Coherent states associated to group representations which are square integrable on a homogeneous space

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

Giorgio Bertotti, Mario Rasetti

Jean-Pierre Antoine

We describe several families of wavelets associated to the Galilei group, extended by space and time dilations. The construction follows a general method based on group representations which are square integrable only on a homogeneous space of the underlying group.

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau +1 more

We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples 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 the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.

Ali, S. T., James A. Brooke, Jean‐Pierre Gazeau

A general definition of squeezed states is proposed and its main features are illustrated through a discussion of the standard optical coherent states represented by 'Gaussian pure states'. The set-up involves representations of groups on Hilbert spaces over homogeneous spaces of the group, and relies on the construction of a square integrable (coherent state) group representation modulo a subgroup. This construction depends upon a choice of a Borel section which has a certain permissible arbitrariness in its selection; this freedom is attributable to a squeezing of the defining coherent states of the representation, and corresponds in this way to a sort of gauging.