an oscillating pattern of electric and magnetic vectors at points of apparently empty space. It is a pattern that propagates, like a waveform on the surface of the ocean. But it propagates at the speed of light.

Furthermore, this field has energy and momentum, quantities previously attributed only to matter, i.e., to stuff that had mass. That is, the field can convey energy and momentum from one place to another. Thus, when you listen to the radio, your aerial is energetically excited, i.e., given energy, by the arriving pattern in the electromagnetic field; and the pattern of excitation is then decoded and amplified into sound.

To sum up, by the end of the nineteenth century, classical physics had a broadly dualist ontology of matter and field. The picture was that matter with mass (and with energy and momentum) is localized in space. It was unknown, and controversial, whether it consisted ultimately of point particles or of continuous, space-filling matter. But in the space between localized pieces of matter, there was not just Newtonian gravity, with its action at a distance, but also an all-pervading electromagnetic field that is the medium by which electromagnetic interactions between bits of charged matter occur, and that also itself possesses energy and momentum.

In the twentieth century, this dualism was overcome, with all-pervading fields getting the upper hand. This happened in various ways. But we need only state two.

First, Einstein’s special relativity theory (from 1905) identified mass with energy, so that one speaks of “mass-energy.” So, the quantity, mass that had from Newton onwards been attributed only to matter was now seen as also an attribute of the electromagnetic field.

Second, and more important for us, from the mid-1920s, quantum theory replaced classical physics’ matter—even a single point particle, not only extended matter—by a field. But it is a very strange field even for a single point particle. And it replaces the classical electromagnetic field with another field that is also strange, in a way exactly parallel to the strange field that it postulates for a piece of matter, such as a point particle.

This strangeness is the source of all the problems about interpreting quantum theory—and it will dominate this chapter.

2. The Quantum State: Probabilities for Classical Alternatives

The clearest way to grasp this strangeness is to return to the idea that a theory attributes to the physical systems it describes instantaneous states.

We saw in section 3 of chapter 3 that in classical physics, specifically, Newtonian mechanics, the state of a point particle is given by an ordered set of six real numbers, a 6-tuple: three numbers for its position in space and three for its momentum. This meant that the state space of a point particle, which can be anywhere and have any momentum, is the set of all 6-tuples of real numbers. This is a six-dimensional space, where we use the word “space” because, although this is not physical space, we can use geometrical ideas to describe it. And we saw that for more complicated systems, the state space rapidly becomes more complicated and intricately structured. Even if we set aside all the momenta and consider only the positions of the component parts—which is called the configuration of the system—the space of configurations (called configuration space) rapidly becomes complicated.

Now that we have the idea of a classical field, we can also talk about an instantaneous state of such a field. Think, for example, of the electric field throughout three-dimensional physical space. Following the discussion above (section 1 of this chapter), its state is, of course, the assignment of an electric field vector at each spatial point. Such a state is also called a field configuration. So, this requires infinitely many real numbers to specify it because, for each of infinitely many points in physical space, we must specify three real numbers. We say the field’s state space, i.e., its set of instantaneous states or configurations, is an infinite-dimensional space. (Again, we say “space” and “dimensional” because we can again use geometrical ideas: much of the intuition, and precise results, about finite-dimensional spaces carries over to infinite-dimensional spaces.)

Now we can state how quantum theory is strange. This lies in a striking contrast between states in classical physics and states in quantum physics. This contrast applies equally to (i) a point particle, to (ii) a finite set of them—any such set would have a finite-dimensional, classical state space—and even to (iii) a field (which has an infinite-dimensional classical state space).

In short, the contrast is this. A classical state is an assignment of specific values to appropriate quantities. For our purposes here, we can neglect ideas about momentum and focus only on position and similar quantities, i.e., on configurations. So, a classical state is an assignment of specific value, either to the positions of a material object’s component parts, or to field quantities such as the electric field at all the points of physical space. But . . .

A quantum state is an assignment of a “square root of a probability” to every possible configuration of the corresponding classical system!

So, a quantum state is a function in the mathematical sense. Its inputs (as we discussed, also called “arguments”) are the classical configurations, and its outputs are “square roots of probabilities.” Here, what matters most—and what is most revolutionary about quantum theory—is not the curious “square root of probability” outputs (which I will discuss shortly), but the fact that a single quantum state mentions, i.e., takes as its domain of inputs, all possible classical configurations.

This fact will be the origin of both the measurement problem and of the Everettian proposal about how to solve it.

Even for a point particle, the proposal is hard to get one’s mind around. For classical physics posited a point particle. Its possible configurations were its possible spatial positions. Agreed, the theory is involved in modality (as discussed in section 3 of chapter 3). But only one configuration is actual: where the point particle happens to be. Now, quantum physics tells us there are no such point particles, each with a single actual position. Each point particle is replaced by what gets called a “quantum particle.” But this entity hardly deserves the name “particle.” For it has no single position. Indeed, it seems thoroughly smeared out in space. For the actual state of this entity, at some time, is an assignment to each point of space—i.e. to each possible configuration of yesterday’s classical point particle—of a number, which (once squared) gives a probability. In short, the state of this so-called quantum particle is a field.

But it is not a field like Newton’s gravity or Maxwell’s electric field. For the so-called particle is not, as I put it, “thoroughly smeared out in space,” in the sense of being a cloud of mass or of electric charge. It is a field of probabilities (or rather, of their square roots). This field, this function on classical configurations that assigns to each configuration a square root of a probability, is called a wave function. It is almost always written as a Greek letter, especially ψ (pronounced “psi”) or φ (pronounced “phi”).

Besides, where classical physics posited two point-particles, and so configurations that are 6-tuples, and so a six-dimensional configuration space, quantum theory says the state is a wave function on this six-dimensional space. So, the “smearing” of what is (undeservedly) called the “quantum two-particle system” is a smearing, not in physical space, but in the abstract space of 6-tuples. And so on, for the quantum replacements of more complicated classical systems. That is, the quantum state, the wave function, has as its domain of inputs (its arguments) the more complicated classical configuration space.

So far, I have summarized the mathematical idea of the quantum state as a function on classical configurations. But the picture gets yet stranger when we ask what the physical meaning of this function is. Again, stranger, even for a point particle—or rather for what replaces the classical point particle and is honorifically labelled “quantum particle.” For I although I said the function gives probabilities, one asks, probability of what? And the answer is a mouthful, which refers to the outcome of a possible measurement, if you were to undertake one, on the system.

For the answer is, for a quantum particle: for each place (i.e., point) x in space, the value (output) of ψ at the argument x gives the probability, were you to measure the quantity position on the system, that you would get the outcome “It is at x.” (Strictly speaking, the value is what I called the curious “square root of probability.” But this idea of square root is a minor aspect, which we can postpone until the next section.)

Equivalently, we can think of measuring a quantity with just two values “Yes” and “No” (or if you prefer: “1” and “0”) that is defined in terms of the place x. In effect, to measure this quantity is to ask the system the question “Are you at x?” Thus, the value of ψ at the argument x gives the probability of getting the answer “Yes” to this question.

The reason why I call this answer “strange” is that it means that the basic interpretation of the theory’s most central mathematical notion, its very concept of state, is in terms of measurement. For think what this implies. Suppose I ask the quantum theorist what their theory of, say, an atom, written in their mathematical language, means in physical terms. I ask: What information about the atom is contained in this mathematical notion ψ that they ascribe to the atom? And their official reply is that ψ gives probabilities of measurement outcomes, measurements using an apparatus that (for an experiment on an atomic system) is typically more than a million million million times bigger than the system being measured.

One naturally asks: How can this interpretation of ψ possibly hold up? For it invokes systems, viz. measurement apparatuses, that are not only utterly different from the system we are concerned with, but also vastly larger—and vastly varied. Can such a grossly extrinsic conception of state, for, e.g., an atom, really be true?

To this, the short answer is that until now, one hundred years after this conception of state was formulated, it is indeed still unrefuted. It is unrefuted for the simple but all-important reason that calculating with it, with due care, delivers the right answers to countless experiments—right answers that underpin countless modern technologies. But, on the other hand, every newcomer, every student of quantum theory, finds this conception of state very hard to believe—indeed, bewildering. Also, most physicists and philosophers who consider this conception in detail, and the questions it raises, conclude that it is not satisfactory.

More precisely, either they conclude that though unsatisfactory, this conception is the best we can now do, and we must hope that the future will bring insight, maybe even a whole theory replacing quantum theory; or they conclude that we already have some special account of the mathematics of quantum theory and/or how we apply this mathematics to the empirical world that vindicates this conception. But there are many such special accounts, which get called “interpretations of quantum theory.” There are about half a dozen main ones, each with many distinctive varieties. The debate between them still rages, decades after quantum theory was formulated—and one such interpretation is the Everettian interpretation, with its multiverse.

But before discussing that, there is more to say about this strange conception of state.

3. Amplitudes and Quantum Fields

We can sum up the exposition so far in terms of how quantum theory replaces the classical physical description of two or more particles. For two particles, the quantum state is an assignment to each pair of points of physical space of a number which, once squared, gives the probability—were you to measure the two position quantities—of getting the answer “Yes” to the two specific questions, “Is one of the particles here?” for the two points. And similarly, for how quantum physics replaces classical physics’ description of N particles. The state is an assignment to each N-tuple of points of physical space, <x₁,y₁,z₁,x₂,y₂,z₂,…,x_N,y_N,z_N> (i.e., each sequence of 3N real numbers) of a square root of a probability.

There are two further comments to make. The first comment is about quantities other than the system’s position, while the second comment is about how quantum theory treats fields rather than particles.

3.1 Quantities Other Than Position

Recall that I called the wave function’s outputs, i.e., the values of the function, “curious square roots of probabilities.” The explanation is that there is a kind of number that this book has so far not mentioned, called a complex number. In effect, a complex number encodes a pair of real numbers in ways that are fruitful. In particular, taking the square of a complex number delivers a third real number (in almost all cases, different from both the given real numbers). So, the values of the wave function are complex numbers. They are called amplitudes (or probability amplitudes).

Using complex numbers is fruitful for quantum theory because it underpins the treatment of quantities apart from position. Recall how our interpretation of the wave function, above, was in terms of probabilities for outcomes of measurements of position. I said nothing about other quantities, such as momentum. But it is natural to expect quantum theory’s conception of state to say something about them. Indeed, it does, by encoding the extra information in its use of complex numbers, rather than real numbers.

Amazingly, the system’s wave function gives, for any quantity (momentum, energy, whatnot), the probabilities of the various possible outcomes of measuring that quantity on the system. So, the conception of state is again bewildering, as regards any quantity. For measuring any quantity, e.g., on an atom, will involve an apparatus that is vastly larger than the atom. But I should also note that the mathematics of how the wave function encodes all the probabilities for all the quantities is unified and very elegant: indeed, very geometrical. For calculating probabilities for various different quantities turns out to be a matter of expressing a vector (not in physical space, but in an abstract space) as a sum of vectors, in various different ways. I will say a bit more about this in the next section.

3.2 Quantum Fields

Finally, let me return to the idea of fields. I introduced this idea for classical fields. Recall the classical description of a fluid, taken as being made, not of atoms jostling each other in a void, but as made—on all length scales, no matter how minuscule—of extended stuff. Or recall the electric field, whose state is an electric vector at each point of physical space. But then we learned that the quantum replacement of a classical point particle is a field of (square roots of) probabilities, defined on configuration space. So, one naturally asks: What about the quantum replacement of a classical field, such as the electric field?

Amazingly, the same strange idea of state works again, as follows. We saw that the configuration of a classical field is given by an infinite number of real numbers (not by 3N real numbers for some whole number N). Thus, the quantum replacement of such a field has as its state an assignment, to each possible configuration of the classical field (one of which, we used to imagine, is physically real), of a complex number, i.e., a probability amplitude. This quantum replacement is called, of course, a quantum field; and the theory of them is called quantum field theory.

Note the dizzying mathematical abstraction. The configuration space of the classical field was itself infinite-dimensional, and now quantum theory posits a state space of functions, each of which has as its arguments that infinite-dimensional space. This makes quantum field theory much more complicated, mathematically, than the theory of quantum particles, i.e., the theory of wave functions on finite-dimensional classical configuration spaces.

Besides, quantum field theory gives a supremely successful replacement not just of what classical physics called “fields,” like the electric field, but also of what classical physics called “particles.” So, there is here another step of conceptual novelty—and again, a dizzying one. For quantum field theory revises what I have said so far about quantum particles—though “revises” means here “extends” rather than “overturns.”

Thus, consider the electron. As we have seen, classical physics treats it as a point particle, located in space at some actual position, moving with some momentum. And similarly for a pair, or any finite number N, of electrons. In short, once you fix the number N, a classical state space is defined. And so far in this chapter, we have learned that quantum physics replaces this with something probabilistic. Namely, the state provides, for any quantity, a probability distribution over its possible values, where “values” are understood to be outcomes of a possible measurement. But we can deduce all these probability distributions from a single representation of the state, the wave function. And as in the classical case, once you fix the number N, the quantum state space—which, to repeat, consists of wave functions assigning amplitudes to each possible configuration of N classical point particles—is defined.

So much by way of summarizing the story so far. Quantum field theory goes beyond this. Its state space is “even bigger” than those we just mentioned. Thus, we have so far mentioned: the space of wave functions for one particle (i.e., complex-valued functions of position in physical space), the space of wave functions for two particles (i.e., complex-valued functions of pairs of positions in physical space), . . ., the space of wave functions for N particles (i.e., complex-valued functions of N-tuples of points of physical space). But for quantum field theory, the state space contains all these infinitely many spaces of wave functions. More precisely, there is a way of adding together two such spaces, or three such spaces, or four of them, or even infinitely many such spaces. (The definition of how to add together spaces is in terms of all the ways of adding together elements of the spaces.) So, for quantum field theory, the state space is the sum of all the spaces just mentioned, for all positive whole numbers N. This vast sum state space is called Fock space.

Thus, quantum field theory envisages the number of particles as a property of the system that can vary from one state to another. That is, it proposes the number of particles as a quantity of the system. This amounts to treating the system as a field, and to treating the quantum particle, discussed above, as itself a state, a state of this field.

So, quantum field theory’s conception of an electron is that it is an excitation (an agitation with an associated energy) of the field. Similarly, two particles are a pair of such excitations, and so on. Thus, the treatment of, say, five electrons that a quantum theorist first learns (using wave functions on 5-tuples of points of physical space, as in section 2 above) turns out to be just the five-particle part of a treatment that encompasses any number of electrons. In common parlance, the elementary quantum state space for five electrons is a subspace of Fock space.

Again, the degree of abstraction is dizzying. But so is the empirical success. The quantum field-theoretic treatment, both of what classical physics called “fields,” e.g., the electric and magnetic fields, and of what classical physics called “particles,” e.g., electrons, yields vastly many precise predictions that have been confirmed. So much so that quantum field theory is now regarded as the lingua franca of physics.

And yet . . . questions remain. For the official interpretation of the state of a quantum field is exactly parallel to what I reported and questioned in the last section for the quantum theory of a fixed number of quantum particles. Thus, for the electric field, the state (i.e., the function on configurations of the classical electric field) encodes probabilities for the various possible values of any quantity, such as energy or momentum of the field, that you might decide to measure on it. So, our interpretative worries at the end of the last section—Can such a grossly extrinsic conception of state really be true?—persist.

