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Quantum Functional Analysis
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Book Synopsis The Functional Analysis of Quantum Information Theory by : Ved Prakash Gupta
Download or read book The Functional Analysis of Quantum Information Theory written by Ved Prakash Gupta and published by Springer. This book was released on 2015-05-28 with total page 139 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides readers with a concise introduction to current studies on operator-algebras and their generalizations, operator spaces and operator systems, with a special focus on their application in quantum information science. This basic framework for the mathematical formulation of quantum information can be traced back to the mathematical work of John von Neumann, one of the pioneers of operator algebras, which forms the underpinning of most current mathematical treatments of the quantum theory, besides being one of the most dynamic areas of twentieth century functional analysis. Today, von Neumann’s foresight finds expression in the rapidly growing field of quantum information theory. These notes gather the content of lectures given by a very distinguished group of mathematicians and quantum information theorists, held at the IMSc in Chennai some years ago, and great care has been taken to present the material as a primer on the subject matter. Starting from the basic definitions of operator spaces and operator systems, this text proceeds to discuss several important theorems including Stinespring’s dilation theorem for completely positive maps and Kirchberg’s theorem on tensor products of C*-algebras. It also takes a closer look at the abstract characterization of operator systems and, motivated by the requirements of different tensor products in quantum information theory, the theory of tensor products in operator systems is discussed in detail. On the quantum information side, the book offers a rigorous treatment of quantifying entanglement in bipartite quantum systems, and moves on to review four different areas in which ideas from the theory of operator systems and operator algebras play a natural role: the issue of zero-error communication over quantum channels, the strong subadditivity property of quantum entropy, the different norms on quantum states and the corresponding induced norms on quantum channels, and, lastly, the applications of matrix-valued random variables in the quantum information setting.
Book Synopsis Quantum Theory for Mathematicians by : Brian C. Hall
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 554 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.
Book Synopsis Methods of Modern Mathematical Physics: Functional analysis by : Michael Reed
Download or read book Methods of Modern Mathematical Physics: Functional analysis written by Michael Reed and published by Gulf Professional Publishing. This book was released on 1980 with total page 417 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modern mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained, although there are occasional references to later volumes. We have included a few applications when we thought that they would provide motivation for the reader. Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, modern physics, and partial differential equations." --Publisher description.
Book Synopsis Quantum Functional Analysis by : Aleksandr I︠A︡kovlevich Khelemskiĭ
Download or read book Quantum Functional Analysis written by Aleksandr I︠A︡kovlevich Khelemskiĭ and published by American Mathematical Soc.. This book was released on 2010 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interpreting ""quantized coefficients"" as finite rank operators in a fixed Hilbert space allows the author to replace matrix computations with algebraic techniques of module theory and tensor products, thus achieving a more invariant approach to the subject.
Book Synopsis Introductory Functional Analysis with Applications by : Erwin Kreyszig
Download or read book Introductory Functional Analysis with Applications written by Erwin Kreyszig and published by John Wiley & Sons. This book was released on 1991-01-16 with total page 706 pages. Available in PDF, EPUB and Kindle. Book excerpt: KREYSZIG The Wiley Classics Library consists of selected books originally published by John Wiley & Sons that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Series: Emil Artin Geometnc Algebra R. W. Carter Simple Groups Of Lie Type Richard Courant Differential and Integrai Calculus. Volume I Richard Courant Differential and Integral Calculus. Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics. Volume II Harold M. S. Coxeter Introduction to Modern Geometry. Second Edition Charles W. Curtis, Irving Reiner Representation Theory of Finite Groups and Associative Algebras Nelson Dunford, Jacob T. Schwartz unear Operators. Part One. General Theory Nelson Dunford. Jacob T. Schwartz Linear Operators, Part Two. Spectral Theory—Self Adjant Operators in Hilbert Space Nelson Dunford, Jacob T. Schwartz Linear Operators. Part Three. Spectral Operators Peter Henrici Applied and Computational Complex Analysis. Volume I—Power Senes-lntegrauon-Contormal Mapping-Locatvon of Zeros Peter Hilton, Yet-Chiang Wu A Course in Modern Algebra Harry Hochstadt Integral Equations Erwin Kreyszig Introductory Functional Analysis with Applications P. M. Prenter Splines and Variational Methods C. L. Siegel Topics in Complex Function Theory. Volume I —Elliptic Functions and Uniformizatton Theory C. L. Siegel Topics in Complex Function Theory. Volume II —Automorphic and Abelian Integrals C. L. Siegel Topics In Complex Function Theory. Volume III —Abelian Functions & Modular Functions of Several Variables J. J. Stoker Differential Geometry
Book Synopsis Linear Functional Analysis by : Bryan Rynne
Download or read book Linear Functional Analysis written by Bryan Rynne and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to the ideas and methods of linear func tional analysis at a level appropriate to the final year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the the ory of metric spaces). Part of the development of functional analysis can be traced to attempts to find a suitable framework in which to discuss differential and integral equa tions. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions defined on some set. In general, such a vector space is infinite-dimensional. This leads to difficulties in that, although many of the elementary properties of finite-dimensional vector spaces hold in infinite dimensional vector spaces, many others do not. For example, in general infinite dimensional vector spaces there is no framework in which to make sense of an alytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now briefly outline the contents of the book.
Book Synopsis Functional Analysis in Mechanics by : Leonid P. Lebedev
Download or read book Functional Analysis in Mechanics written by Leonid P. Lebedev and published by Springer Science & Business Media. This book was released on 2006-04-29 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers functional analysis and its applications to continuum mechanics.; The mathematical material is treated in a non-abstract manner and is fully illuminated by the underlying mechanical ideas.; The presentation is concise but complete, and is intended for specialists in continuum mechanics who wish to understand the mathematical underpinnings of the discipline.; Graduate students and researchers in mathematics, physics, and engineering will find this book useful.; Exercises and examples are included throughout with detailed solutions provided in the appendix.
