The oscillator cranking-model wavefunction can be identified with a set of coherent states of the symplectic group Sp(4,R). A quantum-classical correspondence for the model is, however, established through a set of coherent states of the Heisenberg symplectic group N(2)(X)Sp(4,R). The properties of these states and their usefulness in various fields of physics are studied in detail. Interesting and useful generalisations of these coherent states are also discussed.
The oscillator cranking-model wavefunction can be identified with a set of coherent states of the symplectic group Sp(4,R). A quantum-classical correspondence for the model is, however, established through a set of coherent states of the Heisenberg symplectic group N(2)(X)Sp(4,R). The properties of these states and their usefulness in various fields of physics are studied in detail. Interesting and useful generalisations of these coherent states are also discussed.
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.
We present an explicit construction of coherent states for an arbitrary irreducible representation of the unitary symplectic group USp (4). Three different families of coherent states are obtained, corresponding to the subgroups U (1) × U (1), U (2) and SU (2) × SU (2). The symplectic structure on the manifold of coherent states is obtained, and canonical coordinates are used to express the classical limit of quantum observables. One of the families is seen to provide a trivial classical limit.
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 annihilation-operator coherent states, i.e., the eigenstates of the noncompact lowering generators corresponding to complex eigenvalues. These states generalize the coherent states introduced by Barut and Girardello for Sp(2,R), and later on extended by Deenen and Quesne to the Sp(2d,R) irreducible representations of the type 〈(λ+n/2)d〉. When λ1,...,λd are not all equal, it was shown by Deenen and Quesne that the eigenvalues do not completely specify the eigenstates of the noncompact lowering generators. In the present work, their characterization is completed by a set of continuous labels parametrizing the (unitary-operator) coherent states of the maximal compact subgroup U(d). The resulting coherent states are therefore of mixed type, being annihilation-operator coherent states only as regards the noncompact generators. A realization in a subspace of a Bargmann space of analytic functions shows that such coherent states satisfy a unity resolution relation in the representation space of 〈λd +n/2,...,λ1+n/2〉, and therefore may be used as a continuous basis in such space. The analytic functions and the differential operators representing the representation space discrete bases and the Sp(2d,R) generators, respectively, are found in explicit form. It is concluded that the annihilation-operator coherent state representation provides the mathematical foundation for the use of differentiation operators with respect to the noncompact raising generators in symbolic expressions of the Sp(2d,R) generators. This is to be compared with the habit of replacing a boson annihilation operator by a symbolic differentiation with respect to the corresponding creation operator, accounted for by the Bargmann representation of such operators.
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.