Giuseppe Marmo - Böcker

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.

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.

Introducing a geometric view of fundamental physics, starting from quantum mechanics and its experimental foundations, this book is ideal for advanced undergraduate and graduate students in quantum mechanics and mathematical physics. Focusing on structural issues and geometric ideas, this book guides readers from the concepts of classical mechanics to those of quantum mechanics. The book features an original presentation of classical mechanics, with the choice of topics motivated by the subsequent development of quantum mechanics, especially wave equations, Poisson brackets and harmonic oscillators. It also presents new treatments of waves and particles and the symmetries in quantum mechanics, as well as extensive coverage of the experimental foundations.

2 Elementary systems on G-manifolds. 292 3 Two elementary systems: The(GxG)-manifold and its subma- fold splitting underdiag(GxG). 293 4 Examples of internal coordinates on(GxG). 297 5 Kronecker products and two-particle state decompositions on(Gx G). 298 6 Fusion of two elementary systems on(GxG). 300 7 Elementary systems on the Poincare-manifold. ' 301 7. 1 Mackey and covariant ?elds. 301 7. 2 From covariant to Mackey ?elds. 302 7. 3 Obstruction of Poincare-manifolds ' by covariant ?elds. 303 8 Relativistic position operators and coordinates. 305 8. 1 Position operators for relativistic Mackey ?elds. 305 8. 2 Position operators for ?elds with spin. 306 9 From Dirac ?elds to Bargmann-Wigner ?elds by fusion. 307 10 Elementary systems in interaction. 309 10. 1 Euclidean invariant interactions. 309 10. 2 Interacting Dirac spinor ?elds. 310 11 Scission of an elementary system. 311 12 Conclusion. 313 13 Appendix. 313 13. 1 A: Orthogonality and completeness of unitary represen- tions. 313 13. 2 B: Parameters, cosets and multiplication rules forSl(2,C). 313 13. 3 C: Observables in the relativistic 2-body system. 314 Propagation in crossed electric and magnetic ?elds 317 T. Kramer, C.Bracher 1 Introduction 317 2 Elastic scattering and quantum sources 318 2. 1 Connection to the propagator 319 2. 2 Currents generated by quantum sources 321 2. 3 Density of States 322 2. 4 Construction of the Green function 322 3 Matter waves in crossed electric and magnetic ?elds 324 3. 1 The quantum propagator 324 3.

2 Elementary systems on G-manifolds. 292 3 Two elementary systems: The(GxG)-manifold and its subma- fold splitting underdiag(GxG). 293 4 Examples of internal coordinates on(GxG). 297 5 Kronecker products and two-particle state decompositions on(Gx G). 298 6 Fusion of two elementary systems on(GxG). 300 7 Elementary systems on the Poincare-manifold. ' 301 7. 1 Mackey and covariant ?elds. 301 7. 2 From covariant to Mackey ?elds. 302 7. 3 Obstruction of Poincare-manifolds ' by covariant ?elds. 303 8 Relativistic position operators and coordinates. 305 8. 1 Position operators for relativistic Mackey ?elds. 305 8. 2 Position operators for ?elds with spin. 306 9 From Dirac ?elds to Bargmann-Wigner ?elds by fusion. 307 10 Elementary systems in interaction. 309 10. 1 Euclidean invariant interactions. 309 10. 2 Interacting Dirac spinor ?elds. 310 11 Scission of an elementary system. 311 12 Conclusion. 313 13 Appendix. 313 13. 1 A: Orthogonality and completeness of unitary represen- tions. 313 13. 2 B: Parameters, cosets and multiplication rules forSl(2,C). 313 13. 3 C: Observables in the relativistic 2-body system. 314 Propagation in crossed electric and magnetic ?elds 317 T. Kramer, C.Bracher 1 Introduction 317 2 Elastic scattering and quantum sources 318 2. 1 Connection to the propagator 319 2. 2 Currents generated by quantum sources 321 2. 3 Density of States 322 2. 4 Construction of the Green function 322 3 Matter waves in crossed electric and magnetic ?elds 324 3. 1 The quantum propagator 324 3.

This book describes, by using elementary techniques, how some geometrical structures widely used today in many areas of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics.

