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Principal Bundles and the Aharonov-Bohm Effect

Victoria Radzikowska

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

This thesis reviews fibre-bundle theory for the purpose of showing its equivalence to gauge theory and, in particular, utilizing fibre bundles to explain the origin of a quantum mechanical effect named the Aharonov-Bohm (AB) effect. Initially, the mathematical preliminaries necessary to review fibre-bundle theory are presented. This includes topological manifolds, smooth manifolds, vector fields and 1-forms on smooth manifolds as well as Lie groups and their Lie algebras. Thereafter, fibre bundles are introduced. In particular, principal bundles, connections on principal bundles, parallel transport and associated bundles are studied. The electrons in the setup of the AB effect experience the effect of an electromagnetic potential and form an interference pattern on the screen they hit. The AB effect is described by a principal U(1)-bundle over R2\{0} and an associated complex line bundle with bundle space P xρ C, where P is the bundle space in the principal bundle and ρ a representation of U(1) on C defined ρ(x) = xq, for x in U(1) and q in Z. The vector potential A, on a simply connected subset U ⊂ R2\{0}, is given by A = ∇χU, for a smooth real-valued function χU. The complex scalar field, i.e. wave function, ψU: U → C of an electron with charge q has a local solution ψU = eiqχUψ0, where ψ0 satisfies the free Schrödinger equation. Consequentially, the holonomy is eiqΦ, where Φ is the total flux of the magnetic field.

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This thesis reviews fibre-bundle theory for the purpose of showing its equivalence to gauge theory and, in particular, utilizing fibre bundles to explain the origin of a quantum mechanical effect named the Aharonov-Bohm (AB) effect. Initially, the mathematical preliminaries necessary to review fibre-bundle theory are presented. This includes topological manifolds, smooth manifolds, vector fields and 1-forms on smooth manifolds as well as Lie groups and their Lie algebras. Thereafter, fibre bundles are introduced. In particular, principal bundles, connections on principal bundles, parallel transport and associated bundles are studied. The electrons in the setup of the AB effect experience the effect of an electromagnetic potential and form an interference pattern on the screen they hit. The AB effect is described by a principal U(1)-bundle over R2{0} and an associated complex line bundle with bundle space P xρ C, where P is the bundle space in the principal bundle and ρ a representation of U(1) on C defined ρ(x) = xq, for x in U(1) and q in Z. The vector potential A, on a simply connected subset U ⊂ R2{0}, is given by A = ∇χU, for a smooth real-valued function χU. The complex scalar field, i.e. wave function, ψU: U → C of an electron with charge q has a local solution ψU = eiqχUψ0, where ψ0 satisfies the free Schrödinger equation. Consequentially, the holonomy is eiqΦ, where Φ is the total flux of the magnetic field.
The AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux  through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc     , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.
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​.
The Aharonov-Bohm (A-B) effect has been a major focus of the foundations of physics. And yet, much confusion persists. In particular, the effect purportedly leads to a dilemma: on one horn, we have a non-local action of a gauge-invariant quantity on charged particles; on the other, we get a local action on these particles, but of a non-gauge invariant quantity. This is the folklore, but the folklore is filled with misconceptions. Here, by deploying a recently defended formulation of gauge theory that dispenses with principal bundles, gauge potentials, and explicit gauge symmetries, I argue, with previous authors, that the A-B effect can be understood gauge-independently. But here my argument will go further: I will show that the A-B effect, when expressed in terms of the covariant derivative of a vector bundle, is entirely analogous to the holonomy of spacetime vectors, and can be understood completely locally. The only surprising idea illustrated by the A-B effect is that, in some circumstances, there is more to the covariant derivative than can be accounted for by the curvature and underlying topology of a vector bundle.
Abstract Chapter 9 discusses the rigorous incorporation of electromagnetism in quantum mechanics via minimal substitution and the inclusion of spin, which leads to the Pauli equation. The gauge invariance of this formulation is inspected more closely, which leads us to understanding quantum mechanics as a gauge theory, and on to fascinating geometric effects such as the Aharonov-Bohm effect and the discovery of geometric phases such as the Berry phase and the Aharonov-Anandan phase. To give a better context to these effects, a compact introduction to the mathematical background of differential geometry is provided, introducing concepts such as differentiable manifolds, tangent and cotangent spaces, and maps between them, as well as from their elements to the real numbers, arriving at tensors and differential forms. With these tools at hand, we first define fibre bundles before formulating the previously discussed geometric phases as well as the so-called Dirac monopole in this language
The Aharonov-Bohm (AB) effect fundamentally demonstrates that electromagnetic potentials, rather than strictly classical fields, play a foundational, observable role in quantum mechanics. While standard quantum theory successfully accommodates this phenomenon via the axiomatic insertion of the minimal coupling principle, a purely geometric interpretation of the gauge potential has historically remained elusive. Building on a previously established complex spacetime framework—where conventional spacetime coordinates are extended into real and imaginary domains—this paper proposes a novel, localized geometric origin for the AB effect. We postulate that the imaginary spatial derivative of the complex wavefunction is physically equivalent to the electromagnetic gauge potential, thereby deriving the minimal coupling prescription directly from spacetime geometry rather than treating it as an independent axiom. Under this framework, the topological phase shift experienced by a charged particle in a field-free region arises naturally from its local evolution through the imaginary dimensions of complex spacetime. Furthermore, by treating the gauge potential as an affine connection or cross-metric term that links real and imaginary dimensions, we successfully redefine the complex metric tensor. This allows the classical electromagnetic field tensor to emerge naturally as the Riemann curvature of the complexified space. This approach provides a unified geometric foundation for both gauge potentials and quantum interference phenomena, resolving previous field-tensor metric limitations while offering a promising mathematical framework for future integrations into relativistic quantum mechanics and cosmic geometry.