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Physical Review A

Resolutions of the identity in terms of SU(2) coherent states and their use for quantum-state engineering

Apostol Vourdas

Published Nov 1, 1996• 14 citations
Publisher

Abstract

Resolutions of the identity in terms of line integrals that involve SU(2) coherent states and their complementary states are presented. They are used for the expansion of various states in terms of SU(2) coherent states on a line. The properties of the complementary states are also studied. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.4544

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Resolutions of the identity in terms of line integrals that involve SU(2) coherent states and their complementary states are presented. They are used for the expansion of various states in terms of SU(2) coherent states on a line. The properties of the complementary states are also studied. \textcopyright{} 1996 The American Physical Society.
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.
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.
The SU(2) coherent-state path integral is used to represent the matrix element of a propagator in the SU(2) coherent-state basis. It is argued that the continuum representation of this integral is correct provided the necessary boundary term is taken into account. In the case of the SU(2) dynamical symmetry the path integral is explicitly computed by means of a change of variables, the SU(2) motion of the underlying phase space. The correct stationary-phase expansion for the propagator in terms of the total action including boundary term and classical trajectories is obtained.
The authors examine several topical subjects, commencing with a general introduction to path integrals in quantum mechanics and the group theoretical backgrounds for path integrals. Applications of harmonic analysis, polar coordinate formulation, various techniques and path integrals on SU(2) and SU(1, 1) are discussed. Soluble examples presented include particle-flux system, a pulsed oscillator, magnetic monopole, the Coulomb problem in curved space and others.The second part deals with the SU(2) coherent states and their applications. Construction and generalization of the SU(2) coherent sta
A resolution of the identity due to canonical coherent states is often proven in the weak operator topology. However, such a resolution with an integral symbol is typically supposed to hold in the strong operator topology associated with the framework of the spectral theorem. We provide an elementary proof of the strong convergence for the resolution of the identity due to canonical coherent states starting with a mostly familiar setup. Further, we enjoy a different proof and show that the relevant uniform limit does not exist.