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Theoretical and mathematical physics

Pseudo-Spin Coherent States

Didier Robert, Monique Combescure

Published Jan 1, 2021• 0 citations
Publisher

Abstract

We have seen before that spin coherent states are strongly linked with the algebraic and geometric properties of the Euclidean 2-sphere \(\mathbb S^2\). We shall now consider analogue setting when the sphere is replaced by the 2-pseudosphere i.e. the Euclidean metric in \(\mathbb R^3\) is replaced by the Minkowski metric and the group SO(3) by the symmetry group SO(2, 1). The main big difference with the Euclidean case is that in the Minkowski case the pseudosphere and its symmetry groupe SO(2, 1) are not compact and that all irreducible unitary representations of SO(2, 1) are infinite dimensional.

DOI
10.1007/978-3-030-70845-0_8

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We have seen before that spin coherent states are strongly linked with the algebraic and geometric properties of the Euclidean 2-sphere (\mathbb S^2). We shall now consider analogue setting when the sphere is replaced by the 2-pseudosphere i.e. the Euclidean metric in (\mathbb R^3) is replaced by the Minkowski metric and the group SO(3) by the symmetry group SO(2, 1). The main big difference with the Euclidean case is that in the Minkowski case the pseudosphere and its symmetry groupe SO(2, 1) are not compact and that all irreducible unitary representations of SO(2, 1) are infinite dimensional.
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.
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.
Understanding the geometry of state space is of fundamental interest in the field of quantum information, in particular to study entanglement. The states of a d dimensional system form a convex set in the d2−1 dimensional Euclidean space Rd2−1. For a two dimensional system this convex set is nothing but a solid unit sphere (Bloch sphere) with centre at the origin. The surface S of this unit sphere consists of all the pure states of the system. Every unitary transformation U ∈ SU(2) of the Hilbert space corresponds to an associated SO(3) rotation of this space. This kind of correspondence becomes much richer when we go to higher dimension. For example, in the d = 3 case the corresponding unitary group is the eight parameter group SU(3) whereas the rotation group in R is the 28 parameter group SO(8). Therefore, not every point on or inside the sphere S in R will correspond to a state of the system. So, not every rotation in R will correspond to a valid unitary transformation of the 3 dimensional Hilbert space. This gives us a fairly complex convex object as the set of all states. In order to gain insight into the structure of this state space, we study its two dimensional and three dimensional cross-sections. In particular, we describe in detail the discrete symmetries of a non-trivial 3-section of the convex set of states (density matrices) for three dimensional systems. This group turns out to be the same as that of a tetrahedron.
The quantum mechanics of a particle moving on a pseudosphere under the action of a constant magnetic field is studied from an algebraic point of view. The magnetic group on the pseudosphere is SU(1, 1). The Hilbert space for the discrete part of the spectrum is investigated. The eigenstates of the non-compact operators (the hyperbolic magnetic translators) are constructed and shown to be expressible as continuous superpositions of coherent states. The planar limit of both the algebra and the eigenstates is analyzed. Some possible applications are briefly outlined.
Some basics of su(1,1)su(1,1)su(1,1)
A basic introduction to the su(1,1)su(1,1)su(1,1) algebra is presented, in which we discuss the relation with canonical transformations, the realization in terms of quantized radiation field modes and coherent states. Instead of going into details of these topics, we rather emphasize the existing connections between them. We discuss two parametrizations of the coherent states manifold SU(1,1)/U(1)SU(1,1)/U(1)SU(1,1)/U(1): as the Poincar{é} disk in the complex plane and as the pseudosphere (a sphere in a Minkowskian space), and show that it is a natural phase space for quantum systems with SU(1,1) symmetry.