Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Coherent states and the classical limit on irreducible representations

Sven Gnutzmann∥, Marek Kuś

Published Dec 11, 1998• 58 citations
Publisher

Abstract

We give an explicit construction of the coherent states for an arbitrary irreducible representation. We also construct the symplectic structure on the manifold of coherent states, find canonical variables and discuss various classical limits of quantum-mechanical systems with relevant observables that obey commutation relations.
DOI
10.1088/0305-4470/31/49/011

Related papers

Coherent states for the unitary symplectic group

Marcel Novaes, José Eduardo Martinho Hornos

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.

Journal of Physics A Mathematical and General•Feb 18, 2004•2 citations

6. Coherent quantum physics

Arnold Neumaier

Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.

De Gruyter eBooks•Oct 24, 2019•0 citations

Coherent states for n-dimensional Euclidean groups E(n) and their application

Christopher John Isham, John R. Klauder

The n-dimensional Euclidean group—the group of rotations and translations of Rn —is the natural canonical group for quantizing a system whose configuration space is the sphere Sn−1, n≥2. Since every unitary irreducible representation of E(n) is nonsquare integrable it follows that coherent states involving the whole group do not exist for such representations. Instead, coherent states are defined based on unitary, reducible representations of the group, and it will be shown that, with such states, quantum dynamics on the sphere admits a path-integral representation involving novel first-order classical actions defined on E(n) as well as integrations over the entire group manifold at each time slice. Irreducibility of the quantization is achieved through a final limiting process.

Journal of Mathematical Physics•Mar 1, 1991•54 citations

SU(N) coherent states and irreducible Schwinger bosons

Manu Mathur, Indrakshi Raychowdhury

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.

arXiv:1007.1510•Journal of Physics A Mathematical and Theoretical•Dec 13, 2010•11 citations

Coherent states and projective representation of the linear canonical transformations

Ingrid C. Daubechies

Using a family of coherent state representations we obtain in a natural and coordinate-independent way an explicit realization of a projective unitary representation of the symplectic group. Dequantization of these operators gives us the corresponding classical functions.

Journal of Mathematical Physics•Jun 1, 1980•23 citations

Spin Coherent States

Didier Robert, Monique Combescure

In this chapter we consider the unit sphere \(\mathbb S^2\) of the Euclidean space \(\mathbb R^3\) with its canonical symplectic structure is a phase space. Then coherent states are labelled by points on \(\mathbb S^2\) and allow us to build a quantization of the two-sphere \(\mathbb S^2\). They are defined in each finite dimensional space of an irreducible unitary representation of the symmetry rotation group SO(3) (or its covering SU(2)) of \(\mathbb S^2\) and give a semiclassical interpretation for the spin. As an application we state the Berezin-Lieb inequalities and compute the thermodynamic limit for large spin systems.

Theoretical and mathematical physics•Jan 1, 2021•0 citations
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.
Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.
The n-dimensional Euclidean group—the group of rotations and translations of Rn —is the natural canonical group for quantizing a system whose configuration space is the sphere Sn−1, n≥2. Since every unitary irreducible representation of E(n) is nonsquare integrable it follows that coherent states involving the whole group do not exist for such representations. Instead, coherent states are defined based on unitary, reducible representations of the group, and it will be shown that, with such states, quantum dynamics on the sphere admits a path-integral representation involving novel first-order classical actions defined on E(n) as well as integrations over the entire group manifold at each time slice. Irreducibility of the quantization is achieved through a final limiting process.
In this chapter we consider the unit sphere S2\mathbb S^2S2 of the Euclidean space R3\mathbb R^3R3 with its canonical symplectic structure is a phase space. Then coherent states are labelled by points on S2\mathbb S^2S2 and allow us to build a quantization of the two-sphere S2\mathbb S^2S2. They are defined in each finite dimensional space of an irreducible unitary representation of the symmetry rotation group SO(3) (or its covering SU(2)) of S2\mathbb S^2S2 and give a semiclassical interpretation for the spin. As an application we state the Berezin-Lieb inequalities and compute the thermodynamic limit for large spin systems.