4. The Measurement Problem: Schrödinger’s Cat

The worries raised in the last section about how to interpret the quantum state can be sharpened into an argument, one whose conclusion is that quantum theory makes a wealth of flagrantly false predictions about the macroscopic world around us. This argument is called the measurement problem (also, the reality problem). It is vividly illustrated—indeed, symbolized—by Schrödinger’s cat, which is a thought experiment presented by Schrödinger in 1935.

So, in this section, I will explain the measurement problem, and then the cat. To do this, I first need to explain the idea of superposition, and then how the famous equation of motion of quantum theory, Schrödinger’s equation, “preserves superpositions.”

4.1 Superpositions of Quantum States

So far, we have seen that a quantum state prescribes, for any quantity, a probability distribution over the possible outcomes of measuring that quantity (on a system in the given state). For example, a quantity might have four possible values, whose probabilities are 1/4, 1/3, 1/3, and 1/12 (these add up to 1). Of course, a probability equal to 1 for one value, with probability 0 for any other value, counts as a legitimate probability distribution. It is called a trivial distribution (though dogmatic would be a better name). And for a given quantum state, there are quantities that get such a distribution: one outcome is ascribed probability 1, all the others probability 0. Quantum theory has terminology for this. We say that the state is an eigenstate of the quantity (and the quantity is an eigenquantity of the state—though this second word is less common) and the outcome that is “favoured,” i.e., gets probability 1, is the eigenvalue.

As to the more general situation, viz. each outcome getting a probability less than 1 (maybe some get 0, but all together, they add up to 1), the treatment of this situation turns on the idea that quantum states can be added. This is because states are functions, and any two functions that have numbers as their values can be added by, at each individual argument, adding their values. That is, given any two functions f and g defined on the same domain, i.e., the same set of arguments, their sum is defined to be the function, written f + g, whose value for any argument, a, say, is the number f(a)+g(a). (This is called “point-wise addition” of the functions. But “argument-wise” would be a better name, since there is no need for the arguments to be points in a space, in particular, in physical space.) In particular, this is true for quantum states, i.e., functions ψ and φ. We can add them argument-wise, writing ψ + φ.

This addition of functions is exactly parallel to how we add vectors, which therefore helps to give one geometrical intuition. Think of vectors as directed line segments, with their “tails” at the origin, in the plane or in three-dimensional space. We define addition for them by adding them nose-to-tail, and this corresponds to adding their respective coefficients. Thus, in three-dimensional space (recall the triples in chapter 3, section 3), one adds the two x coefficients to get the sum-vector’s x coefficient. Similarly, one adds the two y coefficients to get the sum vector’s y coefficient. And similarly for the z coefficients. So, given two vectors (x₁, y₁, z₁) and (x₂, y₂, z₂), their sum is defined to be the vector (x₁+x₂, y₁+y₂, z₁+z₂). In short, we add corresponding coefficients of a vector, just like we add corresponding values of a function.

Returning to quantum theory, if you add two eigenstates for a quantity that ascribes probability 1 to two different eigenvalues as possible outcomes of a measurement (i.e., the eigenstates disagree!), the result is a state that ascribes a probability of 1/2 to each eigenvalue. This state is called a superposition of the two given eigenstates. And again, we write a “+” sign for the addition of states.

This example was one of equal weighting. (One could also call it “50-50 weighting.”) But we can also increase one weight and decrease the other. This is like lengthening one directed line segment and shrinking the other, before we add them nose-to-tail. The resulting state is again called a superposition and is written with a “+” sign. It gives the two eigenvalues different probabilities. And again, the probabilities that the state prescribes are the squares of its values for the two possible outcomes, i.e., the two eigenvalues. So, writing √ for “square root” (and setting aside the topic of complex numbers, cf. section 3.1), the values might, for example, be √(6/10) and √(4/10). Besides, thinking of the state as a vector rather than a function, it has coefficients (√(6/10), √(4/10)).

Note that for any given quantity, almost all states are superpositions for that quantity: its eigenstates are a small, special set of states. In other words, for any given quantity in our system, almost all the probability distributions that quantum theory envisages for it are non-trivial. That is, more than one possible value (eigenvalue) of the quantity gets a probability greater than zero.

4.2 The Schrödinger Equation is Deterministic and Linear

Just as classical theories have equations of motion that describe how the system’s state changes over time (cf. section 6 of chapter 2, sections 3 and 8 of chapter 3, and section 1 of chapter 4), so does quantum theory. Its equation is called the Schrödinger equation (he published it in 1926). It has two crucial features.

First, it is deterministic, in the sense of our previous discussions (chapter 3, sections 3 and 8). That is, given the system’s quantum state at a time, and the forces exerted on it in the past and future, the equation prescribes what the state is at all other times. As we put it there, the equation prescribes a unique curve through the quantum state space, i.e., the space of wave functions. Later (in section 5 and especially section 11), we will confront the obvious question: How can this determinism be reconciled with quantum theory’s use of probabilities?

But here in this section, what matters is the second crucial feature of the Schrödinger equation, namely, that it preserves the addition structure of quantum states. That is, if (i) a system’s quantum state, ψ, would evolve (i.e., change), in, say, five seconds, to a state ψ,′ and (ii) another of its states, φ, would evolve in that same five seconds to a state φ,′ then any superposition state of ψ and φ got by, for example, “lengthening” ψ and “shrinking” φ, e.g., the state (√(6/10)ψ+ √(4/10)φ), would evolve in that same five seconds to the corresponding superposition (√(6/10)ψ′ + √(4/10)φ′).

In short, quantum states’ addition structure—including the numbers that are the weights, √(6/10), etc.—is preserved under time-evolution. This property of the Schrödinger equation is called linearity: the Schrödinger equation is linear. (Here, for simplicity, I have written familiar real numbers, √(6/10) and √(4/10), for the weights, rather than the complex amplitudes, i.e., “curious square roots of probabilities,” as mentioned in 3.1. The point about linearity is unaffected.)

4.3 Systems in Superpositions Lack Values

So far, this section has just done some stage-setting: the addition of functions and vectors, and the jargon of eigenstates, eigenvalues, superpositions, and linearity. But now quantum theory, in its orthodox formulation, makes an interpretative claim. It is a very important claim, since it is restrictive, and it leads directly to the measurement problem.

The problem is that quantum theory says that if for a given quantity, the state is a superposition for that quantity (so, not an eigenstate—it ascribes a non-trivial, “non-dogmatic” distribution over possible measurement outcomes), then the system has no value whatsoever if the quantity.

The paradigm case is the quantity position, for a quantum point particle, as in section 2 above. The state is a wave function ψ that assigns a complex number to positions in space, whose squares give probabilities of outcomes of position measurements. In picturesque language, the state gives probabilities of getting the answer “Yes” (“1”) to asking the system “Are you at position x?.” (From now on, I will say “system” rather than “point particle” or “particle,” since the word “particle” strongly connotes a definite position (and definite momentum, etc.), which, as I will stress, quantum theory denies.)

Now imagine that the value of ψ is non-zero only in two separated spatial regions, which we call “L” and “R.” Here, “L” and “R” stand for “Left” and “Right.” For nothing here will depend on space being three-dimensional, so we may as well imagine it as one-dimensional, like “life on a railway line.” So, if we draw a graph of the values of ψ (more precisely, their squares), it looks like two humps with a flat line between them. (Think of the road sign for “Bumps in the road ahead.”)

So, quantum theory says that ψ is a superposition of two states. One of these two states is an eigenstate for being in L. This state says the system would, with probability 1, be found in L if measured. The other state is an eigenstate for being in R; this state says, correspondingly, that the system would, with probability 1, be found in R if measured. This superposition can be written with a “+” sign for the addition of states. We could write mnemonically: ψ = in L + in R.

It is also common to write states between a vertical line and an angle bracket (a notation invented by Dirac (1902–1984), one of the great quantum physicists). So, for the equal or “50-50” weighting of L and R, we write: |ψ > = |in L > + |in R >.

But, says quantum theory, a system in state ψ has literally no position at all. So, the system exists—but it has no location. It only has dispositions to be found in L or in R, if we were to measure position.

Similarly, for other quantities, such as momentum. A system might be in a superposition of momentum eigenstates, for two different values (measurement outcomes) of momentum, say “1” and “2.” So, for equal weighting of “1 and “2,” we can write the state as |momentum 1> + |momentum 2>. Again, quantum theory says that a particle in this state has literally no momentum at all.

To sum up, since for any given quantity, almost all states are superpositions, quantum theory’s denial that systems in superpositions have a value (for that quantity) makes the lack of values endemic. Obviously, this situation prompts the question: How can this lack of values be reconciled with the apparent fact that objects do have values for position, momentum, and other quantities such as energy?

Besides, recall how classical physics gives supremely successful descriptions and explanations of the physical behaviour of macroscopic objects by ascribing them definite values of position, momentum, etc., subject to equations of motion (cf. chapter 2, section 6, chapter 3, section 3 and 8, and chapter 4, section 1). Once we recall this, the question becomes more pointed: How can quantum theory’s denial of values be reconciled with the supreme success of classical physics?

4.4 The Problem

This question is now easily sharpened into an argument, an argument that the lack of values contradicts countless facts that macroscopic objects have definite values. We only need to transmit the lack of values from the microscopic realm of electrons, atoms, etc., for which quantum theory is indeed successful, to the macroscopic realm of tables, chairs, etc., where classical physics, with its definite values, is successful. This is done by describing, within quantum theory, a measurement of a quantity on a microscopic system (say, an electron) that is in a superposition (for that quantity). In such a description, we can see how the lack of values is transmitted to the macroscopic realm—surely contradicting countless facts of definiteness.

So let us assume we have an apparatus for measuring an electron’s momentum that is reliable on each of the two eigenstates, |momentum 1> and |momentum 2>, in the following sense. Starting the apparatus in an appropriate “ready” state, the state of the pair of systems changes over time, so that at the end of the measurement interaction, the pointer on the apparatus reads the corresponding eigenvalue of momentum. Thus, we think of the pointer as being, at the end of the measurement, in front of the digit “1” or “2,” painted on a dial.

Let us use an arrow, →, to symbolize the change of state over the period of the measurement. And let us write the state of the pair of systems by simply juxtaposing their states on the paper. This state of the pair is, in the jargon of logic or philosophy, the conjunction of the two components’ individual states. In physics jargon, it is called a product state (or “the product of the individual states”).

Then our assumption of reliability, that measuring either of the electron eigenstates yields a veridical reading at the end, can be written as:

|momentum 1> |ready> → |momentum 1> |reads “1”>

and

|momentum 2> |ready> → |momentum 2> |reads “2”>.

So far, so good. But now suppose the electron’s initial state is a superposition, for example, of our two eigenstates, |momentum 1> and |momentum 2>. Then the composite system of electron and apparatus has the conjunction or product state, as its initial state: (|momentum 1> + |momentum 2>)|ready>. This can also be written as |momentum 1>|ready> + |momentum 2> |ready>.

Then, because the Schrödinger equation is linear, we must accept:

(|momentum 1> + |momentum 2>)|ready> →
|momentum 1> |reads “1”> + |momentum 2> |reads “2”>.

This is the punchline. For consider the state of the composite system after the measurement, i.e., the second line, or right-hand side, of this formula. Consider what it says about the apparatus, in particular, the position of its pointer. The main point to notice is that it is not an eigenstate of the quantity pointer position, i.e., of the quantity position for the pointer.

(There is also another point to notice about this state. It is not a product state: it is not a conjunction of states for the components. Such a state is called entangled, and the theory allowing for such states is called entanglement. We will return to this in section 7.)

So, quantum theory denies that the pointer has any position. The situation is like in the example above: |ψ > = |in L > + |in R >. Namely, the pointer only has dispositions to be found in front of the numeral “1” on the dial, or to be found in front of the numeral “2” on the dial—if we were to measure it.

But the pointer not having any position surely contradicts the fact that macroscopic apparatuses give definite readings. Besides, the argument is so simple—depending only on microscopic superpositions and the Schrödinger equation being linear—that it suggests more generally that orthodox quantum theory’s denial of values in the microscopic realm will contradict countless facts of definiteness about the macroscopic realm.

So, this is the measurement problem. As I said at the start of this chapter, it has no agreed solution. So, our first job, in the next section, will be to consider some possible solutions, including the Everettian proposal, on which we will then focus.

But before that, it is worth summing up the measurement problem, with Schrödinger’s own description of his eponymous cat. In his paper of 1935—which is still worth reading for many reasons, some of which we will touch on later—he writes (at the end of section 5 of his paper, translation from 1980):

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The wave function of the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.

It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy.

5. Solving the Problem: The Usual Suspects

There are three main strategies for addressing the measurement problem. It will be clear that each is a broad church that includes many versions, and that jointly, they are exhaustive. So, one has to endorse one of the three. But I will not try to formulate the strategies precisely and will only give two or three examples of each strategy. Nor is this trio of strategies, and their versions, original to me. Many surveys of the measurement problem give a similar trio. Hence, this section’s title: “Solving the Problem: The Usual Suspects.”

I especially admire the survey by Bell (1986), which I will follow. It is a brilliant and equation-free introduction to quantum theory and its interpretation. Bell describes, for each of his three strategies, a pair of versions; and for each pair, he gives what he wittily calls a romantic version and an unromantic version. Unsurprisingly, the Everettian interpretation will be the romantic version within its pair.

Bell also makes it clear that for each pair, he prefers its unromantic version. That is perhaps disappointing. But as I discussed in chapter 1, section 4, we each have an intellectual temperament that is hard to change, and about which we are obliged only to be self-aware. (John Bell (1928–1990) was a profound quantum physicist, as well as a gifted writer. He also originated what is now called “Bell’s theorem”: it is about how correlations between quantum systems defy a very natural form of probabilistic explanation.)

The first strategy is to reject, somehow or other, the formulation of the problem. That is, one rejects the premises of the argument leading to the contradiction. The main idea must be to deny that the quantum state is “physically real,” in any sense of that phrase, which makes the final post-measurement state (at the end of the last section) contradict the macroscopic pointer having a definite position.

Bell calls his two versions of this strategy “pragmatism” (the unromantic version) and “complementarity” (the romantic version). Here, “pragmatism” means—not the philosophical tradition launched more than a hundred years ago by American philosophers such as Peirce, James, and Dewey—but “being practical.” That is, using the theory to calculate probabilities of outcomes, without taking its words and concepts to describe any reality other than those outcomes. (Philosophers often call this “instrumentalism”: the theory is an instrument, a tool for predicting observable facts, but not a description of the world, especially not the world beyond observations.) Obviously, pragmatism, in this instrumentalist sense, shades into simply not wishing to ponder whether the theory goes beyond predicting observable facts, rather than the firm view that it does not go beyond such predictions.

On the other hand, “complementarity” is Bohr’s word for his views that attempted to go beyond pragmatism and, by explicitly philosophical argumentation, to solve the measurement problem. His core idea was that since measurement outcomes must be stated using the concepts of classical physics, there is no contradiction with, in Schrödinger’s phrase, “macroscopic determinacy.” And this is so, even though (1) the Schrödinger equation, with its linearity, is always correct, and (2) quantum systems have no values for quantities except the eigenvalue for those quantities for which they are in an eigenstate.

The other two strategies deny, respectively, these claims, (1) and (2). These were, of course, the premises of our formulation of the measurement problem. So, one might say that the first strategy aims to dissolve, rather than solve, the problem, while these other two strategies accept that the problem is genuine and then propose to solve it. Bell himself clearly prefers these two strategies (in their unromantic versions) over the first. As he puts it in another paper, “either the wave-function, as given by the Schrödinger equation, is not everything or it is not right” (1987, 201). Here, “the wave-function is not everything” means that a system has values for quantities additional to the eigenvalues ascribed by orthodoxy, i.e., (2) is wrong. And “the wave-function is not right” means that the system is being attributed the wrong state, because (1) is wrong. In our paradigm case of the pointer, we should attribute to it a state of definite position, not a superposition of position eigenstates, and to do so, we should revise the Schrödinger equation.

