A revolutionary book that for the first time provided a rigorous mathematical framework for quantum mechanics. -- Google books

This graduate-level text introduces fundamentals of classical mechanics; surveys basics of quantum mechanics; and concludes with a look at group theory and quantum mechanics of the atom. 1963 edition.

This is a brief introduction to the mathematical foundations of quantum mechanics based on lectures given by the author to Ph.D.students at the Delhi Centre of the Indian Statistical Institute in order to initiate active research in the emerging field of quantum probability. The material in the first chapter is included in the author's book "An Introduction to Quantum Stochastic Calculus" published by Birkhauser Verlag in 1992 and the permission of the publishers to reprint it here is acknowledged. Apart from quantum probability, an understanding of the role of group representations in the development of quantum mechanics is always a fascinating theme for mathematicians. The first chapter deals with the definitions of states, observables and automorphisms of a quantum system through Gleason's theorem, Hahn-Hellinger theorem and Wigner's theorem. Mackey's imprimitivity theorem and the theorem of inducing representations of groups in stages are proved directly for projective unitary antiunitary representations in the second chapter. Based on a discussion of multipliers on locally compact groups in the third chapter all the well-known observables of classical quantum theory like linear momenta, orbital and spin angular momenta, kinetic and potential energies, gauge operators etc., are derived solely from Galilean covariance in the last chapter. A very short account of observables concerning a relativistic free particle is included. In conclusion, the spectral theory of Schrodinger operators of one and two electron atoms is discussed in some detail.

Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics--a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. In this new edition of this classic work, mathematical physicist Nicholas Wheeler has completely reset the book in TeX, making the text and equations far easier to read. He has also corrected a handful of typographic errors, revised some sentences for clarity and readability, provided an index for the first time, and added prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson. The result brings new life to an essential work in theoretical physics and mathematics.

Mathematical Foundations of Quantum Theory is a collection of papers presented at the 1977 conference on the Mathematical Foundations of Quantum Theory, held in New Orleans. The contributors present their topics from a wide variety of backgrounds and specialization, but all shared a common interest in answering quantum issues. Organized into 20 chapters, this book's opening chapters establish a sound mathematical basis for quantum theory and a mode of observation in the double slit experiment. This book then describes the Lorentz particle system and other mathematical structures with which fundamental quantum theory must deal, and then some unsolved problems in the quantum logic approach to the foundations of quantum mechanics are considered. Considerable chapters cover topics on manuals and logics for quantum mechanics. This book also examines the problems in quantum logic, and then presents examples of their interpretation and relevance to nonclassical logic and statistics. The accommodation of conventional Fermi-Dirac and Bose-Einstein statistics in quantum mechanics or quantum field theory is illustrated. The final chapters of the book present a system of axioms for nonrelativistic quantum mechanics, with particular emphasis on the role of density operators as states. Specific connections of this theory with other formulations of quantum theory are also considered. These chapters also deal with the determination of the state of an elementary quantum mechanical system by the associated position and momentum distribution. This book is of value to physicists, mathematicians, and researchers who are interested in quantum theory.

This book is the first volume of a two-volume work on the Foundations of Quantum Mechanics, and is intended as a new edition of the author's book Die Grundlagen der Quantenmechanik [37] which was published in 1954. In this two-volume work we will seek to obtain an improved formulation of the interpretation of quantum mechanics based on experiments. The second volume will appear shortly. Since the publication of [37] there have been several attempts to develop a basis for quantum mechanics which is, in the large part, based upon the work of J. von Neumann [38]. In particular, we mention the books ofG. W. Mackey [39], J. Jauch [40], C. Piron [41], M. Drieschner [9], and the original work ofS. P. Gudder [42], D.J. Foulis and C.H. Randall [43], and N. Zierler [44]. Here we do not seek to compare these different formulations of the foundations of quantum mechanics. We refer interested readers to [45] for such comparisons.

Arising out of the need for Quantum Mechanics (QM) to be part of the common education of mathematics students, this book formulates the mathematical structure of QM in terms of the C*-algebra of observables, which is argued on the basis of the operational definition of measurements and the duality between states and observables.

This monograph is devoted to quantum statistical mechanics. It can be regarded as a continuation of the book "Mathematical Foundations of Classical Statistical Mechanics. Continuous Systems" (Gordon & Breach SP, 1989) written together with my colleagues V. I. Gerasimenko and P. V. Malyshev. Taken together, these books give a complete pre sentation of the statistical mechanics of continuous systems, both quantum and classical, from the common point of view. Both books have similar contents. They deal with the investigation of states of in finite systems, which are described by infinite sequences of statistical operators (reduced density matrices) or Green's functions in the quantum case and by infinite sequences of distribution functions in the classical case. The equations of state and their solutions are the main object of investigation in these books. For infinite systems, the solutions of the equations of state are constructed by using the thermodynamic limit procedure, accord ing to which we first find a solution for a system of finitely many particles and then let the number of particles and the volume of a region tend to infinity keeping the density of particles constant. However, the style of presentation in these books is quite different.

In the first volume we based quantum mechanics on the objective description of macroscopic devices. The further development of the quantum mechanics of atoms, molecules, and collision processes has been described in [2]. In this context also the usual description of composite systems by tensor products of Hilbert spaces has been introduced. This method can be formally extrapolated to systems composed of "many" ele mentary systems, even arbitrarily many. One formerly had the opinion that this "extrapolated quantum mechanics" is a more comprehensive theory than the objec tive description of macrosystems, an opinion which generated unsurmountable diffi culties for explaining the measuring process. With respect to our foundation of quan tum mechanics on macroscopic objectivity, this opinion would mean that our founda tion is no foundation at all. The task of this second volume is to attain a compatibility between the objective description of macrosystems and an extrapolated quantum mechanics. Thus in X we establish the "statistical mechanics" of macrosystems as a theory more compre hensive than an extrapolated quantum mechanics. On this basis we solve the problem of the measuring process in quantum mechan ics, in XI developing a theory which describes the measuring process as an interaction between microsystems and a macroscopic device. This theory also allows to calculate "in principle" the observable measured by a device. Neither an incorporation of consciousness nor a mysterious imagination such as "collapsing" wave packets are necessary.

This book is the first volume of a two-volume work, which is an improved version of a preprint [47] published in German. We seek to deduce the funda mental concepts of quantum mechanics solely from a description of macroscopic devices. The microscopic systems such as electrons, atoms, etc. must be detected on the basis of the macroscopic behavior of the devices. This detection resembles the detection of the dinosaurs on the basis offossils. In this first volume we develop a general description of macroscopic systems by trajectories in state spaces. This general description is a basis for the special de scription of devices consisting of two parts, where the first part is acting on the second. The microsystems are discovered as systems transmitting the action. Axioms which describe general empirical structures of the interactions between the two parts of each device, give rise to a derivation of the Hilbert space structure of quantum mechanics. Possibly, these axioms (and consequently the Hilbert space structure) may fail to describe other realms than the structure of atoms and mole cules, for instance the "elementary particles". This book supplements ref. [2]. Both together not only give an extensive foundation of quantum mechanics but also a solution in principle of the measuring problem.