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Geometric Phases in Classical and Quantum Mechanics

  • Dariusz Chruściński
  • Andrzej Jamiołkowski

Part of the Progress in Mathematical Physics book series (PMP, volume 36)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Dariusz Chruściński, Andrzej Jamiołkowski
    Pages 1-72
  3. Dariusz Chruściński, Andrzej Jamiołkowski
    Pages 73-109
  4. Dariusz Chruściński, Andrzej Jamiołkowski
    Pages 111-155
  5. Dariusz Chruściński, Andrzej Jamiołkowski
    Pages 157-178
  6. Dariusz Chruściński, Andrzej Jamiołkowski
    Pages 179-238
  7. Dariusz Chruściński, Andrzej Jamiołkowski
    Pages 239-304
  8. Back Matter
    Pages 305-337

About this book

Introduction

This work examines the beautiful and important physical concept known as the 'geometric phase,' bringing together different physical phenomena under a unified mathematical and physical scheme.

Several well-established geometric and topological methods underscore the mathematical treatment of the subject, emphasizing a coherent perspective at a rather sophisticated level. What is unique in this text is that both the quantum and classical phases are studied from a geometric point of view, providing valuable insights into their relationship that have not been previously emphasized at the textbook level.

Key Topics and Features:

• Background material presents basic mathematical tools on manifolds and differential forms.

• Topological invariants (Chern classes and homotopy theory) are explained in simple and concrete language, with emphasis on physical applications.

• Berry's adiabatic phase and its generalization are introduced.

• Systematic exposition treats different geometries (e.g., symplectic and metric structures) living on a quantum phase space, in connection with both abelian and nonabelian phases.

• Quantum mechanics is presented as classical Hamiltonian dynamics on a projective Hilbert space.

• Hannay’s classical adiabatic phase and angles are explained.

• Review of Berry and Robbins' revolutionary approach to spin-statistics.

• A chapter on Examples and Applications paves the way for ongoing studies of geometric phases.

• Problems at the end of each chapter.

• Extended bibliography and index.

Graduate students in mathematics with some prior knowledge of quantum mechanics will learn about a class of applications of differential geometry and geometric methods in quantum theory. Physicists and graduate students in physics will learn techniques of differential geometry in an applied context.

 

 

Keywords

Chern class Homotopy Matrix classical mechanics classical/quantum mechanics differential geometry homotopy theory manifold quantum mechanics topological groups

Authors and affiliations

  • Dariusz Chruściński
    • 1
  • Andrzej Jamiołkowski
    • 1
  1. 1.Nicholas Copernicus UniversityTorunPoland

Bibliographic information

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