This book is about the deep mysteries that lie at the heart of quantum mechanics, the sort that bothered Feynman, and Schrödinger and Einstein before him. Indeed, Feynman's famous quote lends itself to the title of the first chapter of the book. Interest in this subject was once confined to academics, particularly those of a philosophical bent, but that is no longer true. The public has caught wind of the fact that the very puzzles that bothered the greats could hold the keys to some of the transformative technologies of the future, such as quantum computing, and is eager to learn more about them. Many authors have responded to this need by writing books at a variety of levels, ranging from the very elementary (one of which even allows your dog to get a bite out of the subject) to the more demanding. This book is certainly at the more demanding end, but it is still aimed at a non-specialist audience.

The author, Jeffery Bub, is a Distinguished University Professor at the University of Maryland with dual appointments in the Department of Philosophy and the Joint Center for Quantum Information and Computer Science. He has a special interest in the conceptual foundations of the quantum theory and has worked in this area for five decades. He has written numerous research articles as well as two previous books on different aspects of this field. In addition, for many years he has co-organized an annual meeting titled "New Directions in the Foundations of Physics" that has brought together experts from all over the world to discuss, and sometimes debate hotly, problems at the forefront of the field. Thus he is well positioned to write this book.

Bub has two overarching aims in this book. The first is to convey to the reader, as simply as possible, some of the deep mysteries at the heart of quantum mechanics. And the second is to discuss what light our present understanding of them sheds on the nature of our world. Neither task is particularly easy. The first is difficult because most readers will not possess the technical knowledge needed to understand and appreciate many of the mysteries fully. And the second is much, much more difficult but must be attempted all the same, for without it the book might not make much sense to its intended audience.

Bub's approach to the first of these tasks is to introduce the minimum of theoretical equipment needed and to manipulate it skillfully to achieve all his ends. The basic ideas he uses are: a quantum two-state system, or qubit, and the infinite family of states it can assume as a superposition of two of its basis states; the physical realization of a qubit provided by the polarization states of a photon; Malus's law for the result of a polarization measurement on a photon; and a particular type of entangled state of two photons. There is more than just this, of course, but it is introduced at later points in the text when it is needed. The mathematical tools needed to work with qubits are introduced along with the discussion of their physical properties, so that their relevance is always clear. The mathematics involves no more than algebra, geometry, trigonometry, and complex arithmetic, but don't be fooled. The discussion of the physics can quickly lead you into deep waters, and you may find that it takes quite a bit of effort to wrap your head around some of the ideas being discussed.

After a preliminary chapter in which he assembles some basic tools, Bub is off with a bang. Early on he introduces the reader to the PR box, perhaps the most amazing object discussed in the entire book. Named for its discoverers Sandu Popescu and Daniel Rohrlich, it consists of two separate but identical parts that can be placed at any distance from each other. Each part has two switch settings (that I will call S and T), each of which can lead to one of two outputs (that I will call 0 and 1). The PR box has the defining property that if the switch settings of its two parts are one of the combinations SS, ST, or TS, their outputs are the same (being 00 or 11, with equal probability), whereas if it is TT their outputs are different (being 01 or 10, with equal probability). What is so amazing about that, you might ask. The answer, quite simply, is that no one has yet figured out a way to build a PR box. Bub explores a number of plausible schemes, including one based on the spooky actions at a distance that so bothered Einstein, but shows that none of them is up to doing the job (or comes even close). It should be stressed that what makes the PR box so remarkable is that it is non-signaling (i.e., it cannot be used to transmit information between its two parts), but yet exhibits correlations that are much stronger than any we can see in our world.

The discovery of the PR box suggests that the reality we find ourselves in may be a bit like a Russian doll, with a classical domain enclosed inside a larger quantum one and the quantum domain likewise enclosed within a superquantum one. The superquantum domain is the one in which the PR box, and other objects like it, hold sway. Bub shows how some elementary four-dimensional geometry can be used to trace out, at least partially, the boundaries between these different domains. The classical-quantum boundary is of course familiar to most of us, even if we find ourselves hard pressed to define it precisely, but the idea of a quantum-superquantum boundary may come as a surprise to many. Bub introduces the reader to this boundary early in the book, and then crisscrosses it often in the later discussion.

