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Journal of Physics A Mathematical and General

Resolutions of the identity in terms of line integrals ofSU(1, 1) coherent states

Apostol Vourdas, Alfred Wünsche

Published Nov 20, 1998• 18 citations
Publisher

Abstract

Resolutions of the identity in terms of contour integrals that involve coherent states and their `complementary states', are derived. The complementary states are auxiliary states that help in the formulation of these resolutions of the identity. Since the coherent states are normalizable inside the unit disc and the complementary states are normalizable outside the unit disc, enlargements of the Hilbert space are considered which allow the construction of resolutions of the identity in terms of contour integrals in rings . Several examples of our formalism are presented.

DOI
10.1088/0305-4470/31/46/024

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Resolutions of the identity in terms of contour integrals that involve coherent states and their `complementary states', are derived. The complementary states are auxiliary states that help in the formulation of these resolutions of the identity. Since the coherent states are normalizable inside the unit disc and the complementary states are normalizable outside the unit disc, enlargements of the Hilbert space are considered which allow the construction of resolutions of the identity in terms of contour integrals in rings . Several examples of our formalism are presented.
A contour representation in the unit disk based on SU(1,1) coherent states is introduced. The scalar product is given by a contour integral. The regions of convergence of the functions representing ket and bra states are studied. An analytic representation in the unit disk is also considered, where the scalar product is represented by an integral over the unit disk, with the Lobachevsky measure. Various relations which connect these analytic functions with other phase-space quantities are derived.
The resolution of the identity is typically utilized as a passive representation tool in standard quantum mechanics textbooks. In this paper, we reinterpret this traditional perspective by introducing the resolution of the identity associated with Glauber coherent states as a direct generator of exact complex Gaussian integral identities. By systematically projecting the continuous coherent-state completeness relation onto the discrete Fock basis, highly generalized Gaussian integral relations emerge naturally as a direct consequence of state preservation, rather than being introduced as independent mathematical postulates. We demonstrate that this overarching formal structure simultaneously dictates both discrete (Kronecker delta) and continuous (Dirac delta) localization kernels under appropriate basis projections, providing a sharp conceptual contrast between the structural features of overcomplete and strictly orthogonal representations. Crucially, we highlight an intriguing non-commuting behavior in the parameter limits of the master identity. Requiring only the elementary properties of coherent states and the Dirac formalism, this approach offers a transparent and visually intuitive illustration of how Hilbert-space completeness inherently encodes vast libraries of exact, solvable mathematical relations -- providing a unifying perspective suitable for advanced undergraduate and graduate instruction.
In the process of generalizing coherent states, the situation when the measure—which is customarily incorporated in their definition—is indeterminate becomes unavoidable. A more dramatic situation may occur if there is no measure which makes the reproducing kernel Hilbert space, involved in the construction of coherent states, isometrically included in an space. Therefore, it appears that there is a need to redefine coherent states making the definition measure-free. Starting out with the reproducing kernel property, we ensure the basic feature of coherent states—resolution of the identity—to be maintained, cf (15) and remark 2. The only investment in the whole undertaking is a sequence (Φn)dn = 0 satisfying (4); the rest, including the aforesaid resolution of the identity, is a consequence of our choice. The approach is supported by examples which make the circumstances clear under which the sequence (Φn)dn = 0 appears; moment problems or rather orthogonal polynomials are one of them.
The usual construction of coherent states allows a wider interpretation in which the number of distinguishing state labels is no longer minimal; the label measure determining the required resolution of unity is then no longer unique and may even be concentrated on manifolds with positive co-dimension. Paying particular attention to the residual restrictions on the measure, the authors choose to capitalize on this inherent freedom and in formally distinct ways, systematically construct suitable sets of extended coherent states which, in a minimal sense, are characterized by auxiliary labels. Interestingly, they find these states lead to path integral constructions containing auxiliary (essentially unconstrained) path-space variables. The impact of both standard and extended coherent state formulations on the content of classical theories is briefly examined, the latter showing the existence of new, and generally constrained, classical variables. Some implications for the handling of constrained classical systems are given, with a complete analysis awaiting further study.