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Journal of Mathematical Physics

Coherent states and Lie algebras

F. T. Hioe

Published Aug 1, 1974• 16 citations
Publisher

Abstract

The Bloch coherent states for a spin or a system of spins and the Glauber coherent states for bosons are examined from the viewpoint of Lie algebras. It is pointed out that the Bloch coherent states are vectors in the space spanned by the basis functions for an irreducible representation of the unitary unimodular group SU(2), and that the Glauber coherent states are vectors in the space spanned by the basis functions for the infinite-dimensional irreducible representation of a contracted group of SU(2). A deeper understanding of many of the useful properties of these coherent states is gained.

DOI
10.1063/1.1666788

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The Bloch coherent states for a spin or a system of spins and the Glauber coherent states for bosons are examined from the viewpoint of Lie algebras. It is pointed out that the Bloch coherent states are vectors in the space spanned by the basis functions for an irreducible representation of the unitary unimodular group SU(2), and that the Glauber coherent states are vectors in the space spanned by the basis functions for the infinite-dimensional irreducible representation of a contracted group of SU(2). A deeper understanding of many of the useful properties of these coherent states is gained.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.
The application of Glauber's definition (1963) of quantum coherence allows the introduction of neutron fields of partial coherence under the assumption that the complete occupation number space is a direct product of Fermi subspaces. Excitations are described by 'collective' creation and annihilation operators which span an algebra isomorphic to the sl(2,R) algebra. The associated coherent states are of partial coherence and the finite dimensional representation of the su(2,R) algebra is not regularly coherent. In contrast, the transition to an infinite dimensional representation space results in regular coherent properties of the field. This is demonstrated using the representation space of the su(1,1) algebra which has a real isomorphism to the sl(2,R) algebra. The coherent states calculated from Glauber's condition for coherence are completely coherent, and are, moreover, identical to those found by Barut and Girardello (1971) in starting from a far more abstract argument.
If the state space of a quantum system is finite dimensional it may be considered as the carrier space of an irreducible unitary representation of some compact Lie group G. In this case the set of operators may be put in one-to-one correspondence with a class of functions defined on the group parameters (Q representation). For these functions a binary relation is derived that corresponds to the product of two operators. The general formalism is applied to G=SU(2) (spin coherent states) and this result is used to derive a product formula for the non-compact Heisenberg-Weyl group (coherent oscillator states.).
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 review is devoted to an analysis of definite overcomplete non-orthogonal state systems that are connected with irreducible representations of Lie groups–the so called systems of generalized coherent states. These systems, which the author is the first to propose, are generalizations of Glauber's coherentstate system and arise in natural fashion in physical problems that have dynamic symmetry. They permit a considerable simplification of the solution of the quantum problem by reducing it to a simpler classical problem. The review deals with the properties of generalized-coherent-state systems connected with the simplest Lie groups.