So, suppose we deny (1) that the Schrödinger equation is always correct. Again, this strategy comes in various versions.

The simplest version says that at the end of a measurement process (like that at the end of the last section), the troubling superposed final state is replaced by an eigenstate of pointer position, so that indeed, the pointer has a definite position. (Usually, advocates of this version also say that the measured system goes into an eigenstate of the measured quantity—in our example, the electron goes into a momentum eigenstate. But we can concentrate on the measurement apparatus and its pointer.) Besides, which eigenstate replaces the troubling, superposed state is said to be a matter of sheer chance, with each eigenstate occurring with a probability equal to its weight (more precisely, the square of its complex amplitude weight) within the superposed state.

Again, there is jargon: This replacement of the superposition by the eigenstate is called the projection postulate or the collapse of the wave function. (But this second phrase is also used as a vivid label for the measurement problem, not just for this approach to solving it.) And the prescribed probabilities (the squares of the complex amplitude weights) are called Born-rule probabilities. (This is in honour of Max Born (1882–1970), one of the half-dozen co-discoverers of quantum theory who realized this role of probabilities in a 1926 paper.)

This simplest version of denying (1) occurs in many textbooks on quantum theory. But evidently, it is vague and contentious. For when exactly is the state calculated from the Schrödinger equation to be “replaced”? Or in other words: What is the exact definition of “the end of the measurement process”? And since a measurement is, after all, a physical process, how is this suspension of the theory’s equation of motion to be justified? Clearly, this version is close to what Bell called “pragmatism,” and we are back at our initial bewilderment that the notion of a system’s state should invoke the idea of measurement, which is so extrinsic to the system.

Two other versions of this strategy, developed in response to these difficulties, are worth mentioning. The first is very speculative; the second is down to earth. So, Bell calls them, respectively, romantic and unromantic (and as I said, he prefers the latter).

The first says, in a slogan, that consciousness collapses the wave function. The idea is that the Schrödinger equation only falters at the interface between mind and matter. Although the inanimate physical world may get into states superposing macroscopically distinguishable alternatives (e.g., of positions of a pointer); once a conscious being “looks,” the state changes to a macroscopically definite state.

Obviously, people will differ about how plausible they find this proposal. Someone who is a physicalist (cf. section 8 of chapter 3) will almost certainly reject it. Yet it might at first sight appeal to the practically minded physicist, on the grounds that it makes the measurement problem “someone else’s problem.” They might say: “Surely, physics has no responsibility to describe the relation between mind and matter.” But I would say that this response is just a verdict about disciplinary boundaries, or the division of cognitive labour—about who needs to worry. Whichever discipline one takes the problem to fall within, the proposal is obviously hard to make precise, hard to gather evidence for, and indeed, hard to believe. In particular, why should these collapses of the wave function due to consciousness respect the Born-rule probabilities?

The other version of this strategy is much more down to earth, or as Bell says, unromantic. It seeks, as quantum theory’s fundamental equation of motion, a “cousin” of the Schrödinger equation. This equation is to be chosen to (i) agree with the Schrödinger equation for microscopic systems like atoms, so that it also gets the vast amount of confirmation that the Schrödinger equation has gathered over the last hundred years; and yet (ii) to disagree with the Schrödinger equation for macroscopic systems, like pointers and cats. So, according to this version, the wave function of a quantum system does indeed collapse, i.e., transit to an appropriate eigenstate, in suitable—in particular, suitably large—systems. And this occurs in the inanimate world, wholly irrespective of consciousness: the collapse of the wave function is an indeterministic physical process. So, our task is to find the equation that describes these collapses precisely, in a way that meshes with the established successes of the Schrödinger equation. This will include recovering the Born-rule probabilities for experimental outcomes. In short, our task is “theoretical physics as usual: find the right equations.”

In the last forty years, a great deal has been learned about such cousin equations, both mathematically and physically. But no single proposed equation has yet won the allegiance of physicists, on either theoretical grounds or by being confirmed by experiment. So, the question of whether the Schrödinger equation will indeed be overturned remains open. For this, we must wait for the future of physics.

Finally, suppose we deny (2). That is, we say that quantum systems do have values for quantities for which their state is not an eigenstate. The motivation for saying this is, of course, to keep the post-measurement state given at the end of the last section, but nevertheless to ascribe a definite position to the pointer.

One version of this strategy is, again, “theoretical physics as usual.” It was invented by de Broglie (1892–1987). In the mid-1920s, he was another of the half-dozen creators of quantum theory, and his formulation of the theory explicitly attributes values to quantities additional to eigenvalues (of quantities for which the state is an eigenstate). To be precise, let us consider a system that orthodox quantum theory (the textbook version) calls “N quantum particles.” De Broglie proposed that this system’s real physical state includes, in addition to the wave function ψ on configuration space, which always obeys the Schrödinger equation, also our “old friends”: N point particles, each with a definite actual position.

In themselves, these point particles are as described by classical physics: at any instant, each has a definite position in space, and over time each moves in a continuous trajectory. The difference from classical physics lies in how they move. Namely, at each instant, their velocity is determined by a combination of (i) the wave function and (ii) the positions of the other point particles in the system. (The position of another particle contributes to determining the given particle’s velocity in an action-at-a-distance manner, similar to Newtonian gravity—except that the influence does not diminish if there is a larger distance between the particles.) The idea in (i), that the wave-function, though it lacks mass and energy, “guides” all the particles in the system, has given rise to the theory’s name: pilot-wave theory. (Think of how a pilot guides a ship, but he does not do so by the effects of his mass or energy.)

We do not need the details of this theory. What matters for us is that the pilot-wave description of how the point particles move gives the key idea for solving the measurement problem. (This merit became clearer in the work of David Bohm (1917–1992), who in 1952 rediscovered the pilot-wave theory.) In the last section’s measurement scenario, the pilot-wave theory takes the pointer to really be a cloud of point particles. Then it gives a detailed description of how, in each individual measurement, this cloud of particles is guided by the wave function (which always obeys the Schrödinger equation), to be either in front of the digit “1” painted on the dial or in front of the digit “2.” In short, the pilot-wave theory, by ascribing values of position additional to those ascribed by orthodox quantum theory, secures a definite outcome for each individual measurement, solving the measurement problem.

The second version of this strategy, i.e., the second way to claim that quantum systems do have values for quantities for which their state is not an eigenstate, while the Schrödinger equation is nevertheless always correct, is the Everettian proposal, with its multiverse. It will be our topic for the rest of this chapter.

6. Everett’s Proposal: A Bluff?

Let me start by baldly stating the Everettian proposal. This will show how it counts as a version of this last strategy. (From now on, I will talk about “the Everettian,” rather than Everett himself, since the proposal has developed a good deal since Everett’s paper in 1957. Besides, there is controversy about whether the version of the proposal that is nowadays dominant—on which I will concentrate—matches Everett’s own ideas.)

The key ideas are as follows. The cosmos, as a whole, has a quantum state, which always evolves according to the Schrödinger equation: indeed, a very grand version of the equation that describes quantitatively how all the cosmos’ component parts interact, exerting forces on one another. Needless to say, no one has come close to writing down this version of the Schrödinger equation. But the Everettian proposes that it is, as usual, deterministic, so that the wave function of the cosmos never collapses. (In section 11, we will confront the obvious question: How can this determinism be reconciled with quantum theory’s use of probabilities?)

This state is usually written as ψ, where the use of the capital Greek letter (again pronounced “Psi”) is, so to speak, honorific, for no one has the faintest idea how to write it down in detail.

ψ is usually called “the universal state” or “the universal wave function.” But in this book, I have adopted, since the introduction, “multiverse” as the name of all of reality in the most inclusive sense; and “world” or “universe” as the name for its “more familiar” parts, where, as we discussed, each part is understood to mean “throughout all of time and space.” So, using my jargon: the cosmos’ quantum state ψ is the multiverse’s quantum state. But since no one says “the multiversal state” or “the multiversal wavefunction,” I will, in this chapter, speak of “the cosmos’ quantum state.” Hence, this chapter’s title.

Our knowledge of quantum theory, and our empirical success in applying it to small systems, suggests that superpositions will promulgate, in the way we saw in the measurement problem at the end of section 4. So, we have good reason to think that the cosmos’ quantum state is a vast superposition, i.e., a vast sum, of product states. Each of these is a long “conjunctive state” for countless component systems—not just an electron and a measurement apparatus or its pointer, but countless electrons, quarks, molecules, specks of dust, etc.

Again, there is jargon: When items, like numbers, vectors, or these product states, are added together, they are called terms or summands (for “item that gets summed”). So, these product states are jargon of the vast superposition.

The Everettian proposes a literal interpretation of this vast superposition. Just as this state contains vastly many product states, so also the cosmos (i.e., the multiverse) contains a plethora of Everettian “worlds.” (These are also called “branches.”) Each is represented by such a product state. Some (perhaps many) of these worlds are something like the macroscopic realm familiar to us, with all macroscopic objects (pointers, tables, etc.) in definite positions.

But the worlds differ among themselves about these positions; and it is relative to each such world that there are extra values, i.e., values additional to orthodox quantum theory’s attributing only eigenvalues. Think, for example, of section 4’s toy model of a momentum measurement. The two possible outcomes were distinguished by two different positions of the pointer: in front of “1” or in front of “2.” The Everettian proposes that the two outcomes, the two positions, are in different worlds.

This bald statement of the Everettian proposal prompts the obvious question: If this is so, how come I have no evidence of the other worlds? In particular, how come I experience a single definite macrorealm, a macrorealm whose objects are definite in position, and in other quantities like momentum?

As I see matters, there are two main Everettian answers to this question: a traditional one, which was prevalent in the literature between Everett’s original paper in 1957 and about 1990; and a modern one, which has been prevalent since the mid-1990s. The first answer was cast in terms of measurement processes and their outcomes, as in section 4’s toy model. I think this first answer is unsatisfactory. It seems like a bluff or a mere debating tactic. I will discuss it in this section. But the second answer is satisfactory, although not wholly convincing. It is not cast solely in terms of measurement processes and their outcomes. It considers equally all macroscopic objects that classical physics successfully treats as having definite values for position and other quantities, like momentum. And some of these are indeed tiny, for example, specks of dust that are too small to be visible (though I will still say “macroscopic”). This second answer secures definiteness of position, etc. (relative to a world) even for such tiny specks. It does so by appealing to a very important phenomenon that I have so far not mentioned: decoherence. I will explore this phenomenon in the next section.

I say “as I see matters,” because since Everett’s original paper in 1957, many different Everettian answers have, of course, been proposed—and then disagreed with on various grounds. But my claim that there have been two main answers is widely agreed.

So, first, here is what I called the traditional answer. It says that the reason why appearances are indeed definite—just as much as they are on a proposal that the wave function collapses—is that the objects involved in the problematic superposition, say, at the end of a quantum measurement, each split into many copies, corresponding to the various worlds. So, in our toy model of measurement with two outcomes, the apparatus’ pointer splits into copies, some with the “1” outcome and some with the “2” outcome. Commentators proposing this answer differed about how to understand the splitting, in particular, about how many copies there are. Some said that for two outcomes, there are just two copies; others said that for each outcome, there are many copies, perhaps infinitely many. (I will return to this debate in later sections.) But the common idea is that this splitting secures definite appearances. For appearances only “appear” within a world, and within a world, the wave function is, by definition, the corresponding term, which is an eigenstate of the quantity, such as pointer position, that in order to solve the measurement problem, we want to have a definite value.

The trouble with this answer is not that the idea of splitting is plain wrong. It is that, stated so briefly, the answer is too programmatic: it raises more questions than it answers. If the splitting is a bona fide physical process, we need to hear details, for example, about how it can be consistent with laws like the conservation of mass or energy. If it is somehow a conceptual splitting, without a physical description, then there is philosophical work to do to explain what it involves. In particular, if the splitting is “conceptual” in the sense of being a distinction made by a conscious mind, then presumably, there is no splitting in those regions of the cosmos without conscious minds. In that case, the proposal is similar to that considered in the last section, that consciousness collapses the wave function, and it is thereby similarly hard to make precise and to gather evidence for.

In my view, until about 1990, most of the Everettian literature did not adequately answer such questions. Hence my accusation that it seems like a bluff. Of course, I am not alone in my misgivings: many authors pressed such questions. In particular, Bell was very doubtful. In his 1986 paper, which I recommended and summarized in section 5, he says that the Everettian proposal “is surely the most bizarre” and “extravagant, and above all extravagantly vague, hypothesis. I could almost dismiss it as silly” (1986, 192, 194). But, as I announced above, since about 1990 (when Bell died), the Everettian literature has appealed to decoherence, which does address the questions.

That will be the topic of the next three sections. But first, it is worth discussing—and criticising—an analogy that is sometimes suggested in defence of the suspiciously programmatic “splitting” answer above.

The alleged analogy is that the “splitting” answer is like what is surely the right reply to the objection (perhaps apocryphal) against the proposal by Galileo and other advocates of a heliocentric astronomy, i.e., the proposal that the Earth goes around the Sun. The idea of the objection (i.e., in defence of the traditional, Aristotelian geocentric astronomy, that the Sun goes around the Earth) is to appeal to appearances. Namely, it looks (especially at sunrise and sunset) as if the Sun goes around the Earth. So, the objection is that the heliocentrists’ radical proposal seems to conflict with the appearances.

The reply that Galileo and the heliocentrists are said to have made is that, in fact, assuming that the Earth goes around the Sun leads to the very same appearances. That is, the appearance at sunrise is exactly the same whether you describe this as the sun rising above the horizon or as the horizon sinking below the sun. (And similarly, of course, at sunset: the appearance is exactly the same whether you say the sun is sinking, or the horizon is rising.)

Similarly, it is alleged, for the Everettian’s “splitting” answer. If we assume such a splitting, the appearances will be just as the objector says they are, that is, perfectly definite.

But I submit that this analogy has a merely rhetorical force. I agree that the broad logic of the two disputes is the same. In both cases, the new radical proposal (Galileo’s or Everett’s) replies to the objection that it conflicts with appearances by saying: “No, I do not: I accord perfectly well with appearances.” But there is a big difference between the two cases.

For Galileo and the other advocates of a heliocentric astronomy can readily spell out how they accord with appearances. It is a matter of optics, i.e., the paths of light rays. In particular, it is straightforward to argue that (i) the appearance at sunrise is a matter of the angle between a ray of sunlight and one’s line of sight to the horizon increasing (and similarly, at sunset: decreasing); and (ii) this increase (respectively: decrease) depends only on a relative motion of sun and horizon. But the Everettian’s idea of splitting gives no such straightforward argument for recovering definite appearances: it leads only to the questions I pressed above.

7. Doing Better with Decoherence

However, as I said, by appealing to a process called decoherence, the Everettians can make much better sense of their proposed splitting, and since about 1990, they have done so. So, in this section and the next two, I will spell out what decoherence is and how it clarifies what splitting involves. It will also be clear that decoherence is important for all approaches to quantum theory, not just for Everett’s: any of section 5’s interpretations need to accommodate it.

There will be three stages, one in each section. This section gives the basic idea of decoherence. In the next, decoherence helps make the definition of an Everettian world (or “branch”) more precise. After that, decoherence will suggest that macroscopic objects, such as cats or pointers, are not aggregates of stuff, but enduring and stable patterns. (Then in the chapter’s final sections, I will turn to other topics, especially probability.) The plan for this section is that I will state the idea of decoherence in 7.1, and then, in 7.2, state a merit and a demerit of it.

7.1 The Idea

“Decoherence” means, in this context, the diffusion of coherence. Here, “diffusion” means spreading, namely, spreading from the system of interest to its environment with which it is interacting.

“Coherence” is physics’ jargon for some characteristic differences between (i) the probability distributions prescribed by quantum states, especially superpositions, and (ii) those prescribed by classical states. We saw already in section 4 that a quantum superposition prescribes a non-trivial probability distribution over the possible outcomes of a measurement of the quantity concerned. On the other hand, eigenstates are “dogmatic” or “opinionated”—they give probability 1 to just one outcome.

Classical states—meaning states prescribed by classical physics—also prescribe distributions, once we include probabilistic mixtures of states. Thus, imagine being given, say, three states, and taking a mixture of one quarter, one quarter, and one half, of them. This means, for example, taking a thousand systems, of which 250 are in the first state, 250 are in the second state, and 500 are in the third. Then the predicted statistics for measuring any quantity on a randomly selected member of the set of a thousand systems would be the average, with weights 1/4, 1/4, 1/2, of what the three given states prescribe.

The important point here is that superpositions cannot be understood as such classical mixtures. For although for a single quantity, a superposition and a classical mixture might prescribe exactly the same distribution, there will be other quantities (which are said to not commute with the first one), about which their distributions differ. The numerical differences, outcome by outcome, between these distributions (for this other quantity) are called interference terms.

So, in short, interference terms are the signature of a state being a superposition. (The word “interference” comes from the physics of waves. When the peaks of two water waves meet and make a yet higher peak, we say that the waves interfere constructively; when a peak of one wave meets a trough of the other, and they cancel out to give a level surface, we say that the waves interfere destructively.)

So, imagine a quantum system interacting with its environment. Given our interest in the measurement problem, a paradigm case is the pointer of an apparatus (or a cat) interacting with the air around it, by air molecules bouncing off it. So, quantum theory tells us, of course, that the quantum state of the composite system, pointer plus air, prescribes probabilities for the various possible outcomes of measuring any quantity on either component, the pointer or the air. And in general, this composite quantum state will be a superposition of product states, so that the probabilities it prescribes will include interference terms—even for a quantity on a component system. That is, one expects, a priori, that the component system’s state will be superposed, i.e., its probabilities will have interference terms.

Now we can state the punch line about decoherence. Namely, according to many realistic models of how a macroscopic object like a pointer interacts with its environment, such as air molecules, the interaction establishes very rapidly a composite quantum state that falsifies this a priori expectation concerning the macroscopic object. That is, very soon after the interaction starts, the composite quantum state prescribes probabilities for the macroscopic object whose interference terms are negligible. In other words, the composite quantum state determines a state for the pointer that is almost a mixture, in the classical sense above. For it differs from a mixture only by tiny interference terms.

More is true. The states within this mixture, that the pointer is “almost in,” are the states that we intuitively want to solve the measurement problem. For they are, roughly speaking, states of definite position: for example, the position of the centre of mass of the pointer. Thus, for section 4’s toy model of measurement: once we augment the model by including in our analysis the air molecules, then the post-measurement state of the pointer, as determined by the state of the entire composite system (measured system, pointer, and air molecules) is—neglecting tiny interference terms—a mixture of the centre of mass being in front of the numeral “1,” and it being in front of the numeral “2.”

And yet more is true. One does not just get such promising-looking mixtures in situations of explicit measurement, involving everyday-sized objects like pointers. Nowadays, there are detailed models of much smaller objects, immersed in their environment, that give mixtures of states ascribing definite values to the quantities that intuitively we want to be definite. For example, a tiny dust particle, a tenth of a millimetre in diameter, floating in outer space, will be in a mixture of states of definite position, thanks just to its interacting merely with the dim light of the stars.

(Two paragraphs above, I said “roughly speaking, states of definite position,” because in many models, the states in the mixture obtained are so-called coherent states. These are states whose probability distributions are sharply peaked for both position and momentum, so that a system in such a state seems almost definite in both position and momentum. I say “sharply peaked” because the distributions have enough spread to obey quantum theory’s Uncertainty Principle, which vetoes having absolutely precise values for both position and momentum.)

So, to sum up so far, decoherence is the fast and ubiquitous process whereby, for appropriate physical quantities of a system immersed in its environment, the interference terms that are characteristic of a quantum state (a “superposition”) as against classical states (a “mixture”), become tiny. In effect, the coherence has diffused from the system to its environment.

We will see in sections 8 and 9 how, with decoherence, the Everettian can give a much better account of the splitting of worlds. I end this section by stressing two important features of decoherence that apply not just to Everettians, but to all approaches to the measurement problem. The first is positive, while the second is a limitation.

7.2 A Merit and a Limitation

The positive feature is flexibility. The need to be flexible becomes clear when we realize that to solve the measurement problem, we need the classical physical description of the world to be vindicated only approximately, not in every detail, by quantum theory. Thus, we should admit that we need only some subset of quantities, not all quantities, to have definite values. And we should allow that maybe this subset is specified contextually, even vaguely. And besides, maybe the values should only be definite within some margins of error, even vague ones.

The point now is that decoherence secures this sort of flexibility. For which quantity on the system is “preferred” (i.e., rapidly becomes definite, in that the state is a mixture of its eigenstates) is determined by the physical process of interaction—whose definition in the model can be legitimately varied in several ways. Here are three examples of such ways. One can vary the definitions of the system-environment boundary, the time at which the interaction is taken to end, and what counts as the state being “sharply peaked” for a quantity.

I now turn to the limitation. It is very important, since it is often ignored—in the research literature as well as in popular accounts of decoherence.

The limitation is that decoherence does not just by itself solve the measurement problem. More precisely, it does not imply that in any individual case, the macroscopic system actually is in one of the states in the mixture. It implies only that the quantum probabilities for any quantity are as if the system were in one such state.

Furthermore, quantum theory implies that the macroscopic system is in fact not in one of those states. This last is a subtle point, which many textbook discussions miss. Some authors signal this point by using special jargon. They call the mixture that the macroscopic system is in an improper mixture; and for a mixture for which the system in any individual case is indeed in one of the states getting mixed, they use the phrase a proper mixture or an ignorance-interpretable mixture.

So, decoherence ensures that the macroscopic is in an improper mixture. In other words, setting aside this jargon, to say instead that the macroscopic system is in any individual case in one of the states getting mixed would contradict the original hypothesis that the entire composite system, including the environment, is in a superposition, not a mixture.

Though this point is subtle, it is uncontroversial. In fact, it is already clear in Schrödinger’s 1935 paper, in which he introduced his eponymous cat. For in sections 10 to 13 of that paper, he discusses entanglement (indeed, he introduces this word) and expounds the point.

This limitation of decoherence can be made vivid in terms of Schrödinger’s cat. Namely, at the end of the measurement and after the decoherence process, the quantum state still describes two cats, one alive and one dead. It is just that the two cats are correlated with very different microscopic states of the surrounding air molecules. For example, an air molecule will bounce off a wagging upright tail (as in a living cat) and a stationary horizontal one (as in a dead cat) in different directions.

Indeed, we can put the point in cartoon form. Think of the cat being alive as it having vertical legs, a vertical tail, and a smile on its mouth. Similarly, think of it being dead as it having horizontal legs, a horizontal tail, and a frown on its mouth. Then the usual image for the measurement problem, i.e., the Schrödinger cat paradox, is the combination of these two configurations of the cat. Using plastic transparencies, as in a traditional classroom, one would overlay one transparency, portraying the dead “horizontals” cat, on another transparency, portraying the living “verticals” cat—getting the usual paradoxical double image.

Now the advocate of decoherence points out to us that the cat is really in an environment of air molecules, etc. So, they point out that a realistic image for the living cat, standing and smiling, should show many air molecules bouncing off the cat with such-and-such trajectories, while a realistic image for the dead cat, lying down and frowning, should show a similar large number of air molecules bouncing off the cat, but with different, so-and-so, trajectories. And so, the realistic final state of the entire composite system, cat plus air molecules, is the superposition of states corresponding to these two images.

All this we must accept. But it seems blatantly to not solve the basic problem, that quantum theory describes the cat as blurred between being alive (“verticals”) and being dead (“horizontals”). This problem seems wholly unaffected by invoking the air molecules. After all, think again of plastic transparencies. Adding scattered dashes portraying such-and-such trajectories to the transparency for the living “verticals” cat, and adding different scattered dashes portraying so-and-so trajectories to the transparency for the dead “horizontals” cat, does nothing toward unblurring the image. If anything, it makes the situation worse. For it shows the air molecules being blurred and “suffering” indefiniteness of position, just as much as the limbs of the cat.

In section 9, we will see how the Everettian can accommodate this limitation by taking the cat, and other macroscopic objects, to be enduring and stable patterns. But to prepare for that, we need more detail about the Everettian’s concept of a world.

8. A Sketch Definition of “World”

In this section, we will see how decoherence clarifies the Everettian’s idea of worlds and thus of how they “split.” I will first, in 8.1, use decoherence to state a sketch definition of “world.” Then, in 8.2, I will explain two ways in which this sketch definition needs to be improved.

8.1 The Definition

At the beginning of section 6, I introduced the Everettian proposal as saying that the cosmos as a whole has a quantum state ψ that is a vast superposition, i.e., a vast sum, of product states describing countless component systems—not just an electron and a measurement apparatus or its pointer, but countless electrons, quarks, molecules, specks of dust, etc. So, the Everettian now needs to be more precise about how to extract from this state a set of worlds: each of them (or maybe just some of them) like our familiar macroscopic realm, with all its tables, etc., in definite positions.

In this endeavour, decoherence is a promising resource, not least because, as we discussed near the end of the last section, it is flexible. The idea will be that to get something like our familiar macroscopic realm, the Everettian will adopt a definition of “macroscopic object,” or for short “macro system,” and will then take the cosmos as the vast composite system consisting of (i) all the macrosystems, together with (ii) all the countless microscopic systems that are not in a macrosystem, which are treated collectively as a single system, the rest of the cosmos. Accordingly, the quantum state space of the cosmos is broken down, in the mathematics of the theory, into the state spaces of its components: all the state spaces of the macrosystems, and the state-space of the rest of the cosmos. In the mathematics, the state spaces combine rather like numbers being multiplied, rather than added. So, the component state spaces are called factor spaces, and the state space of the cosmos is a product space. I shall assume there is some vast finite number N of macrosystems, and so there are exactly N+1 factor spaces in all.

So far, so mathematical. But now decoherence prompts two distinctively physical suggestions.

First, the Everettian can legitimately define “macro system” along the following lines: “any system whose interaction with its environment rapidly makes its state almost a mixture in quantities such as position (and momentum), i.e., the quantities whose values, as attributed by classical physics, gave such a successful description.” For we learned in the last section that nowadays physicists’ models of decoherence are so many and so varied that a definition along these lines will encompass, not just everyday objects like a table or a pointer of an apparatus, but also a tiny dust particle floating in outer space—and countless objects in between, such as a lock of hair or a dew drop. So, in effect, every object that anyone has described, or could successfully describe, with classical physics will be among the Everettian’s vast set of N macro systems.

Second, I have so far considered only a single macro system interacting with its environment, as the saying goes, being decohered by (i.e., its state becoming a mixture, due to) its environment. But such decoherence interactions are happening continuously to each of the N macro systems. For the rest of the cosmos is a vast, common environment for them all: for the table, the pointer, the lock of hair, the dew drop, the tiny dust particle in outer space. Above the atmosphere, where the air molecules give out—for our purposes, not for human breath, but as a sink into which the quantum interference terms can diffuse—the dim light of the stars can take over.

Now, with a bit more stage-setting, I can state an Everettian definition of “world.” There are two preliminary steps, and then the definition itself. (After stating it, I will comment on how it remains only a sketch definition.)

1. We adopt a definition of “macrosystem” along the lines above. Suppose that throughout the cosmos, there are N macro systems, as thus defined. Then we factorize (break into its factors) the cosmos’ state space into N+1 factor spaces: one factor for each of the N macro systems (dust particles, etc.) that gets decohered by its environment, and one factor for their common environment—the vastly complicated and dispersed rest of the cosmos.
2. Then we express the cosmos’ quantum state, ψ, as a superposition whose components (i.e., summands) are (N+1)-fold product states, i.e., products of N+1 states, one in each factor space. We choose these product states by (i) for each of the first N factors, taking an eigenstate, on the corresponding macro system, for the quantity that gets “preferred” (i.e., selected) by the decoherence process and (ii) for the last factor, i.e., the (N+1)-th factor associated to the rest of the cosmos, taking what is called the relative state of the rest of cosmos.

(We do not need to pause on the notion of relative state, although it was a centrepiece of Everett’s original paper (in 1957). Suffice it to say that, stated for a two-component system (rather than an N+1-component system), the relative state is roughly what the component system “looks like,” assuming the other component system is in a given state. More precisely, it is the state that would be assigned to the component, after a projection-postulate measurement that results in the given state on the other component.)

Finally, a world is defined as the physical reality described by a summand, in this way of writing ψ.

In other words, again using some of the jargon we have introduced, (i) for each of the N macrosystems, we consider a component of its “post-decoherence” improper mixture (i.e., an eigenstate of the quantity selected by decoherence), and then (ii) for the rest of the universe, we consider the relative state. We then consider the product of (i) and (ii), which is an (N+1)-fold product state. This product state defines a world. The world is to have as values for quantities just what that product state ascribes, according to orthodox quantum theory, i.e., just the corresponding eigenvalues.

It is clear that in a world, as I have just defined it, each macro system has, by construction, a definite value for the quantity in that macrosystem that was selected by the decoherence process. Thus, the promise of decoherence for the Everettian lies in the fact that in many models of many sorts of macro systems, the definite-valued quantity is calculated to be position or momentum, or something “close” to these—in short, the sort of quantity that, to solve the measurement problem, we intuitively want to have definite values.

So, this definition makes “world” precise in a way that meshes with the basic ideas of Everett’s proposal. It helps the Everettian answer section 6’s accusation of bluffing.

8.2 Improvements Needed

But I stress that this definition of “world” is very much a sketch definition. There are at least two broad ways in which one would like to see it improved. In listing these ways, I am not being controversial. For the physics of decoherence remains an active area of research, both theoretically and experimentally. Besides, it will be clear that not only the Everettian, but anyone interested in the interpretation of quantum theory, will want to see progress in these two ways.

First, this definition assumed a notion of macro system, which I initially said would be defined in terms of “getting a mixture in quantities . . . whose values, as attributed by classical physics, gave such a successful description.” Such an assumption seems suspiciously close to postulating what one wants, rather than arguing for it. I think this suspicion lessens once one sees the detail in the definition of “world”: the factorization of the cosmos’ state space, and the decoherence of each component system. But I agree that much more detail is needed. Not just more than I have given here, but more than the research literature has so far achieved. As I said, this is an active area of research.

