Geometry of Quantum Theory (eBook)

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.

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It was about four years ago that Springer-Verlag suggested that a revised edition in a single volume of my two-volume work may be worthwhile. I agreed enthusiastically but the project was delayed for many reasons, one of the most important of which was that I did not have at that time any clear idea as to how the revision was to be carried out. Eventually I decided to leave intact most ofthe original material, but make the current edition a little more up-to-date by adding, in the form of notes to the individual chapters, some recent references and occasional brief discussions of topics not treated in the original text. The only substantive change from the earlier work is in the treatment of projective geometry; Chapters II through V of the original Volume I have been condensed and streamlined into a single Chapter II. I wish to express my deep gratitude to Donald Babbitt for his generous advice that helped me in organizing this revision, and to Springer-Verlag for their patience and understanding that went beyond what one has a right to expect from a publisher. I suppose an author's feelings are always mixed when one of his books that is comparatively old is brought out once again.