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Mathematical Aspects Of Weyl Quantization And Phase
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Book Synopsis Mathematical Aspects of Weyl Quantization and Phase by : D. A. Dubin
Download or read book Mathematical Aspects of Weyl Quantization and Phase written by D. A. Dubin and published by World Scientific. This book was released on 2000 with total page 562 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book analyzes in considerable generality the quantization-dequantization integral transform scheme of Weyl and Wigner, and considers several phase operator theories. It features: a thorough treatment of quantization in polar coordinates; dequantization by a new method of "motes"; a discussion of Moyal algebras; modifications of the transform method to accommodate operator orderings; a rigorous discussion of the Dieke laser model for one mode, fully quantum, in the thermodynamic limit; analysis of quantum phase theories based on the Toeplitz operator, the coherent state operator, the quantized phase space angle, and a sequence of finite rank operators.
Book Synopsis Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes by : De Gosson Maurice A
Download or read book Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes written by De Gosson Maurice A and published by World Scientific. This book was released on 2017-11-10 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: The emergence of quantum mechanics from classical world mechanics is now a well-established theme in mathematical physics. This book demonstrates that quantum mechanics can indeed be viewed as a refinement of Hamiltonian mechanics, and builds on the work of George Mackey in relation to their mathematical foundations. Additionally when looking at the differences with classical mechanics, quantum mechanics crucially depends on the value of Planck's constant h. Recent cosmological observations tend to indicate that not only the fine structure constant α but also h might have varied in both time and space since the Big Bang. We explore the mathematical and physical consequences of a variation of h; surprisingly we see that a decrease of h leads to transitions from the quantum to the classical. Emergence of the Quantum from the Classical provides help to undergraduate and graduate students of mathematics, physics and quantum theory looking to advance into research in the field. Contents: Hamiltonian MechanicsHamilton–Jacobi TheoryMatter Waves, Schrödinger's Equation, and Bohm's Theory The MetatronUncertainties and Quantum BlobsQuantum States and the Density MatrixVarying Planck's ConstantAppendices: The Symplectic GroupThe Metaplectic RepresentationBorn–Jordan QuantizationTwisted Product and Convolution Readership: Undergraduate and graduate students of mathematics and physics, interested in analysis and differential equations, probability and statistics, geometry and topology and mathematical physics. Keywords: Classical Mechanics;Hamiltonian Mechanics;Quantum Mechanics;Planck's Constant;Mathematical Physics;Algebraic GeometryReview:0
Book Synopsis Gauge Invariance and Weyl-polymer Quantization by : Franco Strocchi
Download or read book Gauge Invariance and Weyl-polymer Quantization written by Franco Strocchi and published by Springer. This book was released on 2015-11-12 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book gives an introduction to Weyl non-regular quantization suitable for the description of physically interesting quantum systems, where the traditional Dirac-Heisenberg quantization is not applicable. The latter implicitly assumes that the canonical variables describe observables, entailing necessarily the regularity of their exponentials (Weyl operators). However, in physically interesting cases -- typically in the presence of a gauge symmetry -- non-observable canonical variables are introduced for the description of the states, namely of the relevant representations of the observable algebra. In general, a gauge invariant ground state defines a non-regular representation of the gauge dependent Weyl operators, providing a mathematically consistent treatment of familiar quantum systems -- such as the electron in a periodic potential (Bloch electron), the Quantum Hall electron, or the quantum particle on a circle -- where the gauge transformations are, respectively, the lattice translations, the magnetic translations and the rotations of 2π. Relevant examples are also provided by quantum gauge field theory models, in particular by the temporal gauge of Quantum Electrodynamics, avoiding the conflict between the Gauss law constraint and the Dirac-Heisenberg canonical quantization. The same applies to Quantum Chromodynamics, where the non-regular quantization of the temporal gauge provides a simple solution of the U(1) problem and a simple link between the vacuum structure and the topology of the gauge group. Last but not least, Weyl non-regular quantization is briefly discussed from the perspective of the so-called polymer representations proposed for Loop Quantum Gravity in connection with diffeomorphism invariant vacuum states.
Book Synopsis Quantum Mechanics in Phase Space by : Cosmas Zachos
Download or read book Quantum Mechanics in Phase Space written by Cosmas Zachos and published by World Scientific. This book was released on 2005 with total page 560 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wigner's quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence, quantum computing, and quantum chaos. It is also important in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space, or path integral formulations.In this logically complete and self-standing formulation, one need not choose sides ? coordinate or momentum space. It works in full phase space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. This invaluable book is a collection of the seminal papers on the formulation, with an introductory overview which provides a trail map for those papers; an extensive bibliography; and simple illustrations, suitable for applications to a broad range of physics problems. It can provide supplementary material for a beginning graduate course in quantum mechanics.