In broad terms, one would like to see models of decoherence that are more rigorous and of wider scope (i.e., cover more systems) and that make definite the “right” quantities. One aspect of this is a limitation of my sketch definition, as I stated it. Namely, it takes the environment (the rest of the cosmos) to be spatially external to the macro systems, e.g., air molecules external to an apparatus’ pointer, starlight external to a dust particle floating in outer space. But in many cases, the environment is internal. For example, an iron bar can have long-wavelength and short-wavelength vibrations (called “modes”), and when these modes are treated quantum mechanically, they are themselves systems, and the long-wavelength modes get decohered by the short-wavelength modes. In this way, the long-wavelength modes, taken together, form a macro system. Thus, defining the factorization of the cosmos’ state space in terms of N macrosystems will be a lot more subtle and complicated than just “listing” all the N “lumps of matter” (cf. section 1), such as pointers and dust particles.

In this endeavour, one can of course appeal to the flexibility of decoherence that I praised at the end of the last section. Thus, I said that one can vary the definitions of the system-environment boundary, the time at which the interaction is taken to end, and what counts as the state being “sharply peaked” for a quantity. So, in seeking a better definition of “macro system,” in terms of which to make the factorization, one can hope to exploit this variety or “wiggle room.”

So, to sum up this first issue, the Everettian hopes that the physics of decoherence will enable us to avoid taking “macro system” as a primitive concept, but there is much work yet to do. I will return to this challenge at the end of the next section.

Second, there is an issue about the fact that our sketch definition of “world” appeals to a factorization of the cosmos’ state space, prior to and independent of what the state of the cosmos ψ actually is, which seems wrong. This issue is independent of the first. For even if the first issue was completely dealt with by a satisfactory rigorous definition of “macro system,” etc., it will surely still be true that what macro systems there are, and how many there are, will be very much a matter of happenstance, a matter of contingency. This contingency involves two points.

The first is a matter of everyday belief, and independent of quantum theory, Everettian, or otherwise. (It is also independent of the first issue, i.e., how to define “macro system” precisely.) It goes back to section 3 of chapter 3, regarding our being up to our necks in modality. The issue is that there surely could have been different macrosystems than there in fact are, not just different tables, or locks of hair, or stars, but also more mundanely, different dust particles. Secondly, even if one agrees with the Everettian in boldly postulating a quantum state of the cosmos, still it is presumably a contingent fact what that state is. After all, ψ is an element of a vast state space, with countless other elements, and nothing in the Everettian proposal, as so far stated, forbids the cosmos being instead in one of those other states.

Putting these points together, the obvious suggestion is that we should allow the factorization of the cosmos’ state space to depend on the state. By doing that, the second point—the happenstance about what the cosmos’ state in fact is—might perhaps cover, en passant, the first point, viz. that we need to accept happenstance about what, and how many, macro systems there are.

(I say “might perhaps cover,” not “does cover,” because we cannot now be certain that our convictions about the possible varieties and numbers of macro systems will follow from what quantum theory turns out to say about which states of the cosmos ψ are possible. And our uncertainty is all the greater because of the first issue, i.e., our not knowing how to define “macro system” precisely.)

I will return to this second issue in chapter 6. But nothing I say there will undermine the present conclusion: that we should let the Everettian’s factorization of the cosmos’ state space depend on the state, which our sketch definition above does not do.

9. On What There Is: Objects as Patterns

“On What There Is” is the title of a famous article by the philosopher Quine about deciding what truly exists. In this section and the next, I will pursue this theme, as regards the Everettian multiverse. This section will be largely positive, i.e., pro the Everettians. I will start, in 9.1, with an explanation of how objects being patterns implies that at the end of the Schrödinger-cat experiment, there really are two cats. Then, in 9.2, I will state my misgivings. (Section 10 will raise another objection to the Everettians, and it will lead into the last two sections about probability.)

9.1 Two Patterns, and So Two Cats

Section 9’s main idea is in the title “Objects as Patterns.” Nowadays, Everettians build on the previous sections’ account of decoherence to justify their metaphor of the world splitting into many alternatives corresponding to, for example, the various outcomes of a measurement process. In this justification, their main new idea is that a macroscopic object is not a lump of stuff, or an aggregate of tiny lumps, or even a cloud of point particles. (Recall section 1 above, and chapter 2’s discussion of the seventeenth-century mechanical philosophers.) It is really a pattern in the quantum state, in the quantum state of the cosmos.

As we will see, this idea promises to overcome the limitation I explained at the end of section 7 above, that decoherence does not by itself solve the measurement problem, since it does not imply that in any individual case, the system actually is in one of the states in the improper mixture obtained after the decoherence process. I made this limitation vivid in terms of Schrödinger’s cat. Thus, I complained that this improper mixture still describes two (contradictory) states of a cat, being alive and being dead; it is just that the two states of the cat are correlated with very different microscopic states of the surrounding air molecules.

Now, the idea that macroscopic objects are patterns will vindicate the proposal that (i) at the end of a Schrödinger-cat experiment, there are indeed—not just two (contradictory) states of a cat—but two cats; and (ii) the two cats are in different worlds (branches), what section 7 called “cases.” For there are indeed two patterns in the quantum state of the whole system (say: the cat, the apparatus, and air). Therefore, there really are two cats. In other words, the Everettian claims that the final quantum state being an improper mixture describing two states of a cat, being alive and being dead, is a matter of the state encoding two patterns—and, since cats just are patterns, that description is a description of two cats.

This claim is certainly dizzying. But it is, I think, completely coherent. (My misgivings at the end of this section will not refute it but just make it less plausible.) It becomes clearer if we assume that the quantity selected by decoherence (i.e., having negligible interference terms in the final mixture) is a familiar one, and one which, to solve the measurement problem, we intuitively want to have a definite value. Let it be our old friend: position. That assumption is reasonable. For as hinted in section 7, in models of decoherence, the final mixture typically contains coherent states, in which both position and momentum are very close to definite in value.

Of course, there is no single position quantity for the cat. For the cat has very many parts, both macroscopic, like legs and tail, and tiny, like individual cells and molecules. Similarly, of course, for the pointer of a measurement apparatus. When we discussed its being in front of the numeral “1” on the dial, or in front of “2,” we were exploiting the assumed rigidity of the pointer, so as to get by with a description using a single position quantity, viz. the position of the centre of mass.

So we need to recall from section 2 above (and section 3 of chapter 3’s discussion of state spaces in classical physics) that even the classical configurations of a composite system, with say N point particles as its components, form a vast state space. With each point particle placed somewhere in three-dimensional physical space, the classical configuration space is the set of (3N)-tuples of real numbers. For one needs three real numbers for the spatial position of each point particle. So, in all, one needs 3N real numbers. The state of the corresponding quantum system is then a wave function (with complex numbers as values) on this classical configuration space. (As we saw in section 2, this system is called “N quantum particles.” But as I lamented there, the word “particle” is misleading. For it strongly suggests the system is localized, whereas, according to orthodox quantum theory, it in fact only has tendencies, i.e., dispositions (quantified as probabilities) to be found in various locations, if measured by a position apparatus.)

So, we need to ask: How many parts should we take a cat to have? In other words, what is a good guess for the number N, such that a successful quantum description of a cat can use a wave function on the set (3N)-tuples of real numbers? (Again, the factor 3 just encodes the fact that physical space is three-dimensional.) Let us, for simplicity, think of an atom as a single particle. (Again, we mean “particle” in the Pickwickian quantum sense that we take merely to define a location in physical space, in our effort to estimate the number N). But chemistry teaches us that atoms are minuscule. For example, in 12 grams of carbon, the number of carbon atoms is 6, followed by twenty-three zeros. This is written as 6 x 10²³. (This is called “Avogadro’s number.”)

So, for a cat weighing say a thousand grams (1 kilogram: so roughly 12 x 100), the number of atoms—most of which will weigh less than carbon—will be enormous. It will be about 6 x 10²³ x 100. This is roughly 1000 x 10²³. So, we can take as a guess for N: 10²⁶. So, even with our simplifying assumption to think of an atom as a single particle, the classical configuration space is stupendously large. It consists of (3 x 10²⁶)-tuples of real numbers. So, it has dimension 3 x 10²⁶: a vast number, in which the “3” is hardly worth keeping track of. So, the quantum state is a wave function whose arguments (inputs) are elements of this space. Each argument is a (3 x 10²⁶)-tuple, i.e., an exact classical configuration for all the 3 x 10²⁶ atoms.

I can now say how the Everettian claims that in the myriad complexity of such a wave function, there is a pattern that deserves to be called “a living cat,” and another pattern that deserves to be called “a dead cat.” (I say “claims,” since it will be clear that there remains a lot of intellectual work to do.)

The idea is to focus on the fact that in a classical description, there is a set of configurations that would all count as a living cat. Indeed, there are vastly many configurations of about 10²⁶ classical point particles that would count as constituting a living cat. We can put the point in the cartoon form, which I adopted at the end of section 7. Think again of the cat’s being alive as it having vertical legs, a vertical tail, and a smile on its mouth. Similarly, think of it being dead as its having horizontal legs, a horizontal tail, and a frown on its mouth. So, there is a similarly vast set of configurations that would all count as a dead cat. Notice that these two sets, though both vast, do not overlap at all. No configuration of a classical cat with the point particles composing its legs and tail aligned vertically is also a configuration in which the legs and tail are aligned horizontally.

And so—now returning to quantum theory—the Everettian claims that the quantum state at the end of Schrödinger’s experiment is, as regards the cat, a wave function with two peaks. That is, they claim that there are two regions of the classical configuration space, i.e., the set of arguments of the wave function, where the function’s value, i.e., the output or amplitude, is non-negligible.

(And the Everettian claims that for countless other configurations, the amplitude is negligible. Bear in mind that, a priori, these 10²⁶-point particles could be configured to be any classical object of the same total mass. For example, this includes a puddle, or a saucepan with risotto, or a small dachshund, or any of myriadly many nameless and often monstrous combinations that only a horror movie might devise, such as a half-cat-quarter-dachshund-quarter-risotto.)

Summing the amplitudes for all the configurations in each of these two regions, we get a (square root of) a probability that is substantial. In Schrödinger’s original version of the experiment (see the end of section 4), where there is about a 50% probability of an atomic decay causing the release of the poison, the probability for each of the two regions is about 50%. That is, 50% for being alive, and 50% for being dead.

But whether the probabilities are 50-50, or nearly so, doesn’t matter here. What matters is the claim that there are two such patterns in the quantum state, two regions of the classical configuration space where the amplitude is vastly greater (as a ratio) than it is for points outside both regions. So, if we accept that a cat is such a pattern, then there really are two cats.

Note that the essential idea here is independent of quantum theory’s details (as Everettians note). For the idea is closely analogous to one that we all unhesitatingly endorse for several other physical theories. Namely, theories in which states can be added together to give a sum state, in which the component states do not influence each other or only do so negligibly. This is called the state being dynamically isolated from each other. Examples include the theory of water waves and the theory of electromagnetism. (Recall from section 4.1 that just as we add two vectors by adding corresponding coefficients, we add two functions, such as wave functions, by adding their values for each given argument.)

For example, the water in Portsmouth harbour can get into many different states. It can get into (i) a state which we describe as, e.g., a wave passing through the harbour’s centre heading due west; into (ii) a state that we describe as a wave passing through the centre heading due north; or (iii) into a state that is the sum of these. But does this last case (iii) imply that we face a “Portsmouth water paradox”? Should we agonize about how the Portsmouth harbour water system can, in one place (viz. the harbour’s centre), simultaneously have the contrary properties of being westward and being northward?

Of course not! Rather, we say that waves are patterns in the water system. (Agreed, we call such patterns “objects”; in the jargon of philosophy, they are often called higher-level objects.) And so, we say that there are two waves, with the contrary properties, one westward and one northward. Since the contrary properties are possessed by two distinct objects, there is no contradiction.

This is similar to other theories whose states can be added together, with the component states not influencing each other. For example, think of the electromagnetic field in a certain region, and e.g., pulses of laser light travelling in different directions across it—as happens in a light show at a rock concert. There is no “laser light show paradox.”

Analogously, says the Everettian, we should endorse this idea when it is applied to the end of Schrödinger’s experiment. Thus, the Everettian claims to overcome the limitation at the end of section 7 above. For with macroscopic objects as patterns in the quantum state, not lumps of stuff, we see that solving the measurement problem does not require that in any individual case, the quantum system is actually in just one of the states in the improper mixture obtained after the decoherence process. The system is in none of these states. But each state is a pattern, and that—a pattern—is what a macroscopic object such as a cat really is. So, there are two cats, with contrary properties, one in each “case,” i.e., world.

So much by way of expounding the Everettian’s claim that there are two cats. Now for my misgivings.

9.2 Misgivings

Recall that in order to expound the Everettian’s claim, I put it in cartoon form (above and at the end of section 7), with being alive corresponding to vertical legs, a smile, etc. So clearly, there is a vast amount of intellectual work still to do. The Everettian owes us details about which classical configurations are to count as “being alive.” More generally, setting aside cats and Schrödinger’s experiment and biology, we need to be told which classical configurations are to count as the system we are concerned with, having each of a host of properties that we routinely ascribe to macroscopic objects. “Being alive” is, of course, a property ascribed in both everyday life and scientific work. But we can concede that it would be enough for the Everettian to give a “translation manual” from the regions of their vast classical configuration space to just scientific properties, or even just properties used in physics. For example, which regions (and so which quantum states, with a peak, a non-negligible amplitude, over those regions) count as having a density of seven grams per cubic centimetre, or being made of lead, or being a fluid, or being a good electrical conductor? Only if the Everettian can give us such a translation manual will it be plausible that the familiar macroscopic realm—and more precisely, the vast empirical success of classical physics—can be understood as emergent from the quantum state of the cosmos.

Obviously, this is a vast challenge. Besides, the challenge is all the greater when we remember two points made above. First, there are countless configurations of 10²⁶ particles that correspond to no recognizable macroscopic object, not even a horror-movie monstrosity like a half-cat-quarter-dachshund-quarter-risotto. Think of a vastly variegated and spatially very dispersed “sludge”: about a cubic centimetre of cool lead here, a cubic centimetre of hot methane gas there, and so on, and, say, 100 miles away, a (more appealing) hunk of ice cream. Nothing in our Everettian description of a quantum measurement avoids such a sludge getting a non-negligible amplitude. So, why don’t we ever see it?

The second point aggravating the challenge was at the end of section 8. Namely, the example of a how a system’s decohering environment is in some cases spatially internal showed that defining the factorization of the cosmos’ state space in terms of N macro systems will be a lot more subtle and complicated than just “listing” all the N systems that we tend to think of as “lumps of matter,” such as pointers and dust particles. This point is all the sharper once we realize that the macroscopic realm that the Everettian must recover consists of patterned configurations that are a minuscule subset of a configuration space, most of whose elements are nameless and unstructured “sludge.”

But I do not say that this challenge is impossible. And I stress that in fulfilling it, decoherence will undoubtedly play a crucial role. We can illustrate this by going back to the cat, i.e., to biology. Recall from, for example, your biology lessons that biochemistry successfully describes the metabolism of cats (and, of course, all other organisms) in a completely classical way. Its models of chemical reactions in the cell assume that the proteins, DNA sequences, etc., are localized. These molecules are modelled as minuscule cousins of the ball-and-stick models on the biochemist’s tabletop (balls for the atoms, sticks for the bonds between them).

That this classical description of what is, after all, a quantum reality can succeed so well reflects the efficiency and ubiquity of decoherence. For the protein and DNA molecules are, on the atomic scale, very large: they often contain well over 10,000 atoms (and so are often called “macromolecules”). And they are constantly bombarded by tiny molecules, such as water molecules, that decohere them, and so—in the “improper mixture sense” (section 7)—localize them. The upshot is that at the length scale of macromolecules and at longer lengths, a classical description of protein molecules, DNA sequences, etc., as having well-nigh definite positions can succeed. And this success is well illustrated by the models filling biochemistry textbooks.

