This book is an introduction to the foundation of quantum mechanics. As such, this book is perfect: it is the book that my former, physics undergraduate, self would have wanted to read. At the time, like typical physics undergraduates around the globe, I was taught to give up hope of ever understanding what quantum theory claims: at best, the theory is an instrument to predict experimental results. No matter how much we might dislike it, we have to accept it; there is no way out. It was my refusal to give in that led me to become a philosopher of physics. And I was right in being persistent because I later discovered that, as Maudlin clearly explains, there is no reason to be pessimistic: all the quantum mysteries and paradoxes have a solution. Actually, more than one. To have Maudlin’s book at the time would have saved me a lot of time and pain. I am sure it will be a life-saver for many other rightfully puzzled undergraduate physics students wondering if the questions they ask are wrongheaded, as too many of their teachers claim: How can the Schrödinger cat really be dead and alive at the same time? Is reality created by an act of observation or a measurement? How is a measurement not a physical process like any other? Because these are not bad questions at all, and they have straightforward answers: the Schrödinger cat cannot be both alive and dead; reality is not created by observation; a measurement is just a type of interaction between two physical systems.
However, the book is also not perfect (becoming, ironically, a superposition). While Maudlin defends his own position on how to make sense of quantum theory, his book is also potentially misleading about the state of the art in terms of the philosophical foundations of the theory. Maudlin omits to mention (or cite, or give reference to) the majority of the literature generated in the last two decades around the problems he discusses. Although this book may be intended as an introductory textbook to the field, I am disappointed that it skirts so much of the work of the past few decades on the foundations of quantum theory. To be fair, there is a habit in philosophy of physics at large of failing to sufficiently acknowledge other work, but this is why the failure to do much to rectify this trend in even a short introductory textbook like this one troubles me as much as it does. Indeed, this can mislead students who are new to the material, in the same way that I felt misled by my undergraduate physics instructors about the nature of quantum theory. This is also puzzling because, as I will discuss below, the lack of contextualization of his view within the actual literature makes Maudlin’s arguments weaker than they could have been.
Maudlin’s book has seven chapters. The first describes the core quantum phenomena by summarizing them in eight emblematic experiments. He discusses Bell’s inequality and shows how this implies that nature is non-local. Although I wholly agree with Maudlin’s reading of Bell’s theorem, he does not mention the fact that every single aspect of the theorem is debated. This is one of the places where his refusal to mention the existence of disagreement or debate is potentially misleading to newcomers. While I understand that this is not the focus of the book, Maudlin did not have to enter into the debate in order to acknowledge that such a debate exists.
The second chapter focuses on describing the orthodox account, found in physics books, of the experimental results discussed in the previous chapter. This is, in the words of Maudlin, merely a recipe for predicting measurement results, without any attempt to understand how and why these results come about. Why do we see an interference pattern? Why does it disappear? These are questions that do not have an answer in this framework. If one wants more, as Maudlin does (and as I did; and I assume any physicist will be at least interested in whether more is possible), one has to look elsewhere. But where? The third chapter of the book focuses on the wave function, the mathematical object used in the quantum recipe, and asks whether the wave function represents something physical (in the traditional terminology, whether it has an ontic interpretation). Maudlin’s response is that it does: it represents what he calls the ‘quantum state’. The alternative is that the wave function represents only our knowledge of the system (that is, it is epistemic), rather than something about the system itself (ontic). And this, Maudlin claims, is ruled out by the recently proven PBR theorem, which is discussed in detail in this chapter. To my knowledge, this is the first time that this theorem has been presented in a philosophy of physics book, let alone a textbook. However, this is also another instance of Maudlin leaving out information about the state of the art of the discipline. In fact, the debate is still open on whether the epistemic view is actually ruled out by the PBR theorem.
Having analysed the implications of the PBR theorem and concluding that the quantum state has to be ontic, Maudlin recognizes that there is a problem: the quantum state does not exist in physical (three-dimensional) space as the wave function lives in a high-dimensional space called configuration space (which, classically, is the space of the positions of all the particles in the world). So what is the quantum state? Maudlin claims that this question is ‘misguided’ (p. 89) because it assumes that we must have an Aristotelian category in which the quantum state would fit. But why should we assume this? After all, Aristotle did not know anything about quantum theory. The fact is, Maudlin maintains, that the quantum state is ‘a novel feature of reality’ and that ‘there is nothing wrong with allowing it a novel category: the quantum state’ (p. 89).
