This self-contained introduction discusses the evolution of the notion of coherent states, from the early works of Schr?dinger to the most recent advances, including signal analysis. An integrated and modern approach to the utility of coherent states in many different branches of physics, it strikes a balance between mathematical and physical descriptions.

Split into two parts, the first introduces readers to the most familiar coherent states, their origin, their construction, and their application and relevance to various selected domains of physics. Part II, mostly based on recent original results, is devoted to the question of quantization of various sets through coherent states, and shows the link to procedures in signal analysis.

This book originated from a series of advanced lectures on coherent states in physics delivered in Strasbourg, Louvain-la Neuve, Paris, Rio de Janeiro, Rabat, and Bialystok, over the period from 1997 to 2008. In writing this book, I have attempted to maintain a cohesive self-contained content.

Let me first give some insights into the notion of a coherent state in physics. Within the context of classical mechanics, a physical system is described by states which are points of its phase space (and more generally densities). In quantum mechanics, the system is described by states which are vectors (up to a phase) in a Hilbert space (and more generally by density operators).