(With these remarks about biology managing well while treating quantities like the position of a macromolecule classically, without regard to quantum theory, I do not mean to deny that some important biological processes “cue in” to quantum aspects. Examples of this, including crucial processes like photosynthesis, are nowadays a focus of research in the new field called “quantum biology.”)

10. A Reversal of Ideas

So much by way of expounding the Everettians’ claim that indeed there are two cats, just because there are two patterns. Or more generally, that indeed the world splits—there is a multiplicity of objects—at the end of a decoherence process, since the state is then an improper mixture with two or more components corresponding to macroscopic realms with different values for the quantities selected by decoherence. As I said, I think this claim is, though dizzying, coherent. And suppose we now set aside all my misgivings at the end of the last section.

Should we then conclude that the Everettian is home free? That is, does their solution to the measurement problem (specifically, their appeal to decoherence to justify talk of splitting) have no internal conceptual difficulties? I say “internal” because we may prefer a rival solution—perhaps one of those reviewed in section 4 above—for other reasons (including perhaps reasons that boil down to our intellectual temperament, as discussed in section 4 of chapter 1).

I say: no. I submit that there are two main difficulties remaining. Both are distinctively philosophical, or interpretative, rather than physical. One difficulty concerns the topic of probability: I will address it in the next (and final) two sections. In this section, I raise a difficulty about the quantity selected by decoherence.

I do not claim that it is a knock-down objection: It is a conceptual tension or embarrassment facing the Everettian, rather than an outright problem. But it is worth articulating for two reasons. First, so far as I know, it is not addressed in the Everettian literature. Second, there is an interesting analogy between it and a criticism of Bohr’s complementarity interpretation (cf. section 5 above) that Schrödinger made in the great 1935 paper that formulated his cat paradox (and that also, as I mentioned, analyzed entanglement). I will first, in 10.1, state the objection; and then, in 10.2, explain the analogy with Schrödinger’s criticism of Bohr.

10.1 A Worrying Appeal to Approximate Ideas

To explain this difficulty, I need first to stress the striking conceptual unity of classical physics’ successes from the time of Newton until about 1900 (reviewed in sections 3 and 6 of chapter 2, section 3 of chapter 3, and section 1 above). In classical physics, each of a very small set of quantities fulfils two roles that are, a priori, disparate. Namely, (i) being postulated as basic for the description of matter’s tiniest components (whether point particles or small extended pieces of matter), and (ii) being used to describe composite systems with vastly many such components. The paradigm examples are position and momentum within mechanics. In classical physics, a single quantity fulfilling these disparate roles (i) and (ii) was unified by various procedures, especially summing or averaging the values for the tiny components in (i), to get the values for composite systems in (ii). The simplest case is elementary and familiar: the centre of mass of a composite object is a weighted average of the positions of its components, with weights equal to the components’ masses.

Agreed, the development of other branches of physics, especially the rise of field theory (section 1 above), required that quantities other than position and momentum, such as electric charge and electric field, had to also be accepted. Nevertheless, classical physics manages with a very small set of quantities. (Depending on how finely you distinguish quantities, there are between about a dozen and about fifty of them.) And most of the quantities fulfilling role (i) also fulfil role (ii), again with various procedures, especially summing and averaging, unifying the roles. Against this background, we can now see the conceptual tension or embarrassment facing the Everettian. The Everettian makes two claims that are in tension with each other. Namely, they say:

1. Although classical physics took quantities such as position and momentum to be exact, and to always have exact values, such quantities are in fact only definable approximately, through the process of decoherence. (Terminology: philosophers might call them “emergent”; physicists also say “effective”.)
2. But they also say that these classical notions are needed to define the quantum state space. For as we saw, the quantum state is a wave function defined on classical configuration space. Agreed, I have hitherto simplified. For quantum states can be represented as complex-valued functions (also called “wave functions”) with other sets of arguments than components’ positions. The main such alternative representation uses momentum. That is, the set of arguments for an N-particle system are the 3N-tuples of all the possible values of the components’ classical momentum in each spatial direction. But here again, it is the classical notion, viz. of momentum, that needs to be invoked, and so the same conceptual tension arises.

So, the difficulty, or tension, is that notions which, according to the Everettian, are really approximate (emergent, effective) must be appealed to, in order to interpret the theory at the smallest and most basic level.

As I said, I do not think this is a knock-down objection. For example, the Everettian might reply that we often learn new theories by understanding their new concepts through the prism of the old theory, but that in due course there is a gestalt shift, so that we reinterpret the old via the new. Fair comment. But I am uneasy that today, in 2025, a hundred years after the quantum revolution, we (including the Everettians!) have not yet achieved the envisaged gestalt shift.

10.2 The Analogy with Schrödinger’s Objection to Bohr

Finally, an analogy. In section 5’s quick review of the main strategies for solving the measurement problem, I mentioned Bohr’s complementarity interpretation. I can now explain how this difficulty for the Everettian is analogous to a striking criticism of Bohr that Schrödinger made in his “cat” paper of 1935.

Schrödinger’s criticism is based on a historical fact about classical physics being in tension with Bohr’s (as one might say, “the complementarian’s”) claim that classical concepts are indispensable within quantum physics. Adopting a labelling to show the analogy with my 1 and 2 above, the fact and the claim are, respectively, as follows:

(1”) Despite the conceptual unity of classical physics’ successes (described above), classical physicists did not claim that the classical quantities were indispensable for physics. Nor did they claim that such indispensability would be shown by future physics. Indeed, many expected these quantities to be superseded by future physics.

But Bohr claims that:

(2”) Classical quantities are indispensable for physics, although, of course, quantum physics has shown they do not always have exact values. For Bohr, the principal reason for this “lesson” from quantum theory is that position and momentum cannot be measured simultaneously with arbitrary accuracy. (This is the famous Uncertainty Principle (in its best-known form). In the mathematics of quantum theory, this is represented by position and momentum not commuting with each other.) And by “indispensable,” Bohr means, roughly speaking, indispensable for reporting experimental results in an objective language. (But here, we do not need the details of Bohr’s doctrine, or of why he held it, which are controversial.)

So the difficulty, or tension, which Schrödinger is articulating as a criticism of Bohr, is that notions that classical physicists expected to be superseded by future physics, and which, according to the complementarian, are indeed really limited (a lesson taught us by quantum theory), must be appealed to, in order to interpret quantum theory. Indeed, the complementarian says they are needed to report experimental results objectively.

Let me end by quoting Schrödinger’s own words against Bohr (at the end of section 2 of his 1935 paper). He begins with (1”). He praises classical physicists’ intellectual modesty about their theory (which he calls “model” and “picture”), in particular, about whether its quantities (which he calls “determining parts”) can be measured on a microscopic object in nature (which he calls “natural object”). He writes:

Scarcely a single physicist of the classical era would have dared to believe, in thinking about a model, that its determining parts are measurable on the natural object. Only much remoter consequences of the picture were actually open to experimental test. And all experience pointed toward one conclusion: long before the advancing experimental arts had bridged the broad chasm, the model would have substantially changed through gradual adaptation to new facts.

Then he goes on to criticize what he calls the “reigning doctrine” (i.e., Bohr’s complementarity) for declaring that only familiar classical quantities (e.g., position and momentum) are measurable. He writes:

Now while the new theory [i.e., quantum theory: JB] calls the classical model incapable of specifying all details of the mutual interrelationship of the determining parts (for which its creators intended it), it nevertheless considers the model suitable for guiding us as to just which measurements can in principle be made on the relevant natural object. . . . This would have seemed to those who thought up the picture a scandalous extension of their thought-pattern and an unscrupulous proscription against future development. Would it not be pre-established harmony of a peculiar sort if the classical-epoch researchers, those who, as we hear today, had no idea of what measuring truly is, had unwittingly gone on to give us as legacy a guidance scheme revealing just what is fundamentally measurable for instance about a hydrogen atom!?

11. Probabilistic Angst: What Is Objective Probability?

As this chapter’s preamble announced, I have so far emphasized synchronic issues, i.e., issues about the quantum state at a single time. I have neglected issues about time and change, except to say that I postpone until this section the question of how the deterministic Schrödinger equation can be reconciled with quantum theory’s use of probabilities. So, I now focus on how the Everettian answers this question. This will raise the philosophical question: What exactly is objective probability (also known as chance)?

The first point to make is that the question really breaks down into two problems that the Everettian faces. The Everettian literature calls them “the qualitative problem of probability,” and “the quantitative problem of probability.” Discussing the first will lead into the second, which I address in the next section.

The qualitative problem is that probability seems to make no sense if all possible outcomes of a putatively probabilistic process in fact occur. But this is what the Everettian claims, at least for quantum measurements and the other processes, such as radioactive decay (remember the poison for Schrödinger’s cat), in which the quantum state evolves to include a term, i.e., a summand in the sum, for each outcome. (Here, “outcome” was made more precise by section 8’s sketch definition of “world.”)

In short, the Everettian’s answer to this question is that even though all possible outcomes occur, there is subjective uncertainty. Roughly speaking, it is the uncertainty of an experimenter just before doing a quantum measurement, about the question, “Which outcome will I see?” Again, this answer comes in several versions, but I will develop just one main version. I begin, in 11.1, by explaining how subjective uncertainty is invoked in some other physical theories, and then, in 11.2, I return to the Everettian.

11.1 Uncertainty in Classical Physics

It will be clearest to begin by explaining how, in a broadly similar way, probability is taken as subjective uncertainty for deterministic processes of the kind familiar within classical physics, e.g., Newtonian mechanics. (Recall the discussion of determinism in section 8 of chapter 3.) For such a process, a unique future sequence of states is determined by the present state. (More precisely, by the present state, together with the process’ deterministic law, which in mechanics would be a specification of the future forces exerted on the system.) But a person, for example, an experimenter, may not know this future sequence of states in advance, either because she does not know the present state in full detail or because it is too hard to calculate from that state what the future sequence will be. Given our present interest in how to understand probabilities, let us set aside the latter cause of uncertainty, since it is a matter of calculational intractability rather than ignorance of which among several alternatives occurs. It is ignorance of this latter kind which gives scope for the idea of probability. So, in the context of classical physics, probability is reconciled with determinism by subjective uncertainty. This is the idea that a person doesn’t know which alternative really occurs, but has various degrees of belief, i.e., subjective probabilities (cf. section 3.1 of chapter 3), about the matter. In the context of deterministic physics, these will be subjective probabilities about what exactly is the present state.

But here, the phrases “degrees of belief” and “subjective probabilities” should be understood in a logically weak sense. They do not imply that the probabilities, i.e., the numbers 1/2, 1/3, etc., assigned by the person, are a matter of idiosyncratic taste or temperament, that is, are undetermined by all the objective evidence. For there is a branch of classical physics, called “statistical mechanics,” that studies composite systems with vastly many components, a large and important branch, though I have not yet had occasion to mention it. (It was developed from the late nineteenth century onwards, and among its main figures were Maxwell (1831–1879), who we met in section 1, Gibbs (1839–1903), and Boltzmann (1844–1906).) Thus, statistical mechanics studies systems like a sample of gas taken as composed of classical molecules, tiny “lumps in the void.” It surmounts the utter unknowability of the exact microscopic state—the exact positions and momenta of all the classical molecules—by postulating a probability distribution over the possible states and then calculating average (also called expected) values of quantities like energy. Here, the probability distribution gives the weights to be used in calculating the average. The many resulting predictions meet with great empirical success.

Now the point is that a very good case can be made for calling this distribution “objective,” even though it is not determined by the exact microscopic state. Making this case invokes technical notions that go by names like mixing and ergodicity. But we need not go into details. For us, it is enough that some rather natural assumptions about these notions select the empirically successful probability distribution from the countless horde of mathematically possible distributions. a selection that has nothing to do with idiosyncratic taste or temperament.

Besides, a similar strategy for reconciling probabilities with determinism and justifying them as objective occurs in the pilot-wave theory that we mentioned at the end of section 5. Recall that, like the Everettian, the pilot-wave theorist says that the Schrödinger equation is always “right” (as Bell vividly put it in 1987, 201). But unlike the Everettian, the pilot-wave theorist says a quantum system has values for quantities other than its state’s eigenvalues. For there are also point particles with exact positions. With this as background, the pilot-wave theorist goes on to say that the apparent indeterminism of quantum theory arises from the utter unknowability of these positions; and quantum theory’s orthodox (and again, empirically successful) Born-rule probability distribution over those positions can be derived from rather natural assumptions about notions like mixing and ergodicity, applied to the quantum state.

To sum up, both classical statistical mechanics and the pilot-wave theory reconcile the use of probabilities with the future sequence of states being determined by all the details of the present microscopic state, by (i) invoking the utter unknowability of all the details of that state (in short, particles’ positions), and (ii) arguing that the empirically successful probability distribution over these details is not a matter of taste or temperament, but of natural physical assumptions.

11.2 Uncertainty in Everettian Quantum Physics

Now we can say how the Everettian’s answer to the qualitative problem of probability is analogous to all this. They claim that, also in the Everettian framework, probability can be taken as subjective uncertainty about a deterministic process. But now, it will be uncertainty about a deterministic process of the unfamiliar Everettian kind. For such a process, a unique future sequence of states for the composite system—in principle, the entire cosmos—is determined by the present quantum state (together with the Schrödinger equation encoding all the forces that are acting). Yet, says the Everettian, there can still be subjective uncertainty. But the situation differs from the classical one in that this uncertainty arises, even if we assume the person, e.g., the experimenter, does know the present state in all its detail, and also all the forces that are acting, and so also knows how to calculate the entire future sequence of states.

Here, I should clarify that Everettians have an account connecting their state of the cosmos ψ (which, of course, no one knows) with the various quantum states we ascribe, with great empirical success, to objects in the laboratory in real-life experiments. So, although, of course, no one can write down ψ, the Everettian framework can recover the successful real-life ascriptions of quantum states to electrons, atoms, and even dust particles. (We can skip the details of this account: suffice it to say that it uses the ideas in sections 7 and 8.) Thanks to this account, the Everettian can recover the idea, which is realized every day in real-life experiments, of the experimenter ascribing a quantum state to a microscopic system, such as an atom that is about to be measured, and thereby deducing from that state the orthodox Born-rule probabilities, i.e., those numbers.

But do those numbers deserve to be called probabilities? After all, according to the Everettian, each of the various measurement outcomes truly occurs. To this, the Everettian answers: “Yes: the experimenter is uncertain since, thanks to the impending ‘splitting’ during the process of measurement, she will experience, not all the outcomes, but just one—and so she can ask ‘Which outcome will I see?’ And that is enough for the numbers to be called ‘probabilities’.”

I think this answer is tenable. For I think the core meaning of “probability” requires that there are various cases, with numerical weights (or if you prefer, measures or intensities) assigned to them that are to guide, in some suitable way, agents’ beliefs and actions. But this core meaning does not require that only one case really occurs. However, I agree that this answer is incomplete, in that it raises philosophical issues, indeed, at least three.

First, the answer leads to the issue of the identity over time of persons and/or consciousnesses. For it clearly depends on taking the question “Which outcome will I see?” to be analogous in relevant respects to the question in the context of classical physics or pilot-wave theory: “Which alternative (among the many microscopic states compatible with my incomplete knowledge) actually occurs?” So, the analogy involves accepting that the “I” that sees just one outcome could—in some good sense of “could”—see another outcome. Here, we again see section 3 of chapter 3’s theme: that in science, no less than in everyday thought, we are up to our necks in modality. But here, the modality at issue is different from that in chapter 3. For as we have seen, for the Everettian, what could occur, i.e., the unexperienced alternative, is real (if you prefer actual), albeit in an unseen branch. (In section 3 of chapter 6, I will return to this comparison of possible worlds with Everettian branches.)

