Fundamental Mathematical Structures Of Quantum Theory

Download or read book Fundamental Mathematical Structures of Quantum Theory written by Valter Moretti and published by Springer. This book was released on 2019-06-20 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook presents in a concise and self-contained way the advanced fundamental mathematical structures in quantum theory. It is based on lectures prepared for a 6 months course for MSc students. The reader is introduced to the beautiful interconnection between logic, lattice theory, general probability theory, and general spectral theory including the basic theory of von Neumann algebras and of the algebraic formulation, naturally arising in the study of the mathematical machinery of quantum theories. Some general results concerning hidden-variable interpretations of QM such as Gleason's and the Kochen-Specker theorems and the related notions of realism and non-contextuality are carefully discussed. This is done also in relation with the famous Bell (BCHSH) inequality concerning local causality. Written in a didactic style, this book includes many examples and solved exercises. The work is organized as follows. Chapter 1 reviews some elementary facts and properties of quantum systems. Chapter 2 and 3 present the main results of spectral analysis in complex Hilbert spaces. Chapter 4 introduces the point of view of the orthomodular lattices' theory. Quantum theory form this perspective turns out to the probability measure theory on the non-Boolean lattice of elementary observables and Gleason's theorem characterizes all these measures. Chapter 5 deals with some philosophical and interpretative aspects of quantum theory like hidden-variable formulations of QM. The Kochen-Specker theorem and its implications are analyzed also in relation BCHSH inequality, entanglement, realism, locality, and non-contextuality. Chapter 6 focuses on the algebra of observables also in the presence of superselection rules introducing the notion of von Neumann algebra. Chapter 7 offers the idea of (groups of) quantum symmetry, in particular, illustrated in terms of Wigner and Kadison theorems. Chapter 8 deals with the elementary ideas and results of the so called algebraic formulation of quantum theories in terms of both *-algebras and C*-algebras. This book should appeal to a dual readership: on one hand mathematicians that wish to acquire the tools that unlock the physical aspects of quantum theories; on the other physicists eager to solidify their understanding of the mathematical scaffolding of quantum theories.

Download or read book Spectral Theory and Quantum Mechanics written by Valter Moretti and published by Springer. This book was released on 2018-01-30 with total page 962 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses the mathematical foundations of quantum theories. It offers an introductory text on linear functional analysis with a focus on Hilbert spaces, highlighting the spectral theory features that are relevant in physics. After exploring physical phenomenology, it then turns its attention to the formal and logical aspects of the theory. Further, this Second Edition collects in one volume a number of useful rigorous results on the mathematical structure of quantum mechanics focusing in particular on von Neumann algebras, Superselection rules, the various notions of Quantum Symmetry and Symmetry Groups, and including a number of fundamental results on the algebraic formulation of quantum theories. Intended for Master's and PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book also benefits established researchers by organizing and presenting the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly."

Download or read book Quantum Theory for Mathematicians written by Brian C. Hall and published by Springer Science & Business Media. This book was released on 2013-06-19 with total page 566 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrödinger equation in one space dimension; the Spectral Theorem for bounded and unbounded self-adjoint operators; the Stone–von Neumann Theorem; the Wentzel–Kramers–Brillouin approximation; the role of Lie groups and Lie algebras in quantum mechanics; and the path-integral approach to quantum mechanics. The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces. The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.

Download or read book Mathematical Foundations of Quantum Mechanics written by John von Neumann and published by Princeton University Press. This book was released on 1955 with total page 462 pages. Available in PDF, EPUB and Kindle. Book excerpt: A revolutionary book that for the first time provided a rigorous mathematical framework for quantum mechanics. -- Google books

Download or read book Mathematical Concepts of Quantum Mechanics written by Stephen J. Gustafson and published by Springer Science & Business Media. This book was released on 2011-09-24 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline. Starting with an overview of key physical experiments illustrating the origin of the physical foundations, the book proceeds with a description of the basic notions of quantum mechanics and their mathematical content. It then makes its way to topics of current interest, specifically those in which mathematics plays an important role. The more advanced topics presented include many-body systems, modern perturbation theory, path integrals, the theory of resonances, quantum statistics, mean-field theory, second quantization, the theory of radiation (non-relativistic quantum electrodynamics), and the renormalization group. With different selections of chapters, the book can serve as a text for an introductory, intermediate, or advanced course in quantum mechanics. The last four chapters could also serve as an introductory course in quantum field theory.

Download or read book Mathematical Structures Of Quantum Mechanics written by Kow Lung Chang and published by World Scientific Publishing Company. This book was released on 2011-10-31 with total page 209 pages. Available in PDF, EPUB and Kindle. Book excerpt: This marvelous book is aimed at strengthening the mathematical background and sharpening the mathematical tools of students without rigorous training before taking the quantum mechanics course. The abstract construction of quantum postulates in the framework of Hilbert space and Hermitian operators are realized by q-representation in the formulation to demonstrate the conventional approach to quantum theory.Symmetry property is emphasized and extensively explored in this book both in continuous transformations as well as in the discrete ones. The space-time structure is discussed in depth and Dirac equation is formulated by symmetry consideration of Lorentz group.

Download or read book Quantum Theory, Groups and Representations written by Peter Woit and published by Springer. This book was released on 2017-11-01 with total page 659 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.