NB. I was sent this book as a review copy.
I have to admit that I was rather embarrassed to encounter this book, as I had never heard of it, and given the topic, and the author, it seemed that it must be one of the canonical texts in the field. However, it turns out that although Von Neumann wrote this book in 1932 (full German text here), it was not translated until 1955 (by Robert Beyer), and this edition aged quickly, particularly with the limitations of typesetting the equations. It wasn’t until now that a modern edition has been put together, by Nicholas Wheeler, and the result is lovely.
The book is really a collection and expansion of Von Neumann’s previously published works, attempting to put quantum mechanics on a firm mathematical footing. The first chapter is dedicated to the equivalence of Matrix Quantum Mechanics, and Schrodinger’s Wave Mechanics. The reason that this was so hard to put in solid mathematical terms, was because it relied on the use of the Dirac Delta function, and derivatives thereof, which at the time were not seen as well-defined mathematical entities. It is strange now to think of a time where their use indicated that something shady might be going on in a derivation but Von Neumann was able to create a well-argued mathematical derivation of the equivalence of the two theories – something that is absolutely central to our use and understand of quantum mechanics now.
The next section deals with the mathematics of Hilbert spaces in the quantum mechanical context, and in particular how one defines infinite dimensional Hilbert spaces, and unbounded operators (like momentum, position and energy). The key ingredient was the extension of eigenvalues and eigenstates to spectral methods for operators.
He was also able to show that the commutation relations were enough to define the associated operators uniquely, up to a set of transformations. It is for this reason that, just from the canonical commutation relations and the use of the Schrodinger equation, so much can be derived for a quantum system.
After an extremely thorough investigation of these Hilbert Spaces including a complete treatment of invariants (like the trace) of operators, he goes on to look at the measurement problem, and the relationship between probability distributions and the wavefunction. In particular he introduces the density matrix, and thus the Von Neumann entropy, bridging the gap between quantum mechanics and information theory. This of course can then be extended to thermal ensembles, which he develops into the framework of quantum thermodynamics.
The final chapter is about the measurement problem in quantum mechanics, and in this chapter he investigates the ambiguity in the split between a quantum system and an observer and edges ever closer to a many worlds interpretation.
This book is absolutely not for the undergraduate level study of quantum mechanics, unless perhaps for students who are coming from a pure mathematics background and want to get a deeper appreciation for how such a theory can be formally and rigorously defined. However, for anyone interested in truly understanding many of the concepts and methods within quantum mechanics which we so often take for granted, this is an invaluable book.
The writing, in general, is relatively clear, and although the mathematics is certainly dense, it should be possible to follow for students at the graduate level and I would suggest that it would be a very useful exercise for anyone currently working in quantum information theory to dive into chapters of this book to get a real grasp of why we do what we do. This book will, I am sure, be one that I continue to dip into for years to come.