This book describes, by using elementary techniques, how some geometrical structures widely used today in many areas of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics. It is assumed that what can be accessed in actual experiences when studying a given system is just its dynamical behavior that is described by using a family of variables ("observables" of the system). The book departs from the principle that ''dynamics is first'' and then tries to answer in what sense the sole dynamics determines the geometrical structures that have proved so useful to describe the dynamics in so many important instances. In this vein it is shown that most of the geometrical structures that are used in the standard presentations of classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian, Lagrangian) are determined, though in general not uniquely, by the dynamics alone. The same program is accomplished for the geometrical structures relevant to describe quantum dynamics. Finally, it is shown that further properties that allow the explicit description of the dynamics of certain dynamical systems, like integrability and super integrability, are deeply related to the previous development and will be covered in the last part of the book. The mathematical framework used to present the previous program is kept to an elementary level throughout the text, indicating where more advanced notions will be needed to proceed further. A family of relevant examples is discussed at length and the necessary ideas from geometry are elaborated along the text. However no effort is made to present an ''all-inclusive'' introduction to differential geometry as many other books already exist on the market doing exactly that. However, the development of the previous program, considered as the posing and solution of a generalized inverse problem for geometry, leads to new ways of thinking and relating some of the most conspicuous geometrical structures appearing in Mathematical and Theoretical Physics.

This book is an introduction to the role of topology in the quantization of classical systems. It is also an introduction to topological solitons with special emphasis on Skyrmions. As regards the first aspect, several issues of current interest are dealt with at a reasonably elementary level. Examples are principal fibre bundles and their role in quantum physics, the possibility of spinorial quantum states in a Lagrangian theory based on tensorial variables, and multiply connected configuration spaces and associated quantum phenomena like the QCD q angle and exotic statistics. The ideas are also illustrated by simple examples such as the spinning particle, the charge-monopole system and strings in 3+1 dimensions. The application of these ideas to quantum gravity is another subject treated at an introductory level. An attempt has been made in this book to introduce the reader to the significance of topology for many distinct physical systems such as spinning particles, the charge- monopole system, strings, Skyrmions, QCD and gravity. The book is an outgrowth of lectures given by the authors at various institutions and conferences.

This book is an introduction to the role of topology in the quantization of classical systems. It is also an introduction to topological solitons with special emphasis on Skyrmions. As regards the first aspect, several issues of current interest are dealt with at a reasonably elementary level. Examples are principal fibre bundles and their role in quantum physics, the possibility of spinorial quantum states in a Lagrangian theory based on tensorial variables, and multiply connected configuration spaces and associated quantum phenomena like the QCD q angle and exotic statistics. The ideas are also illustrated by simple examples such as the spinning particle, the charge-monopole system and strings in 3+1 dimensions. The application of these ideas to quantum gravity is another subject treated at an introductory level. An attempt has been made in this book to introduce the reader to the significance of topology for many distinct physical systems such as spinning particles, the charge- monopole system, strings, Skyrmions, QCD and gravity. The book is an outgrowth of lectures given by the authors at various institutions and conferences.

This is the proceedings of the 9th conference in this series. In addition to papers presented at the conference proper, it contains some papers delivered at Peter G Bergmann's 75th Birthday meeting (Capri, 24 Sept 1990). Among the subjects covered are cosmology and astrophysics, both theoretical and experimental.

This second workshop on constraint theory aims at reviewing the developments that have taken place in the theory of singular Lagrangians and Dirac-Bergmann Hamiltonian constraints as well as their quantization. Since this theory lies behind all special and general relativistic systems, the topics covered here naturally range from mathematical physics to relativistic system particles, strings and fields and further to general relativity. The variety of topics discussed makes this an important, interesting and informative book.

A P Balachandran has a long and impressive record of research in particle physics and quantum field theory, bringing concepts of geometry, topology and operator algebras to the analysis of physical problems, particularly in particle physics and condensed matter physics. He has also had an influential role within the physics community, not only in terms of a large number of students, research associates and collaborators, but also serving on the editorial boards of important publications, including the International Journal of Modern Physics A.This book consists of articles by students and associates of Balachandran. Most of the articles are scientific in nature, with topics ranging from noncommutative geometry, particle physics phenomenology, to condensed matter physics. Various chapters focus on new perspectives and directions resulting from Balachandran's contributions to physics, as well as some reminiscences of collaborating and working with Balachandran.