Travis Norsen's *Foundations of Quantum Mechanics* could be the spark that ignites a revolution. There is nothing new in it.

If those two sentences sound contradictory, they should. How could a book without a novel thesis change everything?

Welcome to the world of foundations of quantum mechanics. Everyone knows, in some vague way, that there exists such a field as foundations of physics in general, and of quantum theory in particular. But it may be unclear exactly who does this work and what they do. One stereotype is that foundations of physics is what some physicists do on the weekends or after they have run out of real physics to do. Also some philosophers do it full time. This last fact is a huge red flashing warning sign that there is something disreputable about the whole business.

In the case of quantum theory, a terminological marker has been created. Quantum theory is the most predictively accurate theory in history. There is no doubt that it is in some sense correct. But even though we have every reason to trust its predictions, there is still another question: how to *interpret* it.

According to this elucidation, quantum theory has everything one could want from a theory save an “interpretation.” And whatever it is to interpret a theory, it can't be of any importance to physicists in their everyday life. Quantum theory has gone from triumph to triumph without having an “interpretation.” An “interpretation” must be some inessential luxury add-on, like heated seats in a car: it makes you feel warmer and more comfortable, but plays no role in getting you from here to there.

On this understanding, worrying about interpreting quantum theory is inessential to pursuing the basic aims of science.

This is where Norsen comes in. Think of *Foundations of Quantum Mechanics* first and foremost as what it is: a textbook for students. As such, it should not and does not contain any novelty in its content. Textbooks are judged by the logic of their organization, the clarity of their presentation and the lucidity of their style. This one covers many of the topics of a standard introduction to quantum physics, but focuses its attention on the foundational questions: What is there? How does it behave when no one is looking? How does it behave when someone is looking? (Separating these questions indicates that we are doing quantum theory.) Which parts of the mathematical apparatus represent real physical properties and which are merely gauge degrees of freedom? What sort of thing does the wavefunction of a system represent?

Standard textbooks gloss over these questions. Norsen dwells on them. The first chapter covers familiar ground: the structure of pre-quantum theories including Newtonian Mechanics and Maxwellian Electrodynamics. Even here, the presentation foregrounds issues that are commonly ignored. In these seemingly unproblematic theories, how do we determine the physical ontology (i.e., the basic physical entities) postulated by the theory? A familiar example is the scalar and vector potentials of classical electro-magnetism. In certain gauges (e.g., Coulomb gauge) the potentials react instantaneously to distant states of affairs. But the sting of this appearance of action-at-a-distance is drawn if one denies physical reality to the potentials, regarding them instead as mere calculational devices. Already we find ourselves contemplating questions about what is real, and about whether anything physically real goes faster than light.

The second chapter presents basic quantum phenomena involving interference and entanglement. This will be familiar to any student who has had an introduction to quantum mechanics, but playing around with particular examples encourages developing a “feel” for the theory.

Deviation from the standard textbook begins in the next three chapters. Each of these presents a “problem” confronting attempts to understand quantum mechanics as a physical theory. Chapter 3 discusses the Measurement Problem, Chapter 4 the Locality Problem, and Chapter 5 the Ontology Problem.

The Measurement Problem is the best known of the three. Succinctly: is there any fundamental physical difference between interactions that count as “measurements” and those that don't? A “fundamental” difference shows up when articulating the basic laws of the theory.

John von Neumann's axiomatization of quantum mechanics treats measurement as fundamental. The wavefunction evolves by smooth deterministic laws when the system is not being measured and by sudden indeterministic collapses when measured. This approach contradicts the conviction that measurements are physical interactions like any others, governed by the same laws. What's a measurement depends on the physical dynamics rather than the other way around.

The Measurement Problem poses a difficulty if measurement is a trigger for wavefunction collapse. But the collapse itself, no matter how triggered, raises a different puzzle: the Locality Problem. This is what bothered Einstein about quantum theory from the beginning. Collapses, as physical events, are wildly non-local. Thus the famous “spooky-action-at-a-distance” that Einstein could not abide.

Finally, the Ontology Problem concerns the physical significance of the wavefunction. One way to pull the non-local sting from wavefunction collapse is to regard the wavefunction as a mathematical object that does not represent any physical property of an *individual system*. Does it rather represent only statistical features of an *ensemble of systems*? Does it represent any *objective, mind-independent fact*? Or rather reflect just the *information an agent has about the system*?

All of these options have been defended, and it is easy to see their attraction. The wavefunction of an electron spreads out in space. Does that mean the electron itself spread out? Or that a huge collection of electrons spreads out? Or that my information about where the electron is dilutes? But if it is not the single electron physically spreading, how can one explain two-slit interference?

Further, the mathematical wavefunction is not defined over three-dimensional physical space but over the 3N-dimensional configuration space of N particles. Fields in 3N-dimensional space don't have any evident relation to the three-dimensional world we find ourselves in, the world that physics is meant to explain. Norsen recounts how Schrödinger tried to solve this problem by defining a three-dimensional “charge density” for each electron, and then superimposing all of these in a common three-dimensional space. However, the “smeariness” of the charge density could not be quarantined to the microscopic, but amplified up to macroscopic scale. That is the problem of his eponymous cat.

How might one solve the Measurement, Locality and Ontology Problems? These are questions that a typical physics textbook either ignores altogether or tries to finesse. They are also problems that many physics students are intensely interested in. It is here that you least want to hear the command: “Shut up and calculate!”.

