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Journal of Mathematical Physics

SU(2) coherent states on an m-sheeted covering of the sphere

Apostol Vourdas

Published Sep 1, 1995• 5 citations
Publisher

Abstract

SU(2) coherent states on an m-sheeted covering of the sphere are introduced, and properties like overcompleteness and resolution of the identity are studied. Operators J+(m), J−(m), Jz(m) that obey the SU(2) algebra but shift the ‖jn〉 states by m steps are considered, and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the standard SU(2) coherent states with respect to the usual angular momentum operators J+, J−, Jz. An extended SU(2) Bargmann representation on an m-sheeted sphere is introduced in which the transformations of an m-sheeted covering of the SU(2) group are implemented as extended Mobius conformal mappings. The formalism is applied in the study of Hamiltonians that contain the operators J+(m), J−(m), Jz(m) and describe processes where the state of a particle is shifted by m steps. The method is also used in the context of both the Holstein–Primakoff and Schwinger representation of SU(2).

DOI
10.1063/1.530918

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SU(2) coherent states on an m-sheeted covering of the sphere are introduced, and properties like overcompleteness and resolution of the identity are studied. Operators J+(m), J−(m), Jz(m) that obey the SU(2) algebra but shift the ‖jn〉 states by m steps are considered, and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the standard SU(2) coherent states with respect to the usual angular momentum operators J+, J−, Jz. An extended SU(2) Bargmann representation on an m-sheeted sphere is introduced in which the transformations of an m-sheeted covering of the SU(2) group are implemented as extended Mobius conformal mappings. The formalism is applied in the study of Hamiltonians that contain the operators J+(m), J−(m), Jz(m) and describe processes where the state of a particle is shifted by m steps. The method is also used in the context of both the Holstein–Primakoff and Schwinger representation of SU(2).
Coherent states on the m-sheeted complex plane are introduced and properties like overcompleteness and resolution of the identity are studied. Operators am+, am which create and annihilate clusters of m particles are considered and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the ordinary (Glauber) coherent states with respect to a+, a. An extended Bargmann representation in the m-sheeted complex plane and P and Q representations with respect to our coherent states, are studied. Applications of this formalism in the study of Hamiltonians that describe m-particle clustering are also considered.
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.
Susskind–Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coherent state quantization map. In this regard, the modified Susskind–Glogower coherent states have been introduced as an alternative family of states that resolve the identity resolution. In this paper, the quantization map related to the modified Susskind–Glogower coherent states is exploited, which naturally leads to a particular representation of the su(1,1) Lie algebra in its discrete series. The latter provides evidence about further generalizations of coherent states, built from the Susskind–Glogower ones by extending the indices of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind. In this form, the new families of Susskind–Glogower-I and Susskind–Glogower-II coherent states are introduced. The corresponding quantization maps are constructed so that they lead to general representations of elements of the su(1,1) and su(2) Lie algebras as generators of the SU(1, 1) and SU(2) unitary irreducible representations, respectively. For completeness, the optical properties related to the new families of coherent states are explored and compared with respect to some well-known optical states.