The problem is that, in these few sentences, Maudlin quickly dismisses the work of those who have succumbed to the ‘misguided desire to liken the quantum state to anything we are already familiar with’ (p. 89) without having seriously engaged with it. This includes the work of those who think the wave function is part of the laws of nature (for example, Goldstein and Zanghì ; Callender ; Miller ; Bohgal and Perry ; Allori ) or a property of the particles (for example, Monton ; Lewis , , ; Esfeld et al. ; Suarez ); those who propose that the quantum state is a material multi-field in three-dimensional space (Forrest ; Belot ; Hubert and Romano ); those who want to rewrite quantum theory so that the mathematical representation of the quantum state is no longer an object in configuration space but something in three-dimensional space (Norsen ); those who think that the quantum state is, contrary to what Maudlin thinks, what matter is made of (North ; Ney [forthcoming]); and those who are critical of this without necessarily taking a stand on what the quantum state is (for example, Myrvold ). As a result, Maudlin ends up defending the view that the quantum state is the quantum state, primitively so, without any justification other than that it is misguided to follo
w the desire to understand what the quantum state is. But why? No reason is provided and this does not seem to be good advice. In fact, Maudlin’s failure to acknowledge or argue against alternative viewpoints is troublingly reminiscent of my own physics teachers’ failures to acknowledge the possibility of a reasonable understanding of quantum theory. My teachers’ instruction to shut up and calculate was bad advice, as Maudlin agrees; but Maudlin’s equally bad advice seems to be to shut up and take the quantum state as a primitive notion.
Perhaps more importantly, I think that Maudlin’s omissions around the current debate on the nature of the wave function/quantum state hurts his case: he could have argued (presumably by adding just few pages to Chapter 3, already considerably shorter than the other chapters) that the other approaches are fundamentally flawed, and thus he could have better justified why we are left with his position. Instead, he does not engage with any literature on this (with the exception of Albert and Wallace in Chapters 4 and 6, as discussed below) and his argument is left weaker than it might otherwise have been.
The book continues with a presentation of the first solution of the mysteries of quantum theory, namely, the spontaneous collapse theory (the so-called GRW theory). It is presented as a clean and precise formalization of the quantum recipe of Chapter 2, in which the wave function spontaneously collapses in one of the terms of the superposition as a matter of law, rather than as a measurement is performed. In this chapter, I think, comes the main novelty of Maudlin’s book, which sets it apart from the most other books in the philosophical foundations of quantum mechanics: the problem of local beables. This problem was raised by Bell in 1975, and it amounts to the fact that it is hard to see how one can connect the behaviour of stuff in some high-dimensional space with our three-dimensional observations, unless one postulates some ‘local beable’: some ontology in three-dimensional space or four-dimensional spacetime. Maudlin argues that Bell is correct in requiring local beables because ‘to give the theory empirical content, we need some sort of items that exist and move in physical space, influenced by the quantum state’ (p. 110), and that ‘the structure of the wave function must be projected down from configuration space into physical space’ (p. 116).
The chapter continues with a discussion of the two theories proposed to solve the problem of local beables, namely, the matter density version of the collapse theory proposed by Ghirardi (GRWm) and the flash version proposed by Bell (GRWf). However, Maudlin fails to acknowledge that Bell’s suggestion was rediscovered and refined by Dürr et al. (), and then by myself (Allori , ; Allori et al. ), in terms of the primitive ontology approach. Here, instead of local beables, one talks about primitive ontology. In fact, satisfactory beables are more than merely local; they are also ‘primitive’ in the sense that their behaviour is governed by non-primitive entities, such as the wave function. This is what gives the wave function a nomic (or quasi-nomic) status: the wave function does not describe matter, rather it describes how matter moves (thus, it is ‘real’ and objective in the sense that it is part of the laws of nature). There are parts in the text in which Maudlin talks in these terms (for example, ‘the important role of the quantum state is to guide the motions of the particles’, p. 171); however, it is difficult to understand how on Maudlin’s account the quantum state, a primitive without a quasi-nomic status, can play this role. Had he engaged with the primitive ontology view, if only to criticize it, it would have helped me better understand his position.
