Synopses & Reviews
This volume is the result of extensive research into the foundations of quantum mechanics, and presents a new formulation of quantum theory which resolves many existing problems. The formalism is experimentally motivated and is shown to be consistent with recent foundational approaches to quantum mechanics. It is based on new extensions of the theory of group representations, which are developed and illustrated in the text. This new approach establishes connections with quantum logic, philosophy, and the history of science, allowing a broad range of applications not only in physics, but also in such areas as signal processing and neuroscience. Audience: This book will be of interest to researchers and students. It is recommended as a supplementary textbook for advanced courses in quantum mechanics. A background in quantum mechanics and complex analysis is assumed.
Synopsis
In this monograph, we shall present a new mathematical formulation of quantum theory, clarify a number of discrepancies within the prior formulation of quantum theory, give new applications to experiments in physics, and extend the realm of application of quantum theory well beyond physics. Here, we motivate this new formulation and sketch how it developed. Since the publication of Dirac's famous book on quantum mechanics Dirac, 1930] and von Neumann's classic text on the mathematical foundations of quantum mechanics two years later von Neumann, 1932], there have appeared a number of lines of development, the intent of each being to enrich quantum theory by extra- polating or even modifying the original basic structure. These lines of development have seemed to go in different directions, the major directions of which are identified here: First is the introduction of group theoretical methods Weyl, 1928; Wigner, 1931] with the natural extension to coherent state theory Klauder and Sudarshan, 1968; Peremolov, 1971]. The call for an axiomatic approach to physics Hilbert, 1900; Sixth Problem] led to the development of quantum logic Mackey, 1963; Jauch, 1968; Varadarajan, 1968, 1970; Piron, 1976; Beltrametti & Cassinelli, 1981], to the creation of the operational approach Ludwig, 1983-85, 1985; Davies, 1976] with its application to quantum communication theory Helstrom, 1976; Holevo, 1982), and to the development of the C* approach Emch, 1972]. An approach through stochastic differential equations ("stochastic mechanics") was developed Nelson, 1964, 1966, 1967].
Table of Contents
Preface.
I: Basic quantum theory and the necessity for its revision. I.1. Classical mechanics of particles and fluids.
I.2. Structure of a physical model: state, property (observable), measurement.
I.3. Quantum mechanics of a (non-relativistic spinless) particle.
I.4. On the connection between classical mechanics and quantum mechanics.
I.5. Mathematical appendix.
II: Basic experiments suggest generalizing quantum mechanics. II.1. Quantum mechanical descriptions of an experiment.
II.2. Capture on a screen, in a bubble chamber, gel, cloud chamber.
II.3. The Stern-Gerlach experiment.
II.4. Crossed polarizers.
II.5. Single slit experiments and inapplicability of Heisenberg uncertainty relations.
II.6. Spontaneous decay, Breit-Wigner (Cauchy) distributions, and the inapplicability of Heisenberg type uncertainty relations.
II.7. Interferometers.
II.8. Imaging processes and signal analysis.
II.9. Sensory perception and neuroscience.
II.10. Five other subjects and their implications.
II.11. Mathematical appendix.
III: Construction of quantum mechanics on phase space. III.1. Group representation theory.
III.2. The Heisenberg group (Weyl algebra) and the Affine group.
III.3. Representations of the Galilei group.
III.4. Representations of the Poincaré group.
III.5. Remarks on the de Sitter group.
IV: Consequences of formulating quantum mechanics on phase space. IV.1. The quantum/classical connection.
IV.2. Quantum field theory.
IV.3. Spring cleaning in the house of quantum mechanics.
IV.4. Reprise: Expanding the realm of application of quantum mechanics.
IV.5. A discrete (lattice) quantum universe, and computability.
V: Foundational aspects. V.1. Relation to generalized quantum logic.
V.2. P.O.V.M.'s arising on operational manuals.
V.3. Relation to quantum mechanical measurement theory.
V.4. Philosophical and other foundational aspects. References. Index.