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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 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.
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.
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.
This book is addressed to graduate students and research workers in
theoretical physics who want a thorough introduction to group
theory and Hopf algebras. It is suitable for a one-semester course
in group theory or a two-semester course which also treats advanced
topics. Starting from basic definitions, it goes on to treat both
finite and Lie groups as well as Hopf algebras. Because of the
diversity in the choice of topics, which does not place undue
emphasis on finite or Lie groups, it should be useful to physicists
working in many branches. A unique aspect of the book is its
treatment of Hopf algebras in a form accessible to physicists. Hopf
algebras are generalizations of groups and their concepts are
acquiring importance in the treatment of conformal field theories,
noncommutative spacetimes, topological quantum computation and
other important domains of investigation. But there is a scarcity
of treatments of Hopf algebras at a level and in a manner that
physicists are comfortable with. This book addresses this need
superbly. There are illustrative examples from physics scattered
throughout the book and in its set of problems. It also has a good
bibliography. These features should enhance its value to readers.
The authors are senior physicists with considerable research and
teaching experience in diverse aspects of fundamental physics. The
book, being the outcome of their combined efforts, stands testament
to their knowledge and pedagogical skills.
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.
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