Quantum theory arose in the late 19th and early 20th century as a result of experimental results that were paradoxical in light of classical physics. Classical models predicted that a hot body (such as a star) should emit infinite energy, contrary to observation, and the emission of electrons from an illuminated metal surface was found to depend not on the intensity of the illumination (as predicted by classical models), but on its frequency. Quantum mechanics was developed to explain these and similar paradoxes.

In recent years, cognitive psychology has discovered its own set of paradoxes, driven largely by the work of Kahneman and Tversky and embodied in the heuristics and biases school of research. Contrary to classical probabilistic analyses of cognition and decision making, extensive experimental evidence shows that people often assign a higher probability to conjoined statements than to either portion alone. We also vary the probability assigned to a joint state depending on the order in which we consider the components; ignore the impact of base rates and sample sizes; and are heavily biased in our decisions by prior information, among other anomalous effects. These empirical results appear to invalidate the most natural probabilistic models of cognition, and so have created a theoretical vacuum for cognitive psychologists.

This book suggests that quantum theory, which answered the paradoxes of modern physics, also holds the key to understanding the paradoxes of modern experimental psychology. Quantum theory is often characterized in terms of the dual wave-particle nature of subatomic particles, and some researchers speculate that subatomic dynamics in the brain give rise to distinctive cognitive phenomena. However, the authors are agnostic on this point. They focus rather on the mathematical framework of quantum mechanics, Hilbert spaces (abstract vector spaces supporting an inner product). In quantum mechanics, a state is a vector, the probability of a state is the squared length of a vector, superposition effects result from skewing between different sets of basis vectors spanning the same space, and dynamics is defined in terms of unitary transformations of state vectors. The objective here is to show that, unlike classical probabilities based on sets, probabilities defined geometrically using Hilbert spaces are consistent with the modern paradoxes of experimental psychology. For the past ten years, a growing community of cognitive psychologists has been exploring this thesis, and this book presents this body of work in a coherent and unified exposition. In brief, it offers a mathematical theory to explain the observations of the heuristics and biases school.

Chapter 1 motivates the book by pointing out classic problems in cognition and decision making. Chapter 2 provides a brief introduction to quantum theory, contrasting it point by point with classical probability. Chapters 3 through 7 use the theory to explain some of the puzzling results of cognitive experiments, including question order effects, errors in probability judgment, and reasoning about combinations of concepts and the implications for conjoint memory recognition; a general exposition of the human semantic space in terms of quantum theory is also provided. Chapter 8 offers more details on how cognitive dynamics can be described in quantum terms, and chapters 9 through 11 apply these dynamics to decision making, offering a quantum view of human information processing and learning. Chapter 12 reviews the contributions of the quantum approach, surveys current and ongoing work in the field, and addresses a number of objections that the authors have encountered from reviewers.

The title grabs the reader’s attention, but might mislead some into thinking that it advocates the notion of the quantum brain. A more accurate title, but much less exciting, would be *Hilbert space models of cognition and decision*. The authors conclude their preface, “People either love or hate these ideas, but no one remains unaffected.” They have done a good job of motivating the reader to love them. Their proposals are indeed revolutionary, but not unreasonable, and they fill a critical gap by providing an expressive and mathematically rigorous formalism that is consistent with the empirical observations of modern cognitive psychology. The presentation is clear and self-contained, requiring no prior knowledge of quantum mechanics on the part of the reader. Though numerous papers by the authors and others (extensively cited throughout the book) have developed this paradigm, this volume brings these diverse results together in an integrated view to demonstrate how to use Hilbert spaces to model cognition and decision making. Every serious researcher in cognitive science needs to engage with this volume.