Mathematical Foundations of Quantum Theory

in "Foundations 21. Nagel, R. J., "Order unit and base norm spaces, of Quantum Mechanics and Ordered Linear Spaces" (A. Hartkämpfer and H. Neumann, Eds.) pp. 23–29. Lecture Notes in Physics 29, Springer Verlag, New York - Heidelberg, ...

Author: A.R. Marlow

Publisher: Elsevier

ISBN: 9780323141185

Category: Science

Page: 382

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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.
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An Axiomatic Basis for Quantum Mechanics

A. Hartkämper, H. Neumann: Foundations of Quantum Mechanics and Ordered Linear Spaces, Lecture Notes, Vol. 29 (Springer, Berlin, Heidelberg, New York 1974) . A. Kolmogoroff. Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer, ...

Author: G. Ludwig

Publisher: Springer Science & Business Media

ISBN: 9783642700293

Category: Science

Page: 246

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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.
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An Axiomatic Basis for Quantum Mechanics

A Hartkämper, H. Neumann: Foundations of Quantum Mechanics and Ordered Linear Spaces, Lecture Notes, Vol. 29 (Springer, Berlin, Heidelberg, New York 1974) . A. Kolmogoroff: Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer, ...

Author: Günther Ludwig

Publisher: Springer Science & Business Media

ISBN: 9783642718977

Category: Science

Page: 244

View: 861

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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.
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Quantum Theory Informational Foundations and Foils

H. Barnum, A. Wilce, Ordered linear spaces and categories as frameworks for informationprocessing characterizations of quantum theory (2009). arXiv:0908.2354 18. H. Barnum, R. Duncan, A. Wilce, Symmetry, compact closure, ...

Author: Giulio Chiribella

Publisher: Springer

ISBN: 9789401773034

Category: Science

Page: 528

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This book provides the first unified overview of the burgeoning research area at the interface between Quantum Foundations and Quantum Information. Topics include: operational alternatives to quantum theory, information-theoretic reconstructions of the quantum formalism, mathematical frameworks for operational theories, and device-independent features of the set of quantum correlations. Powered by the injection of fresh ideas from the field of Quantum Information and Computation, the foundations of Quantum Mechanics are in the midst of a renaissance. The last two decades have seen an explosion of new results and research directions, attracting broad interest in the scientific community. The variety and number of different approaches, however, makes it challenging for a newcomer to obtain a big picture of the field and of its high-level goals. Here, fourteen original contributions from leading experts in the field cover some of the most promising research directions that have emerged in the new wave of quantum foundations. The book is directed at researchers in physics, computer science, and mathematics and would be appropriate as the basis of a graduate course in Quantum Foundations.
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Foundations of Quantum Mechanics

In: Foundation of Quantum Mechanics and Ordered Linear Spaces. Springer Lecture Notes in Physics, Vol. 29, 1974. [37] G. Ludwig, Quantum theory as a theory of interactions between macroscopic systems which can be described objectively.

Author: Günther Ludwig

Publisher: Springer Science & Business Media

ISBN: 9783662287262

Category: Science

Page: 416

View: 786

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In this second volume on the Foundations of Quantum Mechanics we shall show how it is possible, using the methodology presented in Volume I, to deduce some of the most important applications of quantum mechanics. These deductions are concerned with the structures of the microsystems rather than the technical details of the construction of preparation and registration devices. Accordingly. the only new axioms (relative to Volume I) which are introduced are concerned with the relationship between ensemble operators W, effect operators F, and certain construction principles of the preparation and registration devices. The applications described here are concerned with the measurement of atomic and molecular structure and of collision experiments. An additional and essential step towards a theoretical description of the preparation and registration procedures is carried out in Chapter XVII. Here we demonstrate how microscopic collision processes (that is, processes which can be described by quantum mechanics) can be used to obtain novel preparation and registration procedures if we take for granted the knowledge of only a few macroscopic preparation and registration procedures. By clever use of collision processes we are often able to obtain very precise results for the operators Wand F which describe the total procedures from a very imprecise knowledge of the macroscopic parts of the preparation and regis tration processes. In this regard experimental physicists have done brilliant work. In this sense Chapter XVII represents a general theoretical foundation for the procedures used by experimental physicists.
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