Showing papers on "Coherent states in mathematical physics published in 1989"
TL;DR: This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ in terms of coherent states, focusing on Weyl–Heisenberg coherent states and affine coherent states.
Abstract: This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ in terms of coherent states. Two types of coherent states are considered: Weyl–Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, called ’wavelets,’ which arise as translations and dilations of a single function. In each case it is shown how to represent any function in $L^2 ({\bf R})$ as a sum or integral of these states. Most of the paper is a survey of literature, most notably the work of I. Daubechies, A. Grossmann, and J. Morlet. A few results of the authors are included.
1,121 citations
TL;DR: In this paper, it was shown that the conventional squeezed states of quantum optics, which can be thought of as generalized coherent states for the algebra SU(1,1), are dynamically generated by single-mode hamiltonians characterized by two-photon process interactions.
Abstract: The conventional squeezed states of quantum optics, which can be thought of as generalized coherent states for the algebra SU(1,1), are dynamically generated by single-mode hamiltonians characterized by two-photon process interactions. By the explicit construction of a (highly non-linear) faithful realization of the group $\mathscr G$ of automorphisms of SU(1,1), such hamiltonians are shown to be equivalent — up just to elements of $\mathscr G$ — to that describing quantum mechanically a damped oscillator.
39 citations
TL;DR: In this paper, a generalization of the Perelomov procedure for the construction of coherent states is proposed, which is used to construct coherent states in the carrier spaces of unitary irreducible representations of groups G =S∅V, where V is a vector space and S⊂GL(V).
Abstract: A generalization of the Perelomov procedure for the construction of coherent states is proposed. The new procedure is used to construct systems of coherent states in the carrier spaces of unitary irreducible representations of groups G=S∅V, where V is a vector space and S⊂GL(V). The coherent states are shown to be labeled by the points in cotangent bundles T*O* of orbits O* of S in V*, the dual of V; it is proven that T*O* is a symplectic homogeneous space of G. The generalized procedure for the construction of coherent states presented in this paper is shown to encompass as special cases the constructions known in the literature for the coherent states of the Weyl–Heisenberg, the ‘‘ax+b,’’ and the Galilei and Poincare groups. Moreover, completely new sets of coherent states are constructed for the Euclidean group E(n), where the Perelomov construction fails.
38 citations
18 citations
TL;DR: In this article, generalized coherent states for the one-dimensional harmonic oscillator with maximal symmetry are constructed and the normalization of such states as well as their completeness property are determined and discussed.
Abstract: Generalized coherent states for the one‐dimensional harmonic oscillator with maximal symmetry, i.e., admitting the semidirect sum so(2,1) ⧠ h(2) as the largest invariance Lie algebra pointed out by Niederer are constructed. The normalization of such states as well as their completeness property are determined and discussed. They are also analyzed in the subcontexts of the so(2,1) algebra and of the Heisenberg h(2) algebra. General considerations on Heisenberg relations, on minimal dispersions, and on quantum mechanical entropies are presented in connection with the uncertainty principle.
14 citations
TL;DR: In this article, the possible extension of the notion of generalized coherent state to the case of infinite-dimesional affine Lie algebras is discussed with special attention to the resulting topological structure of the coherent states manifold.
Abstract: The possible extension of the notion of generalized coherent state to the case of infinite-dimesional affine Lie algebras is discussed with special attention to the resulting topological structure of the coherent states manifold, and to its connection with the structure of the algebra. The relevance for the solution of nonlinear dynamical systems equations of motion is briefly reviewed.
5 citations
TL;DR: In this paper, a general framework for vector-like coherent states with the help of nonlinear realization technique as well as with rigged Hilbert space theory is given for the Poincare group and hyperbolic coherent states are found for the SU(1,1) group.
Abstract: An investigation is made of coherent states that differ from the usual ones in two ways: (a) they are connected with the coset space G/H, where the stability subgroup H may be noncompact; and (b) the notion of an H‐invariant ray is replaced by the more general notion of an H‐invariant subspace. A general framework is given for vectorlike coherent states with the help of the nonlinear realization technique as well as with the rigged Hilbert space theory. The vectorlike coherent states are found for the Poincare group and the hyperbolic coherent states are found for the SU(1,1) group.
4 citations
TL;DR: In this article, it was shown that there exists a family of Gaussian wave packets that remain perfectly rigid: while their centre oscillates classically the wave packet rotates around its centre without spreading.
Abstract: It is shown for a two-dimensional oscillator that there exists a family of Gaussian wave packets that remain perfectly rigid: while their centre oscillates classically the wave packet rotates around its centre without spreading. These states include the Glauber coherent states and are a subset of the Perelomov coherent states.
3 citations
TL;DR: In this paper, the authors analyse a possible connection between different quantization schemes and the Bargman-Segal realisation of the Heisenberg algebra H. They show that only a one-parameter subfamily of the family of H(+)Algebras can be rewritten in BargmanSegal form.
Abstract: The authors analyse a possible connection between different quantisation schemes and the Bargman-Segal realisation of the Heisenberg algebra H. They show that only a one-parameter subfamily of the family of Heisenberg algebras HQ subduced from H(+)H can be rewritten in the Bargman-Segal form.
3 citations
TL;DR: In this article, the applicability of the method from the point of view of the time evolution of spin coherent states was examined from the perspective of spin spin states in the Landau-Lifshitzitz model.
Abstract: Method of spin coherent states has been recently extensively applied to different Heisenberg models in order to formulate a correct procedure of the reduction of quantum models to continual Landau-Lifshitz model. This paper deals with the applicability of the method from the point of view of the time evolution of spin coherent states.
2 citations
TL;DR: In this article, the authors formulated the N-representability problem in the context of group-related coherent states, and derived a solution to the problem in terms of a set of positive distributions of one-particle spaces.
Abstract: The N -representability problem is reviewed and formulated for the first time in the context of group-related coherent states. The particular family of states chosen is based on the group of special unitary transformations of one-particle space. Density operators are analyzed in terms of their P and Q symbols associated with these states, and the holomorphic representations of quantum mechanics that they induce. The time-independent Schrodinger equation for the ground state of an electronic system is expressed as a variational problem over a set of positive distributions. This set of distributions solves in an abstract sense the N -representability problem. The computational feasibility of the solution is discussed.