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Journal of Mathematical Physics

Hilbert spaces of analytic functions and generalized coherent states

M. Arık, Darryl D. Coon

Published Apr 1, 1976• 613 citations
Publisher

Abstract

Generalized coherent states which are associated with a generalization of the harmonic oscillator commutation relation are investigated. It is shown that these states form an overcomplete basis in a Hilbert space of analytic functions. The generalized creation and annihilation operators are bounded except in a limit in which they reduce to the usual boson creation and annihilation operators. In this limit the Hilbert space of analytic functions reduces to the Bargmann–Segal Hilbert space of entire functions and in another limit it reduces to the Hardy–Lebesgue space.

DOI
10.1063/1.522937

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Generalized coherent states which are associated with a generalization of the harmonic oscillator commutation relation are investigated. It is shown that these states form an overcomplete basis in a Hilbert space of analytic functions. The generalized creation and annihilation operators are bounded except in a limit in which they reduce to the usual boson creation and annihilation operators. In this limit the Hilbert space of analytic functions reduces to the Bargmann–Segal Hilbert space of entire functions and in another limit it reduces to the Hardy–Lebesgue space.
Recently we have obtained, on the basis of a group approach to quantization, a Bargmann-Fock-like realization of the Relativistic Harmonic Oscillator as well as a generalized Bargmann transform relating fock wave functions and a set of relativistic Hermite polynomials. Nevertheless, the relativistic creation and annihilation operators satisfy typical relativistic commutation relations of the Lie product (vector-z, vector-z(sup dagger)) approximately equals Energy (an SL(2,R) algebra). Here we find higher-order polarization operators on the SL(2,R) group, providing canonical creation and annihilation operators satisfying the Lie product (vector-a, vector-a(sup dagger)) = identity vector 1, the eigenstates of which are 'true' coherent states.
A q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons. The space consists of a class of entire functions of a complex variable z, and has a reproducing kernel. On this space, the q-boson creation and annihilation operators are represented as multiplication by z and q-differentiation with respect to z, respectively. A q-integral analogue of Bargmann's scalar product is defined, involving the q-exponential as a weight function. Associated with this is a completeness relation for the q-coherent states.
It is shown that the completeness problem of the SL(2, R) coherent states proposed by Barut and Girardello leads to a moment problem, not a Mellin transform. This moment problem, which also appears in the theory of para-Bose oscillators, has been solved following the Sharma–Mehta–Mukunda–Sudarshan solution of the problem. The matrix element of finite transformation in the coherent state basis is shown to satisfy a ‘‘quasiorthogonality’’ condition analogous to the orthogonality condition of the matrix element in the canonical basis. Finally, the Barut–Girardello ‘‘Hilbert space of entire analytic functions of growth (1,1)’’ turns out to be only a subspace of Bargmann’s well-known Hilbert space of analytic functions. This subspace, which has been called ‘‘the reduced Bargmann space’’ in a previous paper, is an invariant subspace of SL(2,R). With this identification the generators of the group in this realization turn out to be the well-known boson operators of Holman and Biedenharn.