The second issue is related to the first. In recent decades, quite independently of these quantum conundrums, philosophers of mind and metaphysicians have identified the need for a notion of possibility that generalizes chapter 3’s idea of a possible world. It is sometimes called a centred world. The general notion of possibility is needed to understand the content of sentences that contain (and mental states that are naturally expressed using) words like “I,” “you,” “now,” “then,” “here,” “there,” “this,” and “that”—i.e., words whose referent depends on the context of utterance (cf. section 5 of chapter 3). Such words are called indexicals or token reflexives, and the need for the generalized notion of possibility, which is shown by such sentences, is called “the essential indexical.” We need not go into this issue in detail: both here and in chapter 3, we have had enough to do. I just note that the uncertainty that the Everettian’s answer invokes—viz. uncertainty despite full calculational ability, and full knowledge of the composite system state and of all the forces encoded in the Schrödinger equation—invites comparison with the kind of indexical uncertainty that philosophers nowadays address using centred worlds.

The third issue is, why should this uncertainty be quantified by the Born-rule probabilities derived from the quantum state? Why are they the right, or somehow appropriate, degrees of belief for the experimenter to have about “which outcome I will see”? So, this is what the Everettian literature calls “the quantitative problem of probability.”

12. Subjective Probability to the Rescue?

Indeed, this problem can be made sharper by imagining that a quantum system is subjected to a sequence of measurements. This prompts a tempting line of thought that the Everettian should regard the Born-rule probabilities as wrong. This line of thought was a focus of discussion for what in section 6 I called “the first Everettian answer” (spanning about 1960 to 1990), which formulated the splitting of worlds in terms of measurement outcomes and which made no appeal to decoherence. The line of thought goes as follows.

According to the Everettian, the quantum state evolves over the course of a sequence of measurements to encode all possible sequences of outcomes. Formally, the final state has a term (i.e., a summand in the sum) representing each sequence of outcomes. For example, consider a toy model in which there are ten measurements, each with two outcomes (say, H and T, for “heads” and “tails”). Then there are 2¹⁰ = 1024 sequences of outcomes, and so the Everettian must say there are 1024 terms in the final quantum state.

Since, according to the Everettian, each such sequence actually occurs, it seems at first sight that the Everettian probability of a sequence should be given by the naïve counting measure. That is, the Everettian should say that each sequence has probability 1/1024. And so more generally, it seems that the probability of an event corresponding to a set of sequences, such as three of the ten measurements having outcome H, is the sum of the basic probabilities (each equal to 1/1024) of the sequences in the set. But this amounts to assuming that the two outcomes H and T are equiprobable (and that the measurements form independent trials in the sense of probability theory). And this spells disaster for the Everettian. For the counting measure probabilities bear no relation to the quantum Born-rule probabilities, and so the procedure of counting Everettian worlds just by their sequences of outcomes seems to conflict with quantum theory’s treatment of probability.

So much by way of sharpening the quantitative problem. I will now report what is nowadays the best-known and most developed Everettian answer to it. (But the “Notes and Further Reading” section of this chapter will also mention a recent alternative.) This answer has two parts. The first part builds on preceding sections’ material, but the second part is wholly novel and will occupy the rest of the section.

The first part is to point out that decoherence, thanks to its flexibility, refutes the toy model with its naïve counting measure. (Recall the end of sections 7 and 8.) That is, on any precise definition of “world” for the systems concerned, there will be many trillions of worlds, wholly independently of the number of kinds of outcome registered by the measurement apparatus (in my example, just two, H and T). And more importantly, because one can vary the exact definitions of decoherence’s crucial notions (like “system-environment boundary”), there is no definite number, not even in the trillions, of worlds that we need to—or could!—appeal to, to give an account of probability in terms of counting. In short, the naïve counting measure is a mirage. It is woefully ill-defined, and the Everettian can and should just reject it.

The second part of the Everettian’s answer is a remarkable recent development, that is wholly unlike anything in the previous discussion (either by me in this chapter or in the Everettian literature I have so far drawn on). In terms of the previous section’s discussion, it is an analogue of the arguments within classical statistical mechanics and pilot-wave theory that I mentioned. Recall that they justify those theories’ probability distributions, not by their empirical success, but by their following from natural assumptions. Analogously, Everettians have recently developed theorems that justify the Born rule, not by its empirical success, but by its uniquely following from certain general assumptions. But there are also two striking differences between the two cases.

1. The arguments within statistical mechanics and pilot-wave theory make assumptions about how the system changes over time, albeit general ones. In the jargon: the assumptions about mixing and ergodicity are assumptions about dynamics. But the assumptions of the recent Everettian arguments do not refer to how the system changes: they are synchronic or kinematic. We do not need details, but in short, they turn on the linear structure of the quantum state space.
2. The second difference is even more striking. The arguments within statistical mechanics and pilot-wave theory, and their assumptions, make no appeal to general principles about subjective probability, in particular, to principles about how rationality should govern a person’s subjective probabilities. That is as we saw in the previous section. Although I introduced the reconciliation of probability with determinism in classical physics by invoking subjective probabilities about what the exact state is, it is details of physics, such as assumptions about how a system changes over time, that are the dominant considerations determining which probability is correct (and in particular, which probability is empirically successful). But the Everettians’ assumptions are indeed about what principles of rationality should govern a person’s subjective probabilities.

This is very remarkable since such principles are formulated and compared in a discipline, decision theory (briefly discussed in section 3 of chapter 3), that belongs to psychology and economics, and so prima facie has absolutely nothing to do with physics. So, the Everettians’ idea is remarkably inventive. The idea is, first, to appeal to such principles of rationality as applied to a person’s degrees of belief in the various outcomes of various quantum measurements; and then, to go on to prove that these principles imply that the person’s degrees of belief must be given by the Born rule. So let me end this chapter with some details about this.

In decision theory, there is a tradition of proving what are called representation theorems. They are so called because they show that under certain conditions, a rational person’s behaviour reveals that their degrees of belief can be represented as numerical probabilities. That is, their degrees of belief must conform to the usual rules about probabilities, viz. that the probabilities of all the envisaged alternatives must add up to 1, and that the probability of either alternative A or alternative B, where A and B are incompatible (cannot both be true), is the sum of the individual probabilities. Thus, there are theorems to the following effect. Imagine a person whose preferences for gambles (encoding certain degrees of belief and certain desires) conform to a certain set of axioms: axioms that seem rationally compelling, i.e., compulsory for a rational person. The axioms say, for example, that a person who prefers A to B and B to C must also prefer A to C, and that a person would not enter a bet (or a collection of bets) that is guaranteed, whatever the outcomes, to yield a loss. Then the person must have degrees of belief that conform to the rules of probability. In other words, their degrees of belief are represented by a probability distribution.

We do not need further details, technical or even philosophical, about such theorems. But let us note that these theorems do not dictate a unique probability distribution over the various alternatives. This is, of course, as one would expect. Imagine two people are offered bets on horses in a race. So, which bets they are willing to take, and which bets they decline, reveal their degrees of belief in the alternative propositions about which horse wins. Even if the two people are rational in the sense of the axioms listed in the representation theorems, we surely do not expect the two people to accept bets at exactly the same odds. For they can have legitimate differences of opinion. In short, we, of course, do not expect rationality to dictate specific degrees of belief in arbitrary propositions. (We touched on this in section 5 of chapter 2’s discussion of Hume and inductive logic.)

But the recent Everettian theorems secure precisely this uniqueness about the specific scenario of a person making gambles on the outcomes of various quantum measurements. Besides, the probability distribution that is uniquely dictated by the axioms about the person—which, as in the tradition of decision theory, seem to encode merely the compulsory requirement that they are rational—is indeed quantum theory’s Born-rule probability distribution over the various outcomes.

A bit more precisely, the theorems show that a person who (i) is an Everettian and is about to observe a sequence of quantum measurements, and also (ii) knows the initial state of the quantum system to be measured, and (iii) is forced to gamble on which outcomes she will see (using the Everettian sense of “splitting,” to interpret the phrase “she will see”), and (iv) whose gambles are subject to certain rationality axioms—must apportion her degrees of belief (as shown by her betting behaviour) in accordance with the Born-rule as applied to the initial state that she knows (as assumed in (ii)).

To sum up this section. I began with an objection to the Everettian. It said that they could not answer the quantitative problem of probability, since they seemed obliged to endorse the naïve counting measure, which dooms them to disagreeing with the empirically successful Born-rule probabilities. But the Everettian can and should appeal to decoherence so as to reject the naïve counting measure. And furthermore, Everettians have proved remarkable representation theorems, analogous to those in classical decision theory. But while the classical theorems allow a wide variety of subjective probability distributions, the Everettian theorems dictate that if a rational Everettian knows the relevant quantum state, her degrees of belief about measurement outcomes must be the orthodox Born-rule probabilities. So, the Everettians can say that their framework not only accommodates, but even implies, the Born-rule probabilities. Remarkable indeed!

13. Notes and Further Reading

As for chapters 2 and 3, there is a dauntingly large literature. And as with those chapters, I recommend:

(i) The internet encyclopedias and archives, and the accessible books listed in subsections 7.1 to 7.3 of the “Notes and Further Reading” section 7 for chapter 1. These all cover the problems of interpreting quantum theory and, in particular, the Everettian interpretation.

(ii)Reading the original masterpieces, some of which are indeed very readable.

I divide my more specific suggestions into four groups, following the sequence of topics in the chapter:

1. The measurement problem and the interpretations of quantum theory mentioned in this chapter’s section 5.
2. Other interpretations (again, emphasizing how they propose to solve the measurement problem).
3. The Everettian interpretation, in general terms (cf. sections 5 and 6).
4. Treatments of decoherence (cf. sections 7–10) and probability (cf. sections 11–12).

13.1 The Measurement Problem and Interpretations of Quantum Theory

For quantum theory and its interpretation, in general terms, I suggest four groups of sources, the last of which concerns historical aspects.

Foundational Essays

For the various interpretations of quantum theory prompted by the measurement problem, pride of place must go to the two essays by Bell and Schrödinger on which I relied heavily in the chapter’s exposition:

• Bell, John. “Six Possible Worlds of Quantum Mechanics.” Foundations of Physics 22 (1992). Reprinted in Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 1987; revised 2004). Available at: cambridge.org/core/books/speakable-and-unspeakable-in-quantum-mechanics/E0D032E7E7EDEF4E4AD09F458F2D9DB7
• Schrödinger, Erwin. “The Present Situation in Quantum Mechanics” (1935). English translation by J. Trimmer in Proceedings of the American Philosophical Society 124 (1980): 323–338. Available at: jstor.org/stable/986572. This essay is also reprinted in Wheeler, John Archibald, and Wojciech Hebert Zurek, eds. Quantum Theory and Measurement (Princeton University Press, 1983), an anthology of forty-nine papers that includes many seminal works by Bohr, Wigner, Bohm, Bell, and Everett. Available at: degruyter.com/document/doi/10.1515/9781400854554/html

Another excellent overview, aimed at philosophers, is:

• Lewis, David. “How Many Lives Has Schrödinger’s Cat?” Australasian Journal of Philosophy 82 (2004): 3–22. Available at: https://www.tandfonline.com/doi/abs/10.1080/713659799

Introductory Monographs

There are many fine books that explain quantum theory and its interpretations for newcomers, especially philosophers. Two natural successors to this chapter are:

• Lewis, Peter. Quantum Ontology (Oxford University Press, 2016). Available at: oup.com/academic/product/quantum-ontology-9780190469818
• Barrett, Jeffrey A. The Conceptual Foundations of Quantum Mechanics (Oxford University Press, 2019). Available at:
  oup.com/academic/product/the-conceptual-foundations-of-quantum-mechanics-9780198844693

Advanced Treatments

There are countless more advanced books that address the measurement problem, but presuppose some knowledge of quantum mechanics and facility with mathematics. Here are three books that expound the approaches mentioned in my section 5. They are by eminent leading researchers.

The first two are famous expositions of the pilot-wave theory. The third is P. Pearle’s recent exposition of the programme (usually called “the dynamical reduction programme”) that seeks to solve the measurement problem by modifying the Schrödinger equation: a programme to which Pearle is a pre-eminent contributor. Regrettably, my section 5 gave even less details about this programme than about the pilot-wave theory, simply because it is hard to state briefly how it obtains the right kind of wave-function collapse. But the programme is flourishing and has been worked on by many eminent physicists, such as Diosi, Ghirardi, Gisin, and Penrose, as well as by Pearle.

• Bohm, David, and Basil J. Hiley.The Undivided Universe (Routledge, 1993). Available at: taylorfrancis.com/books/mono/10.4324/9780203980385
• Holland, Peter R. The Quantum Theory of Motion (Cambridge University Press, 1993). Available at: cambridge.org/core/books/quantum-theory-of-motion/EF981BAE6222AE87171908E8DB74AF98
• Pearle, Philip. Introduction to Dynamical Wave Function Collapse (Oxford University Press, 2024). Available at:
  oup.com/academic/product/introduction-to-dynamical-wave-function-collapse-9780198901372

Historical Sources

The history of the interpretations of quantum theory is, of course, fascinating. I pick just five important items.

1. Einstein and Bohr had a famous debate from the mid-1920s onwards. For an introduction, see Bohr’s essay in the extraordinary anthology of essays in honour of Einstein, Albert Einstein: Philosopher-Scientist, Volume 7 of The Library of Living Philosophers, edited by P.A. Schilpp, Open Court, 1949. Bohr’s essay is also reprinted in J. Wheeler and W. Zurek (eds.), Quantum Theory and Measurement, mentioned above.
2. The Solvay conference in 1927 was epoch-making. Its Proceedings were re-published, with a magisterial editorial introduction, edited by G. Bacciagaluppi and A. Valentini, Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference (Cambridge University Press, 2009). It is available at: https://www.cambridge.org/core/books/quantum-theory-at-the-crossroads/0F8A6712D61351E330A4D52C7EC8CC2C
3. A superb survey of the role of philosophical ideas in the historical development of physics, which I already recommended in Chapter 2, and which also covers the quantum theory, is J. Cushing, Philosophical Concepts in Physics (Cambridge University Press, 1998; online 2012). Available at: https://www.cambridge.org/core/books/philosophical-concepts-in-physics/F285F13FE71F225BD8BE01F754F8C2E5
4. A definitive biography of Bell is: A. Whitaker, John Stewart Bell and Twentieth Century Physics (Oxford University Press, 2020). It is available at: https://global.oup.com/academic/product/john-stewart-bell-and-twentieth-century-physics-9780198861263?q=A.%20Whitaker%2C%20Bell&lang=en&cc=gb
5. A recent anthology of commissioned articles about the history is The Oxford Handbook of the History of Quantum Interpretations (Oxford University Press, 2022), edited by O. Freire, G. Bacciagaluppi, O. Darrigol et al. It is available at: https://global.oup.com/academic/product/the-oxford-handbook-of-the-history-of-quantum-interpretations-9780198844495?q=Oxford%20Handbook%20of%20Interpretations%20of%20Quantum%20Theory&lang=en&cc=gb

13.2. Other Interpretations of Quantum Theory

There are several important interpretations (including distinctive approaches to the measurement problem) that this chapter did not mention at all. They are not what section 5 called “the usual suspects.” I give four examples and then end by stressing that physicists recognize the measurement problem as an outstanding problem.