As an aside, in the discussion of local beables in the collapse theory, I did not find very helpful the notion that the quantum state is informationally complete. After all, how can it be that the quantum state is informationally complete given that, for a given theory and in addition to the wave function, we have to specify the map that specifies the local beable for that theory? The very reason why GRWm and GRWf are different theories is that the quantum state is not informationally complete: in the former, in addition to the wave function, the theory also specifies which other function, among infinitely many possibilities, defines the matter density; and in the latter, the theory specifies which, among all possible points in spacetime, are the material points, or the flashes.
The fifth chapter is dedicated to the pilot-wave theory, which states that there are particles moving deterministically and guided by the quantum state. As is generally the case throughout the book, this chapter is very clear. However, I would have liked to see more on how to understand probability in the theory. The usual claim by the Bohmians (as well as by Maudlin) is that empirical frequencies are accounted for if the behaviour of the typical configuration can reproduce them, so that probability should be understood as typicality. This is in agreement with a Boltzmannian understanding of probabilities in statistical mechanics, and so is controversial. This is another instance in which it would have been helpful to have a better picture of the literature on this issue, or at least some mention of additional sources where more discussion can be found. In any case, the discussion of the eight experiments both here and in the context of collapse and many-worlds theories is very illuminating, and I would have loved this presentation as an undergraduate physics student.
Chapter 6 is devoted to the many-worlds theory, which solves the problems of quantum theory with the least deviation from the quantum recipe. Maudlin heavily criticizes the approach pioneered by Wallace on two fronts. The first concerns the problem of understanding the nature of probability in this deterministic theory, and attacks the decision-theoretic approach. In this respect, a notable omission is Barrett (), who discusses many-worlds probabilities in terms of typicality, which is an approach favoured by Maudlin in the context of the pilot-wave theory. The second criticism is more novel and concerns local beables. Maudlin discusses the work of Timpson and Wallace, who (to some extent) agree with Maudlin that local beables are needed. They propose what they call ‘spacetime state realism’; however, Maudlin argues that this is unsatisfactory on the basis that the proposed local beable—the reduced density matrix—is not separable (the whole is not given by the sum of its parts). The problem that Maudlin identifies, and which will be relevant for our discussion below, is that ‘this inversion of the usual relation between spatial parts and wholes means that we cannot infer the macroscopic situation […] from the state of the microscopic terms’ (p. 202).
The final chapter deals with possible relativistic extensions, and it presents theories that have never previously been discussed in a textbook, such as relativistic GRWf, relativistic GRWm, and Bell-type quantum field theories.
To conclude, the main problem that I had with this book was its lack of engagement with current literature, which leaves the (perhaps incorrect) impression that Maudlin’s view is inconsistent. In addition to the examples already mentioned, let me also note the following: In Chapter 3 he tells us that we should not be tempted to connect the quantum state with something familiar, that we should simply accept the quantum state as is. However, he does not seem to follow his own advice when he tells us in Chapter 4 that any fundamental theory should have local beables, and in Chapter 6 that spacetime state realism is inadequate. In fact, the main reason adduced for local beables and against spacetime state realism is familiarity: the quantum state is not in physical space, while the local beables are, in virtue of being in the familiar three-dimensional space; if reduced density matrices are the local beables, then we cannot use familiar notions to recover observations from the theory (because they invert the usual part–whole relation). These arguments use the same notion of familiarity Maudlin warns us about earlier in the book. Instead, on the primitive ontology approach that I defend, we do not resist familiarity and thus there is no such weakness. Indeed, familiarity (or a similar notion) is one of the main reasons why the wave function is not taken to be material on this approach. This is also the main reason why we need local beables, and why we would regard spacetime state realism as unsatisfactory. Had Maudlin explained why familiarity is a problem, he could have criticized our approach and at the same time defend his view that the quantum state is in its own category of quantum state. (Although he would still need to justify why, in the case of local beables, familiarity is a good thing to appeal to , or spell out more clearly why local beables are needed.)
In many respects, this book is a milestone in the philosophy of physics, and it is destined to become a classic. But it also strikes me, at least to some extent, as a lost opportunity. Progress in quantum foundation is hardly a one-man job—or even a four-man job (Albert–Bell–Maudlin–Wallace)—and Maudlin could have done a little more to ensure that everyone participating in solving the quantum puzzle had their place in the story. Not doing so is not only a disservice to the reader, but also makes for a weaker argument.