Book Synopsis Quantum Mechanics for Mathematicians by : Leon Armenovich Takhtadzhi͡an
Download or read book Quantum Mechanics for Mathematicians written by Leon Armenovich Takhtadzhi͡an and published by American Mathematical Soc.. This book was released on 2008 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a comprehensive treatment of quantum mechanics from a mathematics perspective. Including traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin.
Book Synopsis Functional Integration and Quantum Physics by : Barry Simon
Download or read book Functional Integration and Quantum Physics written by Barry Simon and published by American Mathematical Soc.. This book was released on 2005 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focuses on probabilistic foundations of the Feynman-Kac formula. Starting with main examples of Gaussian processes (the Brownian motion, the oscillatory process, and the Brownian bridge), this book presents four different proofs of the Feynman-Kac formula.
Book Synopsis Spectral Theory and Quantum Mechanics by : Valter Moretti
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 950 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."
Book Synopsis Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory by : Guillaume Aubrun
Download or read book Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory written by Guillaume Aubrun and published by American Mathematical Soc.. This book was released on 2017-08-30 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions. Alice and Bob Meet Banach is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information researchers who want to learn AGA or apply its tools; at mathematicians interested in learning QIT, or at least the part of QIT that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. Moreover, this user-friendly book contains numerous tables and explicit estimates, with reasonable constants when possible, which make it a useful reference even for established mathematicians generally familiar with the subject.
Book Synopsis Advanced Quantum Mechanics by : Franz Schwabl
Download or read book Advanced Quantum Mechanics written by Franz Schwabl and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers advanced topics in quantum mechanics, including nonrelativistic multi-particle systems, relativistic wave equations, and relativistic fields. Numerous examples for application help readers gain a thorough understanding of the subject. The presentation of relativistic wave equations and their symmetries, and the fundamentals of quantum field theory lay the foundations for advanced studies in solid-state physics, nuclear, and elementary particle physics. The authors earlier book, Quantum Mechanics, was praised for its unsurpassed clarity.
Book Synopsis Introductory Functional Analysis by : B.D. Reddy
Download or read book Introductory Functional Analysis written by B.D. Reddy and published by Springer Science & Business Media. This book was released on 2013-11-27 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: Providing an introduction to functional analysis, this text treats in detail its application to boundary-value problems and finite elements, and is distinguished by the fact that abstract concepts are motivated and illustrated wherever possible. It is intended for use by senior undergraduates and graduates in mathematics, the physical sciences and engineering, who may not have been exposed to the conventional prerequisites for a course in functional analysis, such as real analysis. Mature researchers wishing to learn the basic ideas of functional analysis will equally find this useful. Offers a good grounding in those aspects of functional analysis which are most relevant to a proper understanding and appreciation of the mathematical aspects of boundary-value problems and the finite element method.
Book Synopsis Real and Functional Analysis by : Serge Lang
Download or read book Real and Functional Analysis written by Serge Lang and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 591 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is meant as a text for a first-year graduate course in analysis. In a sense, it covers the same topics as elementary calculus but treats them in a manner suitable for people who will be using it in further mathematical investigations. The organization avoids long chains of logical interdependence, so that chapters are mostly independent. This allows a course to omit material from some chapters without compromising the exposition of material from later chapters.
Book Synopsis Mathematical Methods in Quantum Mechanics by : Gerald Teschl
Download or read book Mathematical Methods in Quantum Mechanics written by Gerald Teschl and published by American Mathematical Soc.. This book was released on 2009 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. It is well suited for self-study and includes numerous exercises (many with hints).
Book Synopsis Mathematical Quantization by : Nik Weaver
Download or read book Mathematical Quantization written by Nik Weaver and published by CRC Press. This book was released on 2001-05-31 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: With a unique approach and presenting an array of new and intriguing topics, Mathematical Quantization offers a survey of operator algebras and related structures from the point of view that these objects are quantizations of classical mathematical structures. This approach makes possible, with minimal mathematical detail, a unified treatment of a variety of topics. Detailed here for the first time, the fundamental idea of mathematical quantization is that sets are replaced by Hilbert spaces. Building on this idea, and most importantly on the fact that scalar-valued functions on a set correspond to operators on a Hilbert space, one can determine quantum analogs of a variety of classical structures. In particular, because topologies and measure classes on a set can be treated in terms of scalar-valued functions, we can transfer these constructions to the quantum realm, giving rise to C*- and von Neumann algebras. In the first half of the book, the author quickly builds the operator algebra setting. He uses this as a unifying theme in the second half, in which he treats several active research topics, some for the first time in book form. These include the quantum plane and tori, operator spaces, Hilbert modules, Lipschitz algebras, and quantum groups. For graduate students, Mathematical Quantization offers an ideal introduction to a research area of great current interest. For professionals in operator algebras and functional analysis, it provides a readable tour of the current state of the field.
Book Synopsis Quantum Functional Analysis by : Aleksandr I︠A︡kovlevich Khelemskiĭ
Download or read book Quantum Functional Analysis written by Aleksandr I︠A︡kovlevich Khelemskiĭ and published by American Mathematical Soc.. This book was released on 2010 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: This systematic presentation of quantum functional analysis takes a significantly different approach from that of other texts by interpreting quantized coefficients as finite rank operators in a fixed Hilbert space. This approach allows the author to achieve a more invariant approach to the subject. Graduates and researchers interested in functional analysis and related areas of mathematics and mathematical physics will benefit from this presentation.