Pragmatist Interpretations

My first example is very philosophical: the pragmatist interpretation developed by R. Healey, in a series of papers and in his recent book. Here, the label pragmatism is, unlike Bell’s usage reported in section 5, a deliberate echo of the American philosophical movement launched by Peirce, James, and Dewey. Being thus philosophical, Healey’s interpretation is closer to Bohr’s complementarity than to section 5’s “practical instrumentalism.”

Healey’s interpretation is one of a handful that reject one or more presuppositions used in sections 4 and 5 (and in most discussions) to formulate the measurement problem—namely, presuppositions of a realist stripe, especially the assumption that the quantum state straightforwardly describes physical reality, in the manner of the logics and semantics discussed in section 2 of chapter 3.

Being myself a philosophical realist and an enthusiast for those logics and semantics, I disagree with Healey (as I do with the rest of this handful of interpretations, including one called “quantum Bayesianism”). But I applaud the imagination and precision with which he has developed the view. Accordingly, without going into details, I recommend his book, together with a laudatory but dissenting review by a realist Everettian, David Wallace.

• Healey, Richard. The Quantum Revolution in Philosophy (Oxford University Press, 2019). Available at: oup.com/academic/product/the-quantum-revolution-in-philosophy-9780198844679//
• Wallace, David. Review in Analysis 80 (2020): 381–388. Available at: academic.oup.com/analysis/article-abstract/80/2/381/5819198

Modal Interpretations

My second example is the modal interpretation—or rather, modal interpretations, since there are many versions. Their shared strategy is:

1. To maintain that the Schrödinger equation is always correct.
2. To assign to quantum systems definite values of quantities beyond the eigenvalues of their states’ eigenquantities, but not by postulating ab initio a permanently definite quantity (as in pilot-wave theory).

Instead, modal interpretations propose principled rules specifying which additional quantity is definite, which values it may take, and how these change over time. Typically, the definiteness of a quantity, and the value it has, depend (as an interpretative postulate, not causally) on the quantum state. In some versions, this dependence is not merely on the system’s state, but also on that of a larger system of which it is a part.

Details are beyond the scope of this chapter. But here are two expert and craftsman-like monographs and an anthology of important essays, all from the 1990s, and research articles that are, so far as I know, the most recent expert proposal.

• Bub, Jeff. Interpreting the Quantum World (Cambridge University Press, 1997).
• Dieks, Dennis, and Pieter Vermaas, eds. The Modal Interpretation of Quantum Mechanics (Kluwer Academic, 1998).
• Vermaas, Pieter. A Philosopher’s Understanding of Quantum Mechanics (Cambridge University Press, 2000). Available at:
  cambridge.org/core/books/philosophers-understanding-of-quantum-mechanics/179FE190C668C06A217C160DAB8FD2BC
• Barandes, Jacob, and David Kagan. “The Minimal Modal Interpretation of Quantum Theory.” (Successor article in Foundations of Physics 50 (2020): 1189–1218.) Available at: https://arxiv.org/abs/1405.6755. Summary at: https://arxiv.org/abs/1405.6754

One-World Reconciliations

My third and fourth examples are technical and hardly discussed in the philosophical literature about quantum theory. They are due to two physicists, N. Landsman and A. Kent, respectively. I choose them because they have the great merits of (i) invoking powerful ideas (and associated technicalities) from other parts of physics to break out of the “usual suspects” taxonomy; and (ii) being ripe for further development. Thus, I think philosophers should take more notice of them.

Again, this is not the place for details. Suffice it to say that, broadly speaking, they both contrive to retain the two claims (labelled (1) and (2) in section 5) whose combination gave the measurement problem: the claims that the Schrödinger equation is always correct, and that quantum systems only have as definite values of quantities, the eigenvalues. And they achieve this “with only one world,” i.e., not by having an Everettian splitting, with the consequent assignment of eigenvalues only relative to a world. So they both propose a striking one-world reconciliation of section 5’s two claims, (1) and (2).

The differences between the two proposals lie in the ideas invoked to secure the reconciliation. Landsman invokes ideas about perturbation theory and spontaneous symmetry breaking. Kent invokes the idea of a final boundary condition. (Kent also aims for an explicit compatibility with relativity theory—a desideratum for any solution to the measurement problem that, for brevity, I have set aside.)

The main references are, respectively:

• Landsman, Klaas. The Foundations of Quantum Theory (Springer, 2017), esp. chap. 11. Available at: link.springer.com/book/10.1007/978-3-319-51777-3.
• Kent, Adrian. “Solution to the Lorentzian Quantum Reality Problem,” Physical Review A 90 (2014): 012107. Available at: arxiv.org/abs/1311.0249.

Kent has several other articles developing this solution. Further details are in my exposition and assessment (especially as regards quantum non-locality), which is aimed mostly at philosophers:

• Butterfield, Jeremy. “Peaceful Coexistence: Examining Kent’s Relativistic Solution to the Quantum Measurement Problem,” in Reality and Measurement in Algebraic Quantum Theory, ed. M. Ozawa et al. (Springer, 2018), 277–314. Available at: https://link.springer.com/chapter/10.1007/978-981-13-2487-1_11arxiv.org/abs/1710.07844 philsci-archive.pitt.edu/14040.

The Measurement Problem as an Outstanding Problem

I stress that quantum theory’s interpretative problems, especially the measurement problem, are accepted by physicists as outstanding problems. This is illustrated by two accessible essays by A. Leggett and S. Weinberg, both very distinguished winners of the Nobel Prize for Physics. Both essays stress the limitation of decoherence noted at the end of my section 7: in short, that it does not secure a definite outcome at the end of measurements. Both essays are also sympathetic, overall, to the search for a “cousin” of the Schrödinger equation that secures a definite outcome; i.e., to what Bell called the unromantic version of the strategy that the Schrödinger equation “is not right” (section 5). (Nor are these two essays “one-offs”: both authors wrote about their misgivings in several articles.)

• Leggett, Anthony. “Probing Quantum Mechanics Towards the Everyday World: Where Do We Stand?” Physica Scripta T102 (2002): 69–73.
• Weinberg, Steven. “The Trouble with Quantum Mechanics,” The New York Review of Books, January 19, 2017, 51–53. Available at: goldphysics.unm.edu/phys521/features/WeinbergTroubleWithQMe.pdf (accessed January 13, 2026).

13.3. The Everettian Interpretation of Quantum Theory

I postpone the discussion of decoherence and probability to the next subsection 4. Here I confine myself to two topics. First, the basic sources; and second, the idea that the definition of a “world” (or of “branch” or “splitting”) should be given in mentalistic terms, i.e., should be allowed to invoke concepts alien to physics, such as “appearance” and “experience.” The chapter’s discussion gave this idea, often called the “many minds” version of the Everettian interpretation, short shrift. (I only mentioned it obliquely towards the end of section 6 as part of my criticism that, in the early Everettian literature, the idea of splitting was too programmatic.) Indeed, I am not sympathetic, but of course, there are good advocacies of it.

Basic Sources

Pride of place goes to Everett’s original paper:

• Everett, Hugh. “Relative-State Formulation of Quantum Mechanics,” Reviews of Modern Physics 29 (1957): 454–462. Reprinted in Wheeler, John, and Wojciech Zurek, eds. Quantum Theory and Measurement. Princeton University Press, 1983.

As I mentioned at the start of section 6, not only are there different versions of the Everettian interpretation, but there is also controversy about how Everett himself understood it. My discussion did not go into these issues, but focused on what I believe to be the broad version that has been dominant since the 1990s (which invokes decoherence, macroscopic objects being patterns, etc.). For details about Everett’s own views, one should consult:

• Everett, Hugh. The Everett Interpretation of Quantum Mechanics: Collected Works 1955–1980, edited by Barrett, Jeffrey and Peter Byrne (Princeton University Press, 2012). Available at: jstor.org/stable/j.ctt7t2jf
• Byrne, Peter. The Many Worlds of Hugh Everett III. Oxford University Press, 2012. This is a biography of Everett and is available at: oup.com/academic/product/the-many-worlds-of-hugh-everett-iii-9780199659241

Another basic source is the popular book by D. Deutsch, a leading Everettian who pioneered the decision theory approach to deducing the Born-rule probabilities (my sections 12 and 13.4 below):

• Deutsch, David. The Fabric of Reality. Penguin Books, 1997.

The “Many Minds” Approach

The ‘many minds’ version of the Everettian interpretation was first articulated (so far as I know) by H.D. Zeh (a physicist who made profound contributions to the physics of decoherence; cf. below). This was in his paper:

• Zeh, Hans-Dieter, “On the Interpretation of Measurement in Quantum Theory.” Foundations of Physics 1 (1970): 69–76.

A book-length defence is:

• Lockwood, Michael. Mind, Brain and the Quantum. Blackwell, 1989.

Some of my own views on this interpretation are in a journal’s “symposium” on Lockwood’s work. (The paper includes (a) recommending work by M. Donald, and (b) linking the discussion to the topic of probability, as below.) It is at:

• Butterfield, Jeremy. “Whither the Minds?” British Journal for the Philosophy of Science 47 (1996): 200–221.

13.4. Treatments of Decoherence and Probability

Decoherence

I said in section 6 that since about 1990, the Everettian literature has appealed to decoherence. Here I should add that, in fact, the physics of decoherence was much clarified already in the 1970s and 1980s (cf. below). But it was only after 1990, especially in papers by Simon Saunders, that there was a clear philosophical statement that what decoherence theorists usually called “branches” (also “histories”) were a perfect fit for the Everettian’s proposed worlds. Two of Saunders’ papers on this topic are:

• Saunders, Simon. “Decoherence, Relative States, and Evolutionary Adaptation,” Foundations of Physics 23 (1993): 1553–1585.
• Saunders, Simon. “Time, Quantum Mechanics, and Decoherence.” Synthese 102 (1995): 235–266.

For the technical physics of decoherence, two outstanding books are:

• Joos, E. et al. Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed. Springer, 2003. Available at: link.springer.com/book/10.1007/978-3-662-05328-7
• Schlosshauer, Max. Decoherence and the Quantum-to-Classical Transition. Springer, 2007. Available at: link.springer.com/book/10.1007/978-3-540-35775-9

Another two expository technical papers, which are readable by philosophers (and which also address the subtleties about the system-environment split that I raised at the end of sections 7 and 8), are in the Many Worlds? anthology that I cited above. They are:

• Hartle, James. “The Quasiclassical Realms of the Universe.” Available at: https://arxiv.org/abs/0806.3776
• Halliwell, Jonathan. “Macroscopic Superpositions, Decoherent Histories, and the Emergence of Hydrodynamic Behaviour.” Available at: https://arxiv.org/abs/0903.1802

Macroscopic Objects as Patterns

The main theme of philosophical discussion of decoherence in the Everettian context has been the idea that macroscopic objects are patterns. Apart from Wallace’s monograph, The Emergent Multiverse, and Saunders and Wallace’s Introduction and Chapter 1 in the Many Worlds? anthology (both cited above), I also recommend: (i) Wallace’s first article advocating the idea; and (ii) the most recent (so far as I know) critical assessment, by R. Mulder. They are as follows:

• Wallace, David. “Everett and Structure.” Studies in History and Philosophy of Modern Physics 34 (2003): 86–105.
• Mulder, Ruward. “The Classical Stance.” Studies in History and Philosophy of Science 107 (2024): 11–24.

Another philosophical theme about decoherence in the Everettian context is whether the worlds arising from decoherence can accommodate contingency about which macroscopic objects exist. This theme was section 8’s closing issue, i.e., misgivings, about the sketch definition of “world.” We will return to this theme in section 3 of chapter 6’s discussion of A. Wilson’s proposal to understand the philosophical multiverse of chapter 3 in terms of the Everettian multiverse (i.e., in philosophical jargon, his proposal to reduce the former to the latter). A recent critical assessment of the proposal is:

• Harding, Jacqueline. “Everettian Quantum Mechanics and the Metaphysics of Modality.” British Journal for the Philosophy of Science 72, no. 4 (2021).

Probability

About Everettian probability, I will give references for three topics: (a) the decision-theory approach, which, as I reported in section 12, is now dominant; (b) a recent approach by S. Saunders; and (c) controversies about interpreting tiny probabilities or amplitudes

(a) The ground-breaking article that first proposed to derive Born-rule probabilities using axioms from decision theory (section 12) is:

• Deutsch, David. “Quantum Theory of Probability and Decisions,” Proceedings of the Royal Society A (1999). Available at: https://royalsocietypublishing.org/rspa/article-abstract/455/1988/3129/80459/Quantum-theory-of-probability-and-decisions?redirectedFrom=fulltext; arxiv.org/abs/quant-ph/9906015

The idea was much developed by D. Wallace in a series of papers, including chapter 8 in the Many Worlds? anthology (2010), which I cited above, and his own 2012 book:

• Wallace, David. The Emergent Multiverse (Oxford University Press, 2012).
• Saunders, Simon et al., eds. Many Worlds? (Oxford University Press, 2010).

One issue that this did not broach is how an Everettian can make sense of quantum theory getting confirmed by experimental statistics, in view of the obvious worry that in many worlds in the multiverse, experiments will yield statistics far from the Born-rule probabilities. A ground-breaking article addressing this is:

• Greaves, Hilary, and Wayne Myrvold. “Everett and Evidence,” in Many Worlds? (Oxford University Press, 2010).

(b) Recently, Simon Saunders has developed a very different approach to deriving the Born rule in Everettian quantum theory. This approach does not appeal to decoherence or decision theory. But it connects closely to section 11’s topic of probability in classical statistical mechanics, to section 12’s first topic of branch counting, and to frequentism in the philosophy of probability. Two of his papers about this approach are:

• Saunders, Simon. “Finite Frequentism Explains Quantum Probability.” British Journal for the Philosophy of Science (2024). Available at: https://www.journals.uchicago.edu/doi/10.1086/731544; arxiv.org/abs/2404.12954
• Saunders, Simon. “Physical Probability and Locality in No-Collapse Quantum Theory.” https://arxiv.org/abs/2505.06983

(c) The chapter neglected controversies about the interpretation of amplitudes (and so probabilities) that are small, maybe minuscule. Typically, a wave function assigns many possible configurations a small amplitude (these “parts” of the wave function are called “tails,” like the left and right tails of a bell-shaped probability distribution). But interpretation usually focuses on the wave function’s “peaks.” This is especially true of this chapter’s Everettian, who took macroscopic objects to be the peaks of the wave function (or the cluster of classical configurations under such a peak). This prompts an objection to the Everettian’s idea (section 12 and the references just cited) of deducing the probabilistic interpretation of the squares of amplitudes from the betting behaviour of an Everettian experimenter who bets on macroscopic outcomes (relative to worlds, of course) of quantum measurements. Namely, how can the Everettian justify ignoring the tails of the wave function? Surely not by their having a small probability, since the Everettian ignores them en route to their argument justifying the probability interpretation. So how?

In his book, Wallace addresses this objection (2012, 253–254). But not to everyone’s satisfaction. In particular, Dawid and Thebault dissented in a 2015 article:

• Dawid, Richard, and Karim Thebault. “Many Worlds: Decoherent or Incoherent?” Synthese 192 (2015): 1539–1580.
• Dawid, Richard, and Karim Thebault. “Decoherence and Probability.” Available at: https://arxiv.org/abs/2410.01317philsci-archive.pitt.edu/23991