If calculation will not address these problems, what will? Each problem reflects an unclarity about the physical significance of the mathematical formalism. And making precise statements about the physical ontology and dynamical laws is just what it is to *precisely specify a physical theory*. Standard quantum textbooks do not exposit a physical theory that lacks an interpretation: they present a predictive formalism without any accompanying physical theory! “Interpreting quantum theory” is actually constructing alternative physical theories that can account for the accuracy of the predictive formalism.

Chapter Six discusses the most famous “interpretation” of all: the Copenhagen Interpretation. It is not a precisely formulated physical theory. It does not say what physically exists and how it behaves. The contemporary Copenhagen Interpretation is just an attitude: the refusal to ask, much less attempt to answer, foundational questions about quantum theory.

That is not how Bohr saw things. He thought that deep morals about the nature of the world had been revealed by quantum theory. Einstein found Bohr's exposition largely incomprehensible. One lovely thing in these chapters, and indeed throughout the whole book, is the judicious but extensive use of quotations from Einstein, Schrödinger, Heisenberg, Born, Bell, Bohr, etc. Their discussions are sharp and clear, and students will delight at reading the masters debating what they have done. Nothing could be more gratifying to an undergraduate physics student than reading Einstein complain about his difficulties with quantum mechanics.

Chapter 6 ends without any clearly articulated physical theory in hand. Here *Foundations of Quantum Mechanics* departs most dramatically from standard textbook presentations: it presents three clear, mathematically formulated physical theories that aspire to make the same—or nearly the same—predictions as the quantum predictive formalism. Each of these three theories exemplifies a response to Schrödinger's cat problem.

Here's Schrödinger's puzzle. Initially, we assign a wavefunction to the system containing the cat and apparatus. Suppose that wavefunction always evolves in accord with the linear Schrödinger equation. It becomes a superposition of macroscopically different states, some with a live cat and others with it dead. If the wavefunction is *complete* (i.e., if it represents every physical characteristic of the cat) we have a problem. The cat ends up neither simply dead nor simply alive. As John Bell put it: “Either the wavefunction, as given by the Schrödinger equation, is not everything or it is not right.”

Regarding the wavefunction as incomplete—as not everything—yields a *hidden variables* theory. The term is a terrible misnomer. If the extra variables are to determine the health of the cat then they had better not be hidden, else we would not be able to tell if the cat ends up alive or dead. Regarding the wavefunction as complete but not right (as given by Schrödinger's equation) yields a *collapse* theory. The Copenhagen Interpretation is often taken to be a collapse theory that ties the collapses to measurements, an option that highlights the measurement problem.

Chapter 7 presents the most famous “hidden variables” theory: the pilot wave theory or Bohmian mechanics. In this theory “particles” are particles—point objects that have definite positions and follow continuous trajectories through space-time. The wavefunction always evolves by Schrödinger's equation and the point particles also evolve deterministically, in accord with the *guidance equation*. The evolving microscopic particles congregate into macroscopic objects, which are shaped and behave just like ones we see in the real world. At the end of Schrödinger's experiment, for example, there will either be a cat-shaped collection of particles moving like a live cat or a cat-shaped collection inert like a dead cat. No problem.

If Bohmian mechanics solves Schrödinger's problem so cleanly, why has it not been universally adopted? Because the dynamics of the Bohmian particles is wildly non-local: which way a particle *here* goes can depend on the disposition of a piece of matter way over *there*. Bohmian mechanics incorporates the spooky action-at-a-distance that Einstein hated.

Chapter 8 exposits Bell's theorem: John Bell's proof that non-locality is unavoidable given the predictions of standard quantum mechanics. That removes the main objection to Bohmian mechanics, although, as Bell says, in the way Einstein would have liked least.

Chapter 9 presents the most highly developed collapse theory, due to GianCarlo Ghirardi, Alberto Rimini and Tulio Weber, universally known as GRW. GRW avoids the difficulty of tying the collapses to measurements by tying them instead to……nothing. The collapses just happen randomly with fixed probability per unit time.

Finally, Chapter 10 investigates escaping Bell's dilemma by maintaining that the wavefunction evolving by Schrödinger's equation is both everything and is right. This yields the *Many Worlds* or *Everett* Interpretation. It is a famously weird physical theory, not least due to the multiplying worlds. It is, for example, problematic what the *probabilistic predictions* of the quantum predictive apparatus even mean in this setting.

GRW, Many Worlds and Bohmian mechanics are not presented in any standard quantum mechanics textbook. How adequate is Norsen's exposition?

The writing is not just so clear and straightforward that a non-expert can understand it; it is so clear and straightforward that an expert cannot manage to misunderstand it.

What shortcomings does *Foundations of Quantum Mechanics* have? Norsen, like many others, attributes the electromagnetic gauge of Ludvig Lorenz instead to Hendrik Lorentz. And there are many topics that have been omitted: the PBR theorem, the Bohm-Aharonov effect, field theory, the challenge of Relativity, particle creation and annihilation, etc.

But this last complaint is really a call for a successor volume: *Advanced Foundations of Quantum Mechanics*. May this book ignite a revolution in the pedagogy of quantum mechanics. *Vive la Révolution!*

*Tim Maudlin is Professor of Philosophy at New York University. He is the author of Quantum Non-Locality and Relativity, Truth and Paradox, The Metaphysics Within Physics, New Foundations for Physical Geometry, and Philosophy of Physics: Space and Time. He is a member of the American Academy of Arts and Science, the Academie Internationale de Philosophie des Sciences, and a Guggenheim Fellow.*