Book Synopsis Perspectives on Quantization by : Lewis A. Coburn
Download or read book Perspectives on Quantization written by Lewis A. Coburn and published by American Mathematical Soc.. This book was released on 1998 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the proceedings of a 1996 Joint Summer Research Conference sponsored by AMS-IMS-SIAM on "Quantization" held at Mount Holyoke College (Northampton, MA). The purpose of this conference was to bring together researchers focusing on various mathematical aspects of quantization. In the early work of Weyl and von Neumann at the beginning of the quantum era, the setting for this enterprise was operators on Hilbert space. This setting has been expanded, especially over the past decade, to involve C*-algebras - noncommutative differential geometry and noncommutative harmonic analysis - as well as more general algebras and infinite-dimensional manifolds. The applications now include quantum field theory, notable conformal and topological field theories related to quantization of moduli spaces, and constructive quantum field theory of supersymmetric models and condensed matter physics (the fractional quantum Hall effect in particular). The spectrum of research interests which significantly intersects the topic of quantization is unusually broad including, for example, pseudodifferential analysis, the representation theory of Lie groups and algebras (including infinite-dimensional ones), operator algebras and algebraic deformation theory. The papers in this collection originated with talks by the authors at the conference and represent a strong cross-section of the interests described above.
Download or read book Quantum Measurement written by Paul Busch and published by Springer. This book was released on 2016-08-23 with total page 542 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a book about the Hilbert space formulation of quantum mechanics and its measurement theory. It contains a synopsis of what became of the Mathematical Foundations of Quantum Mechanics since von Neumann’s classic treatise with this title. Fundamental non-classical features of quantum mechanics—indeterminacy and incompatibility of observables, unavoidable measurement disturbance, entanglement, nonlocality—are explicated and analysed using the tools of operational quantum theory. The book is divided into four parts: 1. Mathematics provides a systematic exposition of the Hilbert space and operator theoretic tools and relevant measure and integration theory leading to the Naimark and Stinespring dilation theorems; 2. Elements develops the basic concepts of quantum mechanics and measurement theory with a focus on the notion of approximate joint measurability; 3. Realisations offers in-depth studies of the fundamental observables of quantum mechanics and some of their measurement implementations; and 4. Foundations discusses a selection of foundational topics (quantum-classical contrast, Bell nonlocality, measurement limitations, measurement problem, operational axioms) from a measurement theoretic perspective. The book is addressed to physicists, mathematicians and philosophers of physics with an interest in the mathematical and conceptual foundations of quantum physics, specifically from the perspective of measurement theory.
Book Synopsis Symplectic Methods in Harmonic Analysis and in Mathematical Physics by : Maurice A. de Gosson
Download or read book Symplectic Methods in Harmonic Analysis and in Mathematical Physics written by Maurice A. de Gosson and published by Springer Science & Business Media. This book was released on 2011-07-30 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency literature. The topics that are addressed include (but are not limited to) the theory of the Wigner transform, the uncertainty principle (from the point of view of symplectic topology), Weyl calculus and its symplectic covariance, Shubin’s global theory of pseudo-differential operators, and Feichtinger’s theory of modulation spaces. Several applications to time-frequency analysis and quantum mechanics are given, many of them concurrent with ongoing research. For instance, a non-standard pseudo-differential calculus on phase space where the main role is played by “Bopp operators” (also called “Landau operators” in the literature) is introduced and studied. This calculus is closely related to both the Landau problem and to the deformation quantization theory of Flato and Sternheimer, of which it gives a simple pseudo-differential formulation where Feichtinger’s modulation spaces are key actors. This book is primarily directed towards students or researchers in harmonic analysis (in the broad sense) and towards mathematical physicists working in quantum mechanics. It can also be read with profit by researchers in time-frequency analysis, providing a valuable complement to the existing literature on the topic. A certain familiarity with Fourier analysis (in the broad sense) and introductory functional analysis (e.g. the elementary theory of distributions) is assumed. Otherwise, the book is largely self-contained and includes an extensive list of references.
