Ebook: Geometric Phases in Classical and Quantum Mechanics

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.

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Several well-established geometric and topological methods are used in this work on a beautiful and important physical phenomenon known as the 'geometric phase.' Going back to the intense interest in this subject since the mid-1980s and the seminal work of M. Berry and B. Simon, this book examines geometric phases, bringing together different physical phenomena under a unified mathematical scheme. Key background material, beginning with the notion of manifolds and differential forms, as well as basic mathematical tools -- fiber bundles, connections and holonomies -- are presented in Chapter 1. Topological invariants such as Chern classes and homotopy theory are explained in simple, concrete language with emphasis on physical applications. The exposition then unfolds systematically. The adiabatic phases of Berry, the Wilczek--Zee nonabelian factor, and a classical counterpart called Hannay's angles focus on the physical side of the geometric phase problem. Thereafter the geometry of quantum evolution is treated. Here the reader learns about different geometries (such as symplectic and metric structures) living on a quantum phase space in connection with both abelian and nonabelian geometric phases. The concluding section on Examples and Applications paves the way for a continuing study of the geometric PHASES. Throughout the text, material is presented on a level suitable for graduate students and researchers in applied mathematics and physics with an understanding of classical and quantum mechanics.