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arXiv:quant-ph/0303146 · Journal of Physics A Mathematical and General

Coherent states withSU(2) andSU(3) charges

Manu Mathur, Samir K Paul

Published Aug 23, 2005• 1 citation
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Abstract

We define coherent states carrying SU (2) charge by exploiting the Schwinger boson representation of the SU (2) Lie algebra. These coherent states satisfy the continuity property and provide resolution of identity on S 3 . We further generalize these techniques to construct the corresponding SU (3) charge coherent states. The SU ( N ) extension is also discussed.

DOI
10.1088/0305-4470/38/36/005

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We define coherent states carrying SU (2) charge by exploiting the Schwinger boson representation of the SU (2) Lie algebra. These coherent states satisfy the continuity property and provide resolution of identity on S 3 . We further generalize these techniques to construct the corresponding SU (3) charge coherent states. The SU ( N ) extension is also discussed.
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We define coherent states for SU(3) using six bosonic creation and annihilation operators. These coherent states are explicitly characterized by six complex numbers with constraints. For the completely symmetric representations (n,0) and (0,m), only three of the bosonic operators are required. For mixed representations (n,m), all six operators are required. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties. We introduce an explicit parameterization of the group SU(3) and the corresponding integration measure. Finally, we discuss the path integral formalism for a problem in which the Hamiltonian is a function of SU(3) operators at each site.
We define coherent states for SU(3) using six bosonic creation and annihilation operators. These coherent states are explicitly characterized by six complex numbers with constraints. For the completely symmetric representations (n,0) and (0,m), only three of the bosonic operators are required. For mixed representations (n,m), all six operators are required. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties. We introduce an explicit parametrization of the group SU(3) and the corresponding integration measure. Finally, we discuss the path integral formalism for a problem in which the Hamiltonian is a function of SU(3) operators at each site.
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.