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This book presents a comprehensive mathematical study of the operators behind the Born–Jordan quantization scheme. The Schrödinger and Heisenberg pictures of quantum mechanics are equivalent only if the Born–Jordan scheme is used. Thus, Born–Jordan quantization provides the only physically consistent quantization scheme, as opposed to the Weyl quantization commonly used by physicists. In this book we develop Born–Jordan quantization from an operator-theoretical point of view, and analyze in depth the conceptual differences between the two schemes. We discuss various physically motivated approaches, in particular the Feynman-integral point of view. One important and intriguing feature of Born-Jordan quantization is that it is not one-to-one: there are infinitely many classical observables whose quantization is zero.
This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.
Quantum mechanics is arguably one of the most successful scientific theories ever and its applications to chemistry, optics, and information theory are innumerable. This book provides the reader with a rigorous treatment of the main mathematical tools from harmonic analysis which play an essential role in the modern formulation of quantum mechanics. This allows us at the same time to suggest some new ideas and methods, with a special focus on topics such as the Wigner phase space formalism and its applications to the theory of the density operator and its entanglement properties. This book can be used with profit by advanced undergraduate students in mathematics and physics, as well as by confirmed researchers.
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.
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.
This book presents a comprehensive mathematical study of the operators behind the Born–Jordan quantization scheme. The Schrödinger and Heisenberg pictures of quantum mechanics are equivalent only if the Born–Jordan scheme is used. Thus, Born–Jordan quantization provides the only physically consistent quantization scheme, as opposed to the Weyl quantization commonly used by physicists. In this book we develop Born–Jordan quantization from an operator-theoretical point of view, and analyze in depth the conceptual differences between the two schemes. We discuss various physically motivated approaches, in particular the Feynman-integral point of view. One important and intriguing feature of Born-Jordan quantization is that it is not one-to-one: there are infinitely many classical observables whose quantization is zero.
This book provides an in-depth and rigorous study of the Wigner transform and its variants. They are presented first within a context of a general mathematical framework, and then through applications to quantum mechanics. The Wigner transform was introduced by Eugene Wigner in 1932 as a probability quasi-distribution which allows expression of quantum mechanical expectation values in the same form as the averages of classical statistical mechanics. It is also used in signal processing as a transform in time-frequency analysis, closely related to the windowed Gabor transform.Written for advanced-level students and professors in mathematics and mathematical physics, it is designed as a complete textbook course providing analysis on the most important research on the subject to date. Due to the advanced nature of the content, it is also suitable for research mathematicians, engineers and chemists active in the field.
Fighting to reclaim the French crown for the Bourbons, the duchesse de Berry faces betrayal at the hands of one of her closest advisors in this dramatic history of power and revolution. The year was 1832, a cholera pandemic raged, and the French royal family was in exile, driven out by yet another revolution. From a drafty Scottish castle, the duchesse de Berry -- the mother of the eleven-year-old heir to the throne -- hatched a plot to restore the Bourbon dynasty. For months, she commanded a guerilla army and evaded capture by disguising herself as a man. But soon she was betrayed by her trusted advisor, Simon Deutz, the son of France's Chief Rabbi. The betrayal became a cause célèbre for Bourbon loyalists and ignited a firestorm of hate against France's Jews. By blaming an entire people for the actions of a single man, the duchess's supporters set the terms for the century of antisemitism that followed. Brimming with intrigue and lush detail, The Betrayal of the Duchess is the riveting story of a high-spirited woman, the charming but volatile young man who double-crossed her, and the birth of one of the modern world's most deadly forms of hatred. !--EndFragment--
Paul Adrian Maurice Dirac (1902-84) is one of the icons of modern physics. His work provided the mathematical foundations of quantum mechanics. He also made key contributions to quantum field theory and quantum statistical mechanics. He is perhaps best known for formulating the Dirac equation, a relativistic wave equation which described the properties of the electron, and also predicted the existence of anti-matter. The Dirac Centennial Symposium commemorated the contributions of Dirac to all areas of physics, and assessed their impact on frontier research. This book constitutes the proceedings of the symposium, containing articles by Leopold Halpern, Pierre Ramond, Frank Wilczek, Maurice Goldhaber, Jonathan Bagger, Joe Lykken, Roman Jackiw, Stanley Deser, Joe Polchinski, Andre Linde and others. A special contribution from Dirac's daughter Monica Dirac presents a portrait of Paul Dirac as father and family man.
