This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.
Focusing on the mathematical foundations of quantum mechanics, this book rigorously explores harmonic analysis tools crucial for its modern formulation. It introduces innovative ideas and methods, particularly emphasizing the Wigner phase space formalism and its implications for density operators and entanglement properties. Aimed at advanced undergraduate students in mathematics and physics, as well as experienced researchers, the text bridges theoretical concepts with practical applications in various scientific fields.
This book presents a comprehensive mathematical study of the operators behind the Born–Jordan quantization scheme. The Schrödinger and Heisenberg pictures of quantum mechanics are equivalent only if the Born–Jordan scheme is used. Thus, Born–Jordan quantization provides the only physically consistent quantization scheme, as opposed to the Weyl quantization commonly used by physicists. In this book we develop Born–Jordan quantization from an operator-theoretical point of view, and analyze in depth the conceptual differences between the two schemes. We discuss various physically motivated approaches, in particular the Feynman-integral point of view. One important and intriguing feature of Born-Jordan quantization is that it is not one-to-one: there are infinitely many classical observables whose quantization is zero.
The Maslov classes have been playing an essential role in various parts of applied and pure mathematics, and physics, since the early 1970s. This volume intends to provide a thorough treatment of the Maslov classes and of their relationship with the metaplectic group.