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Journal of Mathematical Physics

The Jordan–Schwinger representations of Cayley–Klein groups. II. The unitary groups

N. A. Gromov, Vladimir I. Man’ko

Published May 1, 1990• 16 citations
Publisher

Abstract

The unitary Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the special unitary groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

DOI
10.1063/1.528782

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The unitary Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the special unitary groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.
We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.
Abstract This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.
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).