Valia Allori Department of Philosophy Northern Illinois University email@example.com
 See, for example, (Myrvold et al. ) and references therein.
 See, for instance, (Ben-Menahem ) and references therein for discussion.
 See, for example, (Wilhelm [forthcoming]) and references therein.
Allori V. : ‘Fundamental Physical Theories: Mathematical Structures grounded on a Primitive Ontology’, Ph.D. Thesis, Rutgers University.
Allori, V. : ‘Primitive Ontology and the Structure of Fundamental Physical Theories’, in D. Albert and A. Ney (eds), The Wave Function: Essays in the Metaphysics of Quantum Mechanics, Oxford: Oxford University Press, pp. 58–75.
Allori, V. : ‘A New Argument for the Nomological Interpretation of the Wave Function: The Galilean Group and the Classical Limit of Nonrelativistic Quantum Mechanics’, International Studies in the Philosophy of Science, 31, pp. 177–88.
Allori, V., Goldstein, S., Tumulka, R. and Zanghì, N. : ‘On the Common Structure of Bohmian Mechanics and the Ghirardi–Rimini–Weber Theory’, British Journal for the Philosophy of Science, 59, pp. 353–89.
Barrett, J. : ‘Typical Worlds’, Studies in History and Philosophy of Modern Physics, 58, pp. 31–40.
Belot, G. : ‘Quantum States for Primitive Ontologists: A Case Study’, European Journal for Philosophy of Science, 2, pp. 67–83.
Ben-Menahem, Y. : ‘The PBR Theorem: Whose Side Is It On?’, Studies in the History and Philosophy of Modern Physics, 57, pp. 80–8.
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Callender, C. : ‘One World, One Beable’, Synthese, 192, pp. 3153–77.
Dürr, D., Goldstein, S. and Zanghì, N. : ‘Bohmian Mechanics and the Meaning of the Wave Function’, in R. S. Cohen, M. Horne and J. Stachel (eds), Experimental Metaphysics: Quantum Mechanical Studies for Abner Shimony, Volume 1, Boston: Kluwer Academic Publishers, pp. 25–38.
Esfeld, M. A., Lazarovici, D., Huber, M. and Dürr, D. : ‘The Ontology of Bohmian Mechanics’, British Journal for the Philosophy of Science, 65, pp. 773–96.
Forrest, P. : Quantum Metaphysics, Oxford: Blackwell.
Goldstein, S. and Zanghì, N. : ‘Reality and the Role of the Wavefunction in Quantum Theory’, in D. Albert and A. Ney (eds), The Wave Function: Essays in the Metaphysics of Quantum Mechanics, Oxford: Oxford University Press, pp. 96–109.
Hubert, M. and Romano, D. : ‘The Wave-Function as a Multi-field’, European Journal for Philosophy of Science, pp. 1–17.
Lewis, P. J. : ‘Dimension and Illusion’, in D. Albert and A. Ney (eds), The Wave Function: Essays in the Metaphysics of Quantum Mechanics, Oxford: Oxford University Press, pp. 110–25.
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Miller, E. : ‘Quantum Entanglement, Bohmian Mechanics, and Humean Supervenience’, Australasian Journal of Philosophy, 92, pp. 567–83.
Monton, B. : ‘Against 3N-Dimensional Space’, in D. Z. Albert and A. Ney, The Wave-Function: Essays in the Metaphysics of Quantum Mechanics, New York: Oxford University Press, pp. 154–67.
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Myrvold, W. C., Genovese, M. and Shimony, A. : ‘Bell’s Theorem’, in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy.
Ney, A. [forthcoming]: Finding the World in the Wave Function, Oxford: Oxford University Press.
Norsen, T. : ‘The Theory of (Exclusively) Local Beables’, Foundations of Physics, 40, pp. 1858–84.
North, J. : ‘The Structure of the Quantum World’, in D. Z. Albert and A. Ney (eds), The Wave Function: Essays in the Metaphysics of Quantum Mechanics, New York: Oxford University Press, pp. 184–202.
Suarez, M. : ‘Bohmian Dispositions’, Synthese, 192, pp. 3203–28.