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Reviews in Mathematical Physics

QUANTIZATION AND SUPERSELECTION SECTORS II: DIRAC MONOPOLE AND AHARONOV-BOHM EFFECT

Nicolaas P. Landsman

Published Jan 1, 1990• 20 citations

Abstract

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.

DOI: 10.1142/s0129055x90000041

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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.

Quantization is defined as the act of assigning an appropriate C*-algebra [Formula: see text] to a given configuration space Q, along with a prescription mapping self-adjoint elements of [Formula: see text] into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here [Formula: see text] is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of [Formula: see text]. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of [Formula: see text].

We discuss the Aharonov-Bohm (

A-B

) effect and the Dirac (

D

) monopole of magnetic charge

g={{1}\over{2}}

in the context of bundle theory, exhibiting a purely geometric relation between them. If

ξ_{A-B}

and

ξ_D

are the respective

U(1)

-bundles, we show that

ξ_{A-B}

is isomorphic to the pull-back of

ξ_D

induced by the inclusion of the corresponding base spaces

ι:(D_0^2)^*\to S^2

}. The fact that the

A-B

effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux

Φ_0={{2π}\over{\vert e\vert}}

associated with the electric charge

\vert e\vert

, reflects here as a consequence of the pull-back by

ι

of the Dirac connection in

ξ_D

to

ξ_{A-B}

, and the Dirac quantization condition. We also show the necessary vanishing in

ξ_{A-B}

of the pull-back of the Chern class

c_1

in

ξ_D

.

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.

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.