Coherent states over symplectic homogeneous spaces. (English) Zbl 0686.46049

De Bièvre, S.

J. Math. Phys. 30, No. 7, 1401-1407 (1989).

Summary: A generalization of the Perelomov procedure for the construction of coherent states is proposed. The new procedure is used to construct systems of coherent states in the carrier spaces of unitary irreducible representations of groups \(G=S/V\), where V is a vector space and \(S\subset GL(V)\). The coherent states are shown to be labeled by the points in cotangent bundles \(T^*{\mathcal O}^*\) of orbits \({\mathcal O}^*\) of S in \(V^*\), the dual of V; it is proven that \(T^*{\mathcal O}^*\) is a symplectic homogeneous space of G. The generalized procedure for the construction of coherent states presented in this paper is shown to encompass as special cases the constructions known in the literature for the coherent states of the Weyl-Heisenberg, the \(``ax+b''\), and the Galilei and Poincaré groups. Moreover, completely new sets of coherent states are constructed for the Euclidean group E(n), where the Perelomov construction fails.

MSC:

46N99	Miscellaneous applications of functional analysis
81R10	Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations

Keywords:

Perelomov procedure for the construction of coherent states; carrier spaces of unitary irreducible representations of groups; cotangent bundles; symplectic homogeneous space

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