D. Chruściński, A. Jamiołkowski,
Geometric Phases in Classical and Quantum Mechanics,
Birkhaüser Boston 2004
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.
- Problems at the end of each chapter.
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. [Źródło: Amazon.com]
[Dr hab. Andrzej Jamiołkowski, prof. UMK oraz Dr hab. Dariusz Chruściński, prof. UMK są pracownikami Instytutu Fizyki UMK]