The most fascinating question raised by the PR box is why we can conceive of a (superquantum) world in which it exists and yet not be able to set foot in that world. After two decades of pondering this question, the experts finally seem to be coming around to the view that the PR box may be an impossible object, like Escher's Waterfall, that we can conceive of in our minds but that cannot exist in reality. The reason for this feeling is that if a PR box existed, it could be used to perform a number of feats that are all too good to be true. What these feats are, and how the PR box can be used to accomplish them, are matters that Bub discusses at length. And because physicists believe there is no such thing as a free lunch, they conclude that a PR box cannot exist. But they have not been content to let matters rest there. Some researchers have shown that even if one had an imperfect PR box, it could still be used to perform magical feats (although of a lesser degree), and they then asked how much more the box would have to be weakened before its magic disappeared completely. The answer, many hope, is that it would have to be weakened to the point where it can be simulated perfectly with the resources available in our (quantum) world. We do not have a proof of this conjecture yet, but the pursuit of this Holy Grail continues to inspire a great deal of current research.

The story of the PR box is just one of the subplots of this book, but I have talked about it at some length because it is one of the things that makes the book stand out to me. There are no other books at this level, to my knowledge, that discuss this topic at all, and Bub's treatment of it is masterful; he begins from the basics and takes you all the way up to the research frontiers of the field. Many readers, from non-specialists to professional physicists, will find themselves caught up in the excitement of this quest as they read Bub's riveting account of it.

I will mention just two of the other topics discussed in the book. In a chapter entitled "Quantum Magic," Bub introduces the reader to the ideas of contextuality and nonlocality (i.e., Bell's theorem) and the subtle interplay between them. He does this by presenting the reader with a number of magic tricks, embedded in a triangular or pentagrammic framework, and showing how an analysis of their results leads to all the conclusions. I can best convey the flavor of Bub's demonstrations by saying that they are a bit like the "magic eye" pictures (or random dot stereograms) you may see in your Sunday newspaper: exactly as you might have to stare at your paper for a while before the three-dimensional image floats into view, so might you have to stare at Bub's diagrams a bit before the patterns hidden in them become clear. But the patterns you see here are more remarkable than any you will see in your Sunday newspaper, because they lay to rest the possibility of a finer and more definite reality underlying the quantum theory.

How do physicists react to all the quantum weirdness they are awash in (forgetting the superquantum madness of the PR box, which many hope will simply be squeezed out of existence)? They have come up with their own interpretations of what nature is really like. Except that when some say “really,” others say “not really” and come up with alternative interpretations. So physicists are divided into camps—Bohmians, Everettians, Qubists, and many others—that coexist somewhat warily with each other. Bub ventures into this territory by providing overviews of some of the more popular interpretations, while also pointing out their limitations. His purpose here is not to be exhaustive but rather to convey a feeling of what the interpretations can and cannot do. In one notable passage, he shows how the Bohmian viewpoint can be used to account perfectly for the action of a PR box; the trick is to use the “hidden variables” of the box, which are always present in the Bohmian scheme, to signal instantaneously between its two parts. This explanation shows how the spookiness that is woven naturally into the fabric of the Bohmian scheme can be used to account for the superquantum weirdness of the PR box. The book abounds in nuggets of insight like this.

Bub uses a number of props to help his readers navigate their way through the book. Each chapter has an appendix entitled “More,” where elaborations and extensions of the material in the main text can be found. A mathematical supplement provides the background for some of the more demanding calculations in the book. The end of each section contains a summary, in the form of bullet points, of all the important results obtained in it; readers will find these particularly helpful in refreshing their memories when they are further along in the book. Finally, professionals will find a wealth of interesting commentary tucked away in the footnotes at the end of the chapters, as well as a large number of references to the original literature (some of it quite recent). And, oh yes, there are the bananas too. But I'll let you discover why they are there when you take your first bite out of the book.

In addition to discussing foundational questions and their philosophical implications, the book also talks about some of their surprising modern day applications (such as quantum cryptography and teleportation). A book of this length obviously cannot discuss everything, but Bub has made a wise choice of topics that reflects itself in the thematic unity and coherence of his presentation. Above all, he has taken great pains to ensure that his treatment of the topics makes them accessible to as broad an audience as possible. It is therefore my hope that this book finds the wide readership it deserves.

*P. K. Aravind is a Professor of Physics at Worcester Polytechnic Institute. He was worked on the Bell and Kochen-Specker theorems and problems in quantum information theory.*