COHERENT STATES AND THEIR GENERALIZATIONS - A MATHEMATICAL OVERVIEW

We present a survey of the theory of coherent states (CS); and some of their generalizations, with emphasis on the mathematical structure, r ather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS ar e associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the gr oup context altogether, and thus obtain the so-called reproducing trip les and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and th e quantization/dequantization problem, that is, the transition from th e classical to the quantum level and vice versa.