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−B) effect and the Dirac (
D) monopole of magnetic charge
g=21 in the context of bundle theory, exhibiting a purely 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
ι:(D02)∗→S2}. The fact that the
A−B effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux
Φ0=∣e∣2π associated with the electric charge
∣e∣, reflects here as a consequence of the pull-back by
ι of the Dirac connection in
ξD to
ξA−B, and the Dirac quantization condition. We also show the necessary vanishing in
ξA−B of the pull-back of the Chern class
c1 in
ξD.
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.
In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, A=A(s,z)ϕ̂, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations.
The quantization procedure of the preceding paper is applied to study two generic topological quantum effects, viz. the charge quantization induced by (abelian) magnetic monopoles, and the Aharonov-Bohm effect. Prior to these applications, a general procedure is given for reducing unitary representations of a locally compact G which are induced by nontrivial unitary representations of H⊂G. This involves the use of spherical trace functions, and is useful in the determination of the eigenfunctions of the Hamiltonian of the particle in a given superselection sector. Such Hamiltonians, implementing the time-evolution on the given abstract C*-algebra, are explicitly constructed and analyzed. The relevant quantum effects are found to be a consequence of the representation theory of the appropriate algebras of observables. In this way a group- and operator-theoretic elucidation of the mathematical structure of the given systems is attempted. This paper may be read independently of its predecessor.