Geometry of Quantum Theory

This is a book about the mathematical foundations of quantum theory. Its aim is to develop the conceptual basis of modern quantum theory from general principles using the resources and techniques of modern mathematics.

There are two aspects to this book. The first deals with the geometrical structure of the logic of quantum mechanics, which is very different from that of classical mechanics, and which has as its source the complementarity principle that goes back to Bohr and Heisenberg. The treatment builds on the approach of Von Neumann. The second aspect is that of symmetry which plays a much greater role in quantum mechanics than in classical mechanics. Symmetry in quantum theory manifests itself through unitary representations of the symmetry groups, and the book presents a systematic treatment which is adequate for all applications.

All the fundamental results are proved here: Wigner’s theorem on quantum symmetries, the equivalence of wave and matrix mechanics, namely, the identity of the Schrodinger and the Heisenberg–Dirac pictures of the quantum world, impossibility of a hidden variables explanation of quantum theory (Von Neumann–Mackey–Gleason theorem), classification of free relativistic particles through their mass and spin, the structure of quantum systems of many identical particles (bosons and fermions), and so on.

The reader of this book will be thoroughly prepared for studying modern theories such as gauge theories, quantum field theory, and super symmetry.