Maurice A De Gosson's Books

Born-Jordan Quantization

Theory and Applications

By: Maurice A. de Gosson

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Born-Jordan Quantization

Theory and Applications

By: Maurice A. de Gosson

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.

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Page Count

226

Publisher

Springer

Published Date

ISBN 10

3319279025

ISBN 13

9783319279022

The Wigner Transform

By: Maurice A De Gosson

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The Wigner Transform

By: Maurice A De Gosson

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.

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Page Count

250

Publisher

World Scientific Publishing Company

Published Date

ISBN 10

1786343118

ISBN 13

9781786343116

Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes

By: Maurice A De Gosson

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Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes

By: Maurice A De Gosson

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.

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Page Count

306

Publisher

World Scientific

Published Date

ISBN 10

1786344165

ISBN 13

9781786344168

Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition)

By: Maurice A De Gosson

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Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition)

By: Maurice A De Gosson

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.

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Page Count

423

Publisher

World Scientific

Published Date

ISBN 10

9813200987

ISBN 13

9789813200982

Symplectic Methods in Harmonic Analysis and in Mathematical Physics

By: Maurice A. de Gosson

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Symplectic Methods in Harmonic Analysis and in Mathematical Physics

By: Maurice A. de Gosson

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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351

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3764399929

ISBN 13

9783764399924

Quantum Harmonic Analysis

An Introduction

By: Maurice A. de Gosson

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Quantum Harmonic Analysis

An Introduction

By: Maurice A. de Gosson

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.

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Page Count

240

Publisher

Walter de Gruyter GmbH & Co KG

Published Date

ISBN 10

3110722771

ISBN 13

9783110722772

Symplectic Geometry and Quantum Mechanics

By: Maurice A. de Gosson

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Symplectic Geometry and Quantum Mechanics

By: Maurice A. de Gosson

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.

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Page Count

375

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3764375752

ISBN 13

9783764375751

Born-Jordan Quantization

Theory and Applications

By: Maurice A. de Gosson

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Born-Jordan Quantization

Theory and Applications

By: Maurice A. de Gosson

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226

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Springer

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ISBN 10

3319279025

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9783319279022

Symplectic Geometry and Quantum Mechanics

By: Maurice A. de Gosson

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Symplectic Geometry and Quantum Mechanics

By: Maurice A. de Gosson

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375

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Springer Science & Business Media

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ISBN 10

3764375752

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9783764375751

Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes

By: Maurice A De Gosson

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Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes

By: Maurice A De Gosson

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Page Count

306

Publisher

World Scientific

Published Date

ISBN 10

1786344165

ISBN 13

9781786344168

Symplectic Geometry and Quantum Mechanics

By: Maurice A. de Gosson

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Symplectic Geometry and Quantum Mechanics

By: Maurice A. de Gosson

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Page Count

368

Publisher

Birkhäuser

Published Date

ISBN 10

3764391251

ISBN 13

9783764391256

Symplectic Methods in Harmonic Analysis and in Mathematical Physics

By: Maurice A. de Gosson

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Symplectic Methods in Harmonic Analysis and in Mathematical Physics

By: Maurice A. de Gosson

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351

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3764399929

ISBN 13

9783764399924

Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition)

By: Maurice A De Gosson

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Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition)

By: Maurice A De Gosson

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423

Publisher

World Scientific

Published Date

ISBN 10

9813200987

ISBN 13

9789813200982

The Principles of Newtonian and Quantum Mechanics

The Need for Planck's Constant, H

By: Maurice A. De Gosson

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The Principles of Newtonian and Quantum Mechanics

The Need for Planck's Constant, H

By: Maurice A. De Gosson

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 physe space unit. The mathematical tools developed in this book are the theory of the symplectic and metaplectic group, the Maslov index in a rigrous 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"--

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Page Count

423

Published Date

ISBN 10

9813200979

ISBN 13

9789813200975

The Wigner Transform

By: Maurice de Gosson

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The Wigner Transform

By: Maurice de Gosson

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.

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Page Count

229

Publisher

Wspc (Europe)

Published Date

ISBN 10

1786343096

ISBN 13

9781786343093

Maslov Classes, Metaplectic Representation and Lagrangian Quantization

By: Maurice de Gosson

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Maslov Classes, Metaplectic Representation and Lagrangian Quantization

By: Maurice de Gosson

The Maslov Classes have been playing an essential role in various parts of applied and pure mathematics, and physics, since the early 70's. Their correct definition is due to V. I. Arnold and J. Leray, in the transversal case, and to P. Dazord and the author in the general case. The aim of this book is to give a thorough treatment of the theory of the Maslov classes and of their relationship with the metaplectic group. It is (among other things) shown that these classes can be reconstructed, modulo 4, using only the analytic properties of the metaplectic group. In the last chapter the author sketches a scheme for geometric quantization by introducing two new concepts, that of metaplectic half-form and that of Lagrangian catalogue, the latter generalizes and simplifies the notion of "Lagrangian function" introduced by J. Leray. A Lagrangian catalogue is a collection of metaplectic half-forms which are themselves "cohomological wave functions", whose definition is made possible by using the combinatorial properties of the Maslov classes. The transformation of Lagrangian catalogues under the metaplectic group and of Hamiltonian flows is studied, and it is shown that one thus recovers very easily the so-called "quasi-classical approximation" to the solutions of Schr?dinger equation if one introduces a natural concept, that of projection of a Lagrangian catalogue. An application to geometric phase shifts, including Berry's phase, is given.

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Page Count

188

Publisher

Wiley-VCH

Published Date

ISBN 10

3527400877

ISBN 13

9783527400874

Book's by Maurice A De Gosson

Born-Jordan Quantization

Born-Jordan Quantization

The Wigner Transform

The Wigner Transform

Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes

Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes

Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition)

Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition)

Symplectic Methods in Harmonic Analysis and in Mathematical Physics

Symplectic Methods in Harmonic Analysis and in Mathematical Physics

Quantum Harmonic Analysis

Quantum Harmonic Analysis

Symplectic Geometry and Quantum Mechanics

Symplectic Geometry and Quantum Mechanics

Born-Jordan Quantization

Born-Jordan Quantization

Symplectic Geometry and Quantum Mechanics

Symplectic Geometry and Quantum Mechanics

Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes

Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes

Symplectic Geometry and Quantum Mechanics

Symplectic Geometry and Quantum Mechanics

Symplectic Methods in Harmonic Analysis and in Mathematical Physics

Symplectic Methods in Harmonic Analysis and in Mathematical Physics

Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition)

Principles Of Newtonian And Quantum Mechanics, The: The Need For Planck's Constant, H (Second Edition)

The Principles of Newtonian and Quantum Mechanics

The Principles of Newtonian and Quantum Mechanics

The Wigner Transform

The Wigner Transform

Maslov Classes, Metaplectic Representation and Lagrangian Quantization

Maslov Classes, Metaplectic Representation and Lagrangian Quantization

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