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AIP conference proceedings

Coherent states associated to the real Jacobi group

Stefan Berceanu, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmeier, Theodore Voronov

Published Jan 1, 2007• 12 citations
Publisher

Abstract

Using the coherent states attached to the complex Jacobi group—the semi‐direct product of the Heisenberg‐Weyl group with the real symplectic group—we obtain the resolution of unity of coherent states defined on the Kähler manifold which is the product of the n‐dimensional complex plane with the Siegel upper half‐plane.
DOI
10.1063/1.2820972

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The Jacobi group the semidirect product of the Heisenberg group and the symplectic group [1–3] is an important object in quantum mechanics, geometric quantization, optics [4–6]. The Jacobi group is responsible for the squeezed states [7–12] in quantum optics [5], [6], [13], [14]. Some coherent state systems based on SiegelJacobi domains [15], [16] have been investigated in the framework of quantum mechanics, geometric quantization, dequantization, quantum optics, nuclear structure, and signal processing [17], [18], [19], [20][23]. We have attached in [20] Perelomov’s coherent states [24] to the Jacobi group G1∗= H1(R)nSU(1,1), based on the Siegel-Jacobi disk D1 =C×D1, where D1 = {z ∈ C||z| . ez ∈ H denotes the Perelomov coherent state vector indexed by the points z of the homogeneous manifold M = G/H , where H is compact subgroup of G. The generalized squeezed states of the type introduced in [22], [23], [25] are not new [8, 26, 27]. The paper is organized as follows. §2 collects the main notation concerning
In the present series of papers, the coherent states of Sp(2d,R), corresponding to the positive discrete series irreducible representations 〈λd+n/2,...,λ1+n/2〉 encountered in physical applications, are analyzed in detail with special emphasis on those of Sp(4,R) and Sp(6,R). The present paper discusses the unitary-operator coherent states, as defined by Klauder, Perelomov, and Gilmore. These states are parametrized by the points of the coset space Sp(2d,R)/H, where H is the stability group of the Sp(2d,R) irreducible representation lowest weight state, chosen as the reference state, and depends upon the relative values of λ1,...,λd, subject to the conditions λ1≥λ2≥ ⋅ ⋅ ⋅ ≥λd≥0. A parametrization of Sp(2d,R)/H corresponding to a factorization of the latter into a product of coset spaces Sp(2d,R)/U(d) and U(d)/H is chosen. The overlap of two coherent states is calculated, the action of the Sp(2d,R) generators on the coherent states is determined, and the explicit form of the unity resolution relation satisfied by the coherent states in the representation space of the irreducible representation is obtained. The Hilbert space of analytic functions arising from the coherent state representation is studied in detail. Finally, some applications of the formalism developed in the present paper are outlined. In particular, its relevance to the study of boson realizations of the Sp(2d,R) algebra is stressed.
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation TTT of a real algebraic Lie group GGG. This requires defining an inner product on the Hilbert space mathbbH\\mathbb{H}mathbbH that carries the representation TTT. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of TTT, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (mathfrakFmathbbHD\\mathfrak{F}_{\\mathbb{H}_D}mathfrakFmathbbHD​​, mathbbHD\\mathbb{H}_DmathbbHD​, mathfrakFmathfrakHD\\mathfrak{F}^{\\mathfrak{H}_D}mathfrakFmathfrakHD​) of Bargmann spaces. Coherent state wave functions -- the elements of mathfrakFmathbbHD\\mathfrak{F}_{\\mathbb{H}_D}mathfrakFmathbbHD​​ and of mathfrakFmathbbHD\\mathfrak{F}^{\\mathbb{H}_D}mathfrakFmathbbHD​ -- are used to define the inner product on mathbbHD\\mathbb{H}_DmathbbHD​ in a way that simplifies the calculation of the overlaps. This inner product and the group action Gamma\\GammaGamma of GGG on mathfrakFmathbbHD\\mathfrak{F}^{\\mathbb{H}_D}mathfrakFmathbbHD​ are used to formulate expressions for the matrix elements of TTT with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for Gamma\\GammaGamma relies on matrix factorizations in the complexification of GGG even though the representation TTT does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and Gamma\\GammaGamma action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in mathbbH\\mathbb{H}mathbbH. The scalar and vector coherent state methods will both be applied to Sp(nnn) and Sp(n,mathbbRn,\\mathbb{R}n,mathbbR).