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International Journal of Modern Physics B

MAGNETIC GROUP ON THE PSEUDOSPHERE

Vincenzo Barone A, Vittorio Penna, Pasquale Sodano

Published Nov 10, 1992• 3 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979292001559

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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.
We consider a particle moving on a 2-sphere in the presence of a constant magnetic field. Building on our earlier work in the nonmagnetic case we construct coherent states for this system. The coherent states are labeled by points in the associated phase space, the (co)tangent bundle of S 2 . They are constructed as eigenvectors for certain annihilation operators and expressed in terms of a certain heat kernel. These coherent states are not of Perelomov type but rather are constructed according to the ‘complexifier’ approach of Thiemann. We describe the Segal–Bargmann representation associated with the coherent states which is equivalent to a resolution of the identity. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
We have seen before that spin coherent states are strongly linked with the algebraic and geometric properties of the Euclidean 2-sphere S2\mathbb S^2S2. We shall now consider analogue setting when the sphere is replaced by the 2-pseudosphere i.e. the Euclidean metric in R3\mathbb R^3R3 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.
We find that in a supersymmetric quantum mechanics (SUSY QM) system, in addition to supersymmetric algebra, an associated SU(2) algebra can be obtained by using semiunitary (SUT) operator and projection operator, and the relevant constants of motion can be constructed. Two typical quantum systems are investigated as examples to demonstrate the above finding. The first example is the quantum system of a nonrelativistic charged particle moving in x-y plane and coupled to a magnetic field along z-axis. The second example is provided with the Dirac particle in a magnetic field. Similarly there exists an SU τ (2) ⊗ SU σ (2) symmetry in the context of the relativistic Pauli Hamiltonian squared. We show that there exists also an SU(2) symmetry associated with the supersymmetry of the Dirac particle.
The (group and spin space) matrix Hamiltonian describing the dynamics of a nonrelativistic spin 1/2 particle moving in a static, but spatially dependent, non-Abelian magnetic field in two spatial dimensions is shown to take the form of an anticommutator of a nilpotent operator and its hermitian conjugate. Consequently, the (group space) matrix Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra is exploited to explicitly construct the exact zero energy ground state wavefunction(s) for the system. The remaining eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs.