Coherent States in Quantum Physics

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).
There exist superpositions of quantum states which have many features (properties
or dynamical behaviors) analogous to those of their classical counterparts:
they are the so-called coherent states, already studied by Schrödinger in 1926 and
rediscovered by Klauder, Glauber, and Sudarshan at the beginning of the 1960s.
The phrase “coherent states” was proposed by Glauber in 1963 in the context of
quantum optics. Indeed, these states are superpositions of Fock states of the quantized
electromagnetic field that, up to a complex factor, are not modified by the
action of photon annihilation operators. They describe a reservoir with an undetermined
number of photons, a situation that can be viewed as formally close to the
classical description in which the concept of a photon is absent.
The purpose of these lecture notes is to explain the notion of coherent states and
of their various generalizations, since Schrödinger up to some of the most recent
conceptual advances and applications in different domains of physics and signal
analysis. The guideline of the book is based on a unifying method of construction
of coherent states, of minimal complexity. This method has a substantially
probabilistic content and allows one to establish a simple and natural link between
practically all families of coherent states proposed until now. This approach embodies
the originality of the book in regard to well-established procedures derived
essentially from group theory (e.g., coherent state family viewed as the orbit under
the action of a group representation) or algebraic constraints (e.g., coherent states
viewed as eigenvectors of some lowering operator), and comprehensively presented
in previous treatises, reviews, an extensive collection of important papers, and
proceedings.