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: โ{128}ข Background material presents basic mathematical tools on manifolds and differential forms. โ{128}ข Topological invariants (Chern classes and homotopy theory) are explained in simple and concrete language, with emphasis on physical applications. โ{128}ข Berry's adiabatic phase and its generalization are introduced. โ{128}ข 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. โ{128}ข Quantum mechanics is presented as classical Hamiltonian dynamics on a projective Hilbert space. โ{128}ข Hannay's classical adiabatic phase and angles are explained. โ{128}ข Review of Berry and Robbins' revolutionary approach to spin-statistics. โ{128}ข A chapter on Examples and Applications paves the way for ongoing studies of geometric phases. โ{128}ข Problems at the end of each chapter. โ{128}ข 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.
CONTENT
1 Mathematical Background -- 2 Adiabatic Phases in Quantum Mechanics -- 3 Adiabatic Phases in Classical Mechanics -- 4 Geometric Approach to Classical Phases -- 5 Geometry of Quantum Evolution -- 6 Geometric Phases in Action -- A Classical Matrix Lie Groups and Algebras -- B Quaternions
