Foundations o f Physics Letters, Vol. 2, No. 5, 1989
Book Review
A n I n t r o d u c t i o n to H i l b e r t S p a c e a n d Q u a n t u m Logic. By David W. Cohen. Springer-Verlag, New York, 1989, XII + 149pp., $39.80 (hardcover). The quantum logic approach to quantum physics was initiated by G. Birkhoff and J. yon Neumann ["The logic of quantum mechanics," Ann. Math. 37, 823-843 (1936)]. Their idea was to develop an axiomatic basis for the foundations of quantum mechanics. They attempted to present only physically motivated axioms which ideally could be subjected to laboratory tests. Their goal was to eventually derive the Hilbert-space model, which was (and still is) the dominant framework for quantum mechanical studies. Their efforts were only a partial success, and since that time hundreds of articles and many books have been written on the subject. Quantum logic has been studied by a host of investigators across a number of disciplines including physics, mathematics, philosophy, physical chemistry, electrical engineering, and computer science. Some critics claim that we have reached Birkhoff and von Neumann's goal. Others claim that the goal is attainable but has not yet been reached. Still others suggest that the goal is unattainable; in fact, the Hilbert-space model is not the correct one for quantum mechanics. In the past 25 years, at least 20 books have been written on quantum logic. The first of these was by G. Mackey [The Mathematical Foundations of Quantum Mechanics (Benjamin, New York, 1963)] while the latest was by I. Pitowski [Quantum Probability-Quantum Logic ISpringer-Verlag, Berlin, 1989)]. Other quantum-logic book writers incmde V. Varadarajan, J. Jauch, C. Piton, M. Jammer, E. Beltrametti and G. Cassinelli, and the reviewer. If there is nothing new to say, why write another book on this subject? The book under review has nothing really new to say, but there was a reason to write it. All of these previous books were written for the mathematically or physically sophisticated reader. In short, they were intended for the expert. The present book is quite different. First, the presentation is at the undergraduate level. The only prerequisites are an undergraduate course in linear algebra and one in mathematical analysis or advanced calculus. No background in physics is assumed. Second, the material is presented using a tutorial style. Most of the proofs in the main text are omitted, and many of the examples are in 503
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Book Review
the form of projects for the reader to complete. The coaching manual, which follows the main text, contains hints for completing the easier projects and proofs, while entire solutions are given for the others. The main reason for writing this book is to bridge the growing gap between mathematics and physics and to bridge this gap at an early stage of the student's development. The book explores quantum physics in three stages. The assumptions and equations of orthodox quantum mechanics are developed. The model based on the mathematics of Hilbert space is presented. Finally, the previous stages are abstracted to the quantumlogic approach. This approach is derived from the more basic framework of operational statistics due to C. Randall and D. Foulis ["The operational approach to quantum mechanics," in Physical Theory as Logico-Operational Structure, C. Hooker, ed. (Reidel, Dordrecht, 1978)]. One of the highlights of the book is the last chapter, which contains an interesting discussion of the EPR problem. This book can be used either as a course text or for self-study. It would be a good first exposure for students interested in the foundations of quantum mechanics, and even the expert can find some unexpected insights. The reviewer's only minor criticism is that the bibliography is not very comprehensive, thus making it less useful for further study. Stanley P. Gudder
Department of Mathematics and Computer Science University of Denver Denver, Colorado 80208
