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arXiv:1007.1510 · Journal of Physics A Mathematical and Theoretical

SU(N) coherent states and irreducible Schwinger bosons

Manu Mathur, Indrakshi Raychowdhury

Published Dec 13, 2010• 11 citations
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Abstract

We exploit the SU( N ) irreducible Schwinger boson to construct SU( N ) coherent states. This construction of an SU( N ) coherent state is analogous to the construction of the simplest Heisenberg–Weyl coherent states. The coherent states belonging to irreducible representations of SU( N ) are labeled by the eigenvalues of ( N − 1) SU( N ) Casimir operators and characterized by ( N − 1) complex orthonormal vectors describing the SU( N ) group manifold.

DOI
10.1088/1751-8113/44/3/035203

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We exploit the SU( N ) irreducible Schwinger boson to construct SU( N ) coherent states. This construction of an SU( N ) coherent state is analogous to the construction of the simplest Heisenberg–Weyl coherent states. The coherent states belonging to irreducible representations of SU( N ) are labeled by the eigenvalues of ( N − 1) SU( N ) Casimir operators and characterized by ( N − 1) complex orthonormal vectors describing the SU( N ) group manifold.
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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.
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The Schwinger oscillator operator representation of SU(3), studied in a previous paper from the representation theory point of view, is analysed to discuss the intimate relationships between standard oscillator coherent state systems and systems of SU(3) coherent states. Both SU(3) standard coherent states, based on choice of highest weight vector as fiducial vector, and certain other specific systems of generalised coherent states, are found to be\nrelevant. A complete analysis is presented, covering all the oscillator coherent states without exception, and amounting to SU(3) harmonic analysis of these states.\n\n
Considering the present situation of Lie-algebraic studies of high-temperature superconductivity, a Schwinger-type boson realization of the su(4) algebra is investigated. The basic framework comes from a possible form of the su(n) and the su(n,1) algebras in the Schwinger representation, which was recently proposed by two of the present authors (M. Y. & A. K.), together with J. da Providência. The form investigated in this paper is, in some sense, a natural extension of the Schwinger boson representation for the so(5), so(4) and su(2)⊗su(2) algebras, which was also proposed by the present authors. The orthogonal set for the irreducible representation of the su(4) algebra is completely analogous to that expressed in terms of fermion operators for pairing correlation.