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Modern Physics Letters A

TOPOLOGICAL QUANTIZATION OF THE MAGNETIC FLUX

DANIEL WISNIVESKY

Published Aug 10, 2000• 1 citation
Publisher

Abstract

We discuss the quantum problem of a charged particle in a multiply connected region encircling a magnetic tube, using a theory in which space and internal coordinates are derived from the parameters of a linear group of transformations (group space quantum theory). Based only on symmetry considerations, we show that, the magnetic flux in the tube must be quantized in multiples of the Dirac monopole charge.

DOI
10.1142/s021773230000181x

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•Jan 1, 1997•2 citations
We discuss the quantum problem of a charged particle in a multiply connected region encircling a magnetic tube, using a theory in which space and internal coordinates are derived from the parameters of a linear group of transformations (group space quantum theory). Based only on symmetry considerations, we show that, the magnetic flux in the tube must be quantized in multiples of the Dirac monopole charge.
Abstract The problem of quantizing a particle on a two-sphere has been treated by numerous approaches, including Isham’s global method based on unitary representations of a symplectic symmetry group that acts transitively on the phase space. Here we reconsider this simple model using Isham’s scheme, enriched by a magnetic flux through the sphere via a modification of the symplectic form. To maintain complete generality we construct the Hilbert space directly from the symmetry algebra, which is manifestly gauge-invariant, using ladder operators. In this way, we recover algebraically the complete classification of quantizations, and the corresponding energy spectra for the particle. The famous Dirac quantization condition for the monopole charge follows from the requirement that the classical and quantum Casimir invariants match. In an appendix we explain the relation between this approach and the more common one that assumes from the outset a Hilbert space of wave functions that are sections of a nontrivial line bundle over the sphere, and show how the Casimir invariants of the algebra determine the bundle topology.
We discuss the Aharonov-Bohm (A -B) effect and the Dirac (D) monopole of magnetic charge g = 1 2 in the context of bundle theory, which allows to exhibit a deep geometric relation between them.If ξA-B and ξD are the respective U (1)-bundles, we show that ξA-B is isomorphic to the pull-back of ξD induced by the inclusion of the corresponding base spaces.The fact that the A -B effect disappears when the magnetic flux in the solenoid equals an integer number of times the quantum of flux associated with the unit of electric charge, reflects here as a consequence of the pull-back of the Dirac connection in ξD to ξA-B, and the Dirac quantization condition.
We study the quantum mechanics of a charged particle confined to the surface of a cube enclosing a magnetic monopole. The magnetic field is chosen to have a constant magnitude on each face and to point along the outward normal, preserving the rotational symmetry of the cube. We formulate the continuum problem using two gauge patches on the cube surface and show that consistency of the wavefunction gives the Dirac quantization condition. Since an explicit vector potential does not remain invariant under ordinary rotations, we construct gauge-modified rotation operators and use them to classify the eigenstates. Even monopole charges are described by the irreducible representations of the cubic rotation group O, while odd monopole charges require the spinorial representations of the binary octahedral group 2O. We compute the spectrum with a gauge-covariant finite-difference discretization and find Landau-level-like manifolds whose degeneracies are split by the discrete cubic symmetry. We also study the corresponding tight-binding Hofstadter problem on the discretized cube. The resulting spectrum contains the usual magnetic subband structure together with additional gap states localized near the cube corners.
Within the context of infinite-dimensional representations of the rotation group, the Dirac monopole problem is studied in details. We argue that an arbitrary magnetic charge is allowed, and the Dirac quantization condition can be replaced by a generalized quantization rule yielding a new quantum number, the so-called topological spin, which is related to the weight of the Dirac string. We find that the emerging difficulties in explanation of Aharonov-Bohm effect for an arbitrary Dirac string may be removed by introducing nonassociative path-dependent wave functions. The latter provides the absence of the AB effect for an arbitrary magnetic charge. Nevertheless, a singular string can be observed in the quantum-mechanical scattering process.
We discuss the Aharonov-Bohm (A−BA-BA−B) effect and the Dirac (DDD) monopole of magnetic charge g=12g={{1}\over{2}}g=21​ in the context of bundle theory, exhibiting a purely geometric relation between them. If ξA−Bξ_{A-B}ξA−B​ and ξDξ_DξD​ are the respective U(1)U(1)U(1)-bundles, we show that ξA−Bξ_{A-B}ξA−B​ is isomorphic to the pull-back of ξDξ_DξD​ induced by the inclusion of the corresponding base spaces ι:(D02)∗→S2ι:(D_0^2)^*\to S^2ι:(D02​)∗→S2}. The fact that the A−BA-BA−B effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux Φ0=2π∣e∣Φ_0={{2π}\over{\vert e\vert}}Φ0​=∣e∣2π​ associated with the electric charge ∣e∣\vert e\vert∣e∣, reflects here as a consequence of the pull-back by ιιι of the Dirac connection in ξDξ_DξD​ to ξA−Bξ_{A-B}ξA−B​, and the Dirac quantization condition. We also show the necessary vanishing in ξA−Bξ_{A-B}ξA−B​ of the pull-back of the Chern class c1c_1c1​ in ξDξ_DξD​.
On the basis of the principle that topological quantum phases arise from the scattering around space-time defects in higher dimensional unification, a geometric model is presented that associates with each quantum phase an element of a transformation group. 1 Introduction Within a physical theory there are often effects or phenomena studied in idealized form in order to gain insight into the peculiarities of the theory. In Quantum Mechanics one such effect is the appearance of topological quantum phases in the wave functions of particles moving freely in multiply connected space-times the prototype of this effect being the Aharonov-Bohm (AB) effect [1], the appearance of a phase factor in the wave function of an electron which moves around a magnetic flux line. A similar effect is the Aharonov-Casher (AC) effect [2] which is obtained from the AB effect by replacing the flux line and the electron by a charged line and a neutral particle with magnetic moment, respectively. There have al...