The 1987 Cargese Summer Institute on Partiele Physies was organized by the Universite Pierre et Marie Curie, Paris (M. LEVY and J.-L. BASDEVANT), CERN (M. JACOB), the Universite Catholique de Louvain (D. SPEISER and J. WEYERS), and the Katholieke Universiteit te Leuven (R. GASTHANS), whieh, sinee 1975, have joined their efforts and worked in eommon. It was the 25th summer institute held at Cargese and the ninth one organized by the two institutes of theoretieal physics at Leuven and Louvain-la-Neuve. The 1987 school was centered around two main themes: the re cent developments in string theory and the physics of high energy colliders. As the standard model of the fundamental interaetions has repeatedly proved to be suecessful in explaining the experimental findings in par tiele physies, more attention was given in this school to possible new features arising from string inspired models. This led us to inelude in the program aseries of lectures devoted to string theory per se. They eovered the more mathematical aspects of the theory as weIl as the phenomenological implications. The second theme concerns high energy collider physics and was meant to prepare young physicists for the future experimental results to be expected from the pp and e+e- colliders. It brought theorists and ex perimentalists actively together in their search for a better understand ing of the high energy phenomena.
The 1981 Cargese Summer Institute on Fundamental Interactions was organized by the Universite Pierre et Marie Curie, Paris (M. LEVY and J-L. BASDEVANT), CERN (M. JACOB), the Universite Catholique de Louvain (D. SPEISER and J. WEYERS), and the Kotholieke Universiteit te Leuven (R. GASTMANS), which, since 1975 have joined their efforts and worked in common. It was the 24th Summer Institute held at Cargese and the 8th one organized by the two institutes of theoretical physics at Leuven and Louvain-Ia-Neuve. The 1985 school was centered around two main themes : the standard model of the fundamental interactions (and beyond) and astrophysics. The remarkable advances in the theoretical understanding and experimental confirmation of the standard model were reviewed in several lectures where the reader will find a thorough analysis of recent experiments as well as a detailed comparaison of the standard model with experiment. On a more theoretical side, supersymmetry, supergravity and strings were discussed as well. The second theme concerns astrophysics where the school was quite successful in bridging the gap between this fascinating subject and more conventional particle physics. We owe many thanks to all those who have made this Summer Institute possible ! Thanks are due to the Scientific Committee of NATO and its President and to the "Region Corse" for a generous grant. .. We wish to thank Miss M-F. HANSELER, Mrs ALRIFRAI, Mr and Mrs ARIANO, and Mr BERNIA and all others from Paris, Leuven, Louvain-la-Neuve and especially Cargese for their collaboration.
Black holes and gravitational radiation are two of the most dramatic predictions of general relativity. The quest for rotating black holes - discovered by Roy P. Kerr as exact solutions to the Einstein equations - is one of the most exciting challenges facing physicists and astronomers. Gravitational Radiation, Luminous Black Holes and Gamma-Ray Burst Supernovae takes the reader through the theory of gravitational radiation and rotating black holes, and the phenomenology of GRB-supernovae. Topics covered include Kerr black holes and the frame-dragging of spacetime, luminous black holes, compact tori around black holes, and black-hole spin interactions. It concludes with a discussion of prospects for gravitational-wave detections of a long-duration burst in gravitational-waves as a method of choice for identifying Kerr black holes in the Universe. This book is ideal for a special topics graduate course on gravitational-wave astronomy and as an introduction to those interested in this contemporary development in physics.
This textbook provides students with a solid introduction to the techniques of approximation commonly used in data analysis across physics and astronomy. The choice of methods included is based on their usefulness and educational value, their applicability to a broad range of problems and their utility in highlighting key mathematical concepts. Modern astronomy reveals an evolving universe rife with transient sources, mostly discovered - few predicted - in multi-wavelength observations. Our window of observations now includes electromagnetic radiation, gravitational waves and neutrinos. For the practicing astronomer, these are highly interdisciplinary developments that pose a novel challenge to be well-versed in astroparticle physics and data-analysis. The book is organized to be largely self-contained, starting from basic concepts and techniques in the formulation of problems and methods of approximation commonly used in computation and numerical analysis. This includes root finding, integration, signal detection algorithms involving the Fourier transform and examples of numerical integration of ordinary differential equations and some illustrative aspects of modern computational implementation. Some of the topics highlighted introduce the reader to selected problems with comments on numerical methods and implementation on modern platforms including CPU-GPU computing. Developed from lectures on mathematical physics in astronomy to advanced undergraduate and beginning graduate students, this book will be a valuable guide for students and a useful reference for practicing researchers. To aid understanding, exercises are included at the end of each chapter. Furthermore, some of the exercises are tailored to introduce modern symbolic computation.
This book deals with the foundations of classical physics from the ?symplectic? point of view, and of quantum mechanics from the ?metaplectic? point of view. The Bohmian interpretation of quantum mechanics is discussed. Phase space quantization is achieved using the ?principle of the symplectic camel?, which is a recently discovered deep topological property of Hamiltonian flows. The mathematical tools developed in this book are the theory of the metaplectic group, the Maslov index in a precise 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. A precise form of Feynman's integral is introduced in connection with the extended metaplectic representation.
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.
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