Book Synopsis A Concise Treatise on Quantum Mechanics in Phase Space by : Thomas L Curtright
Download or read book A Concise Treatise on Quantum Mechanics in Phase Space written by Thomas L Curtright and published by World Scientific Publishing Company. This book was released on 2013-11-11 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a text on quantum mechanics formulated simultaneously in terms of position and momentum, i.e. in phase space. It is written at an introductory level, drawing on the remarkable history of the subject for inspiration and motivation. Wigner functions — density matrices in a special Weyl representation — and star products are the cornerstones of the formalism. The resulting framework is a rich source of physical intuition. It has been used to describe transport in quantum optics, structure and dynamics in nuclear physics, chaos, and decoherence in quantum computing. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative way to formulate and understand quantum mechanics, independent of the conventional Hilbert space or path integral approaches to the subject. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space variables. It works in full phase space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory. The observables in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, only composed together in novel algebraic ways using star products. This treatise provides an introductory overview and supplementary material suitable for an advanced undergraduate or a beginning graduate course in quantum mechanics.
Book Synopsis Compendium of Quantum Physics by : Daniel Greenberger
Download or read book Compendium of Quantum Physics written by Daniel Greenberger and published by Springer Science & Business Media. This book was released on 2009-07-25 with total page 901 pages. Available in PDF, EPUB and Kindle. Book excerpt: With contributions by leading quantum physicists, philosophers and historians, this comprehensive A-to-Z of quantum physics provides a lucid understanding of key concepts of quantum theory and experiment. It covers technical and interpretational aspects alike, and includes both traditional and new concepts, making it an indispensable resource for concise, up-to-date information about the many facets of quantum physics.
Book Synopsis Harmonic Analysis in Phase Space by : G. B. Folland
Download or read book Harmonic Analysis in Phase Space written by G. B. Folland and published by Princeton University Press. This book was released on 1989-03-21 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides the first coherent account of the area of analysis that involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and related concepts. This circle of ideas comes principally from mathematical physics, partial differential equations, and Fourier analysis, and it illuminates all these subjects. The principal features of the book are as follows: a thorough treatment of the representations of the Heisenberg group, their associated integral transforms, and the metaplectic representation; an exposition of the Weyl calculus of pseudodifferential operators, with emphasis on ideas coming from harmonic analysis and physics; a discussion of wave packet transforms and their applications; and a new development of Howe's theory of the oscillator semigroup.
Book Synopsis Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition) by : De Gosson Maurice A
Download or read book Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition) written by De Gosson Maurice A and published by World Scientific. This book was released on 2016-11-10 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second edition of this book deals, as the first, with the foundations of classical physics from the "symplectic" point of view, and of quantum mechanics from the "metaplectic" point of view. We have revised and augmented the topics studied in the first edition in the light of new results, and added several new sections. The Bohmian interpretation of quantum mechanics is discussed in detail. Phase space quantization is achieved using the "principle of the symplectic camel", which is a deep topological property of Hamiltonian flows. We introduce the notion of "quantum blob", which can be viewed as the fundamental phase space unit. The mathematical tools developed in this book are the theory of the symplectic and metaplectic group, the Maslov index in a rigorous form, and the Leray index of a pair of Lagrangian planes. The concept of the "metatron" is introduced, in connection with the Bohmian theory of motion. The short-time behavior of the propagator is studied and applied to the quantum Zeno effect.
Book Synopsis Quantum Mechanics of Non-Hamiltonian and Dissipative Systems by : Vasily Tarasov
Download or read book Quantum Mechanics of Non-Hamiltonian and Dissipative Systems written by Vasily Tarasov and published by Elsevier. This book was released on 2008-06-06 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum Mechanics of Non-Hamiltonian and Dissipative Systems is self-contained and can be used by students without a previous course in modern mathematics and physics. The book describes the modern structure of the theory, and covers the fundamental results of last 15 years. The book has been recommended by Russian Ministry of Education as the textbook for graduate students and has been used for graduate student lectures from 1998 to 2006. • Requires no preliminary knowledge of graduate and advanced mathematics • Discusses the fundamental results of last 15 years in this theory • Suitable for courses for undergraduate students as well as graduate students and specialists in physics mathematics and other sciences
Book Synopsis The Mathematica GuideBook for Symbolics by : Michael Trott
Download or read book The Mathematica GuideBook for Symbolics written by Michael Trott and published by Springer Science & Business Media. This book was released on 2007-04-03 with total page 1490 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides reader with working knowledge of Mathematica and key aspects of Mathematica symbolic capabilities, the real heart of Mathematica and the ingredient of the Mathematica software system that makes it so unique and powerful Clear organization, complete topic coverage, and an accessible writing style for both novices and experts Website for book with additional materials: http://www/MathematicaGuideBooks.org Accompanying DVD containing all materials as an electronic book with complete, executable Mathematica 5.1 compatible code and programs, rendered color graphics, and animations
Book Synopsis Quantum Mechanics in Phase Space by : Cosmas Zachos
Download or read book Quantum Mechanics in Phase Space written by Cosmas Zachos and published by World Scientific. This book was released on 2005 with total page 562 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wigner's quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence, quantum computing, and quantum chaos. It is also important in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space, or path integral formulations.In this logically complete and self-standing formulation, one need not choose sides ? coordinate or momentum space. It works in full phase space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. This invaluable book is a collection of the seminal papers on the formulation, with an introductory overview which provides a trail map for those papers; an extensive bibliography; and simple illustrations, suitable for applications to a broad range of physics problems. It can provide supplementary material for a beginning graduate course in quantum mechanics.
Book Synopsis From Classical to Quantum Mechanics by : Giampiero Esposito
Download or read book From Classical to Quantum Mechanics written by Giampiero Esposito and published by Cambridge University Press. This book was released on 2004-03-11 with total page 612 pages. Available in PDF, EPUB and Kindle. Book excerpt: This 2004 textbook provides a pedagogical introduction to the formalism, foundations and applications of quantum mechanics. Part I covers the basic material which is necessary to understand the transition from classical to wave mechanics. Topics include classical dynamics, with emphasis on canonical transformations and the Hamilton-Jacobi equation, the Cauchy problem for the wave equation, Helmholtz equation and eikonal approximation, introduction to spin, perturbation theory and scattering theory. The Weyl quantization is presented in Part II, along with the postulates of quantum mechanics. Part III is devoted to topics such as statistical mechanics and black-body radiation, Lagrangian and phase-space formulations of quantum mechanics, and the Dirac equation. This book is intended for use as a textbook for beginning graduate and advanced undergraduate courses. It is self-contained and includes problems to aid the reader's understanding.
Book Synopsis Philosophy of Physics by : Jeremy Butterfield
Download or read book Philosophy of Physics written by Jeremy Butterfield and published by Elsevier. This book was released on 2007 with total page 1481 pages. Available in PDF, EPUB and Kindle. Book excerpt: The ambition of this volume is twofold: to provide a comprehensive overview of the field and to serve as an indispensable reference work for anyone who wants to work in it. For example, any philosopher who hopes to make a contribution to the topic of the classical-quantum correspondence will have to begin by consulting Klaas Landsman's chapter. The organization of this volume, as well as the choice of topics, is based on the conviction that the important problems in the philosophy of physics arise from studying the foundations of the fundamental theories of physics. It follows that there is no sharp line to be drawn between philosophy of physics and physics itself. Some of the best work in the philosophy of physics is being done by physicists, as witnessed by the fact that several of the contributors to the volume are theoretical physicists: viz., Ellis, Emch, Harvey, Landsman, Rovelli, 't Hooft, the last of whom is a Nobel laureate. Key features - Definitive discussions of the philosophical implications of modern physics - Masterly expositions of the fundamental theories of modern physics - Covers all three main pillars of modern physics: relativity theory, quantum theory, and thermal physics - Covers the new sciences grown from these theories: for example, cosmology from relativity theory; and quantum information and quantum computing, from quantum theory - Contains special Chapters that address crucial topics that arise in several different theories, such as symmetry and determinism - Written by very distinguished theoretical physicists, including a Nobel Laureate, as well as by philosophers - Definitive discussions of the philosophical implications of modern physics - Masterly expositions of the fundamental theories of modern physics - Covers all three main pillars of modern physics: relativity theory, quantum theory, and thermal physics - Covers the new sciences that have grown from these theories: for example, cosmology from relativity theory; and quantum information and quantum computing, from quantum theory - Contains special Chapters that address crucial topics that arise in several different theories, such as symmetry and determinism - Written by very distinguished theoretical physicists, including a Nobel Laureate, as well as by philosophers
Book Synopsis Theoretical Foundations of Nanoscale Quantum Devices by : Malin Premaratne
Download or read book Theoretical Foundations of Nanoscale Quantum Devices written by Malin Premaratne and published by Cambridge University Press. This book was released on 2021-01-07 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nanooptics which describes the interaction of light with matter at the nanoscale, is a topic of great fundamental interest to physicists and engineers and allows the direct observation of quantum mechanical phenomena in action. This self-contained and extensively referenced text describes the underlying theory behind nanodevices operating in the quantum regime for use both in advanced courses and as a reference for researchers in physics, chemistry, electrical engineering, and materials science. Presenting an extensive theoretical toolset for design and analysis of nanodevices, the authors demonstrate the art of developing approximate quantum models of real nanodevices. The rudimentary mathematical knowledge required to master the material is carefully introduced, with detailed derivations and frequent worked examples allowing readers to gain a thorough understanding of the material. More advanced applications are gradually introduced alongside analytical approximations and simplifying assumptions often used to make such problems tractable while representative of the observed features.
