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arXiv:math-ph/0209006 · Journal of Physics A Mathematical and General

Characterization of SU(1, 1) coherent states in terms of affine group wavelets

J. Bertrand, Mich egrave le Irac-Astaud

Published Aug 13, 2002• 7 citations
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Abstract

The Perelomov coherent states of SU (1, 1) are labelled by elements of the quotient of SU (1, 1) by the compact subgroup. Taking advantage of the fact that this quotient is isomorphic to the affine group of the real line, we are able to parametrize the coherent states by elements of that group or equivalently by points in the half-plane. Such a formulation permits to find new properties of the SU (1, 1) coherent states and to relate them to affine wavelets.

DOI
10.1088/0305-4470/35/34/308

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The Perelomov coherent states of SU (1, 1) are labelled by elements of the quotient of SU (1, 1) by the compact subgroup. Taking advantage of the fact that this quotient is isomorphic to the affine group of the real line, we are able to parametrize the coherent states by elements of that group or equivalently by points in the half-plane. Such a formulation permits to find new properties of the SU (1, 1) coherent states and to relate them to affine wavelets.
Although coherent states and wavelets have developed by now into full-fledged subdisciplines, with plenty of applications in physics, mathematics and engineering, both have deep roots in group theory,. The construction of coherent states was identified as a group-theoretical problem in the pioneering work of Perelomov and Gilmore, whereas wavelets are simply the coherent states associated to the affine group of the line (or the similitude group of the plane for the two-dimensional case). It was therefore appropriate to have both topics represented in the GROUP-24 Colloquium.
: n-dimensional coherent states systems generated by translations, modulations, rotations and dilations are described. Starting from unitary irreducible representations of the n-dimen- sional affine Weyl-Heisenberg group, which are not square-integrable, one is led to consider systems of coherent states labeled by the elements of quotients of the original group. Such systems can yield a resolution of the identity, and then be used as alternatives to usual wavelet or windowed Fourier analysis. When the quotient space is the phase space of the representation, different embeddings of it into the group provide different descriptions of the phase space. R'esum'e: On d'ecrit des syst`emes d"etats coh'erents engendr'es par des translations, dilatations et rotations en n-dimensions. Partant de repr'esentations unitaires irr'eductibles du groupe de Weyl-Heisenberg affine en dimension n, qui ne sont pas de carr'e integrable, on est naturellement conduit `a consid'erer des syst`emes d"etats coh...
Abstract We quantify the nonclassicality of quantum states associated with the Lie group SU(1,1) by regarding states as observables and considering their variances in the SU(1,1) Perelomov coherent states. Combining the resolution of identity induced by the SU(1,1) Perelomov coherent states, we propose a quantifier for nonclassicality of a state based on the average uncertainty (variance) of the state (regarded as an observable) in the SU(1,1) Perelomov coherent states. This quantifier is easy to calculate and possesses several operational interpretations. We reveal its basic properties and illustrate it by several prototypical examples.
Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU (2) is studied in particular and the orbit spaces of its j = 1/2 and 1 representations completely determined. The orbits of SU (2) in CP N can be either two- or three-dimensional, the former being either isomorphic to S 2 or to RP 2 and the latter being isomorphic to quotient spaces of RP 3 . We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.