Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

On Quantum Effects of Vector Potentials and Generalizations of Functional Analysis

Ismael L. Paiva

Published May 1, 2020• 0 citations
Publisher

Abstract

This is a dissertation in two parts. In the first one, the Aharonov-Bohm effect is investigated. It is shown that solenoids (or flux lines) can be seen as barriers for quantum charges. In particular, a charge can be trapped in a sector of a long cavity by two flux lines. Also, grids of flux lines can approximate the force associated with continuous two-dimensional distributions of magnetic fields. More, if it is assumed that the lines can be as close to each other as desirable, it is explained how the classical magnetic force can emerge from the Aharonov-Bohm effect. Continuing, the quantization of the source of the magnetic field, and not just of the degrees of freedom of the particle interacting with it, is considered. Special attention is given to the cases where the source has a relatively small spreading and is post-selected. As it will be discussed, in those cases, the weak value plays a role in the determination of the effective vector potential "experienced" by the particle. In the second part of this work, notions from functional analysis are extended to Banach algebras and completions of Grassmann algebras. A notion of analyticity is given to the functions of a single Banach algebra variable. This notion allows the introduction of holomorphic polynomials, power series, and rational functions. With that, the analogous of Hilbert spaces of power series are also considered. Finally, closures of Grassmann algebras with respect to the 1 and the 2-norms are explored. The analogous of the complex analysis in the open disk or a half-plane (usually referred to as Schur analysis) is presented in the 1-norm closure. Also, a Wiener-like algebra, interpolation problems, and a process known as the Schur algorithm are studied in this setting. Now, an inner product between two elements can be introduced in the 2-norm closure, revealing similarities between this space and the non-commutative Fock-Bargamann-Segal space. It is, then, defined a class of stochastic processes. To conclude, the derivatives of these processes are analyzed in an analogous of the space of stochastic distributions.

DOI
10.36837/chapman.000159

Related papers

Quantized vector potential and alternative views of the magnetic Aharonov-Bohm phase shift

Philip Pearle, Anthony Rizzi

We give a complete quantum analysis of the Aharonov-Bohm (AB) magnetic phase shift involving three entities: the electron, the charges constituting the solenoid current, and the vector potential. The usual calculation supposes that the solenoid's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized electron moving in this vector potential. Recently, Vaidman presented a semiclassical calculation [L. Vaidman, Phys. Rev. A 86, 040101 (2012)], later confirmed by a fully quantum calculation of Pearle and Rizzi [preceding paper, Phys. Rev. A 95, 052123 (2017)], where it is supposed that the electron's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized solenoid charges moving in this vector potential. Here we present a third calculation, which supposes that the electron and solenoid currents may be well approximated as classical sources. The AB phase shift is then shown to be acquired by the quantized vector potential. We next show these are three equivalent alternative ways of calculating the AB shift. We consider the exact problem where all three entities are quantized. We approximate the wave function as the product of three wave functions: a vector potential wave function, an electron wave function, and a solenoid wave function. We apply the variational principle for the exact Schr\"odinger equation to this approximate form of solution. This leads to three Schr\"odinger equations, one each for vector potential, electron, and solenoid, each with classical sources for the other two entities. However, each Schr\"odinger equation contains an additional real $c$-number term, the time derivative of an extra phase. We show that these extra phases are such that the phase of the total wave function produces the AB shift. Since none of the three entities requires different treatment from any of the others, this leads to three alternative views of the physical cause of the AB magnetic effect.

arXiv:1605.04324•Physical Review A•May 25, 2017•28 citations

The Aharonov-Bohm Effect: An Exploration of Quantum Interference and Electromagnetic Potentials

Bahman Zohuri

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.

Journal of Energy and Power Engineering•Jun 28, 2024•1 citation

Finite and half-infinite solenoids and the Aharonov-Bohm effect

Joel Franklin, Gopal Goel

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.

American Journal of Physics•Oct 18, 2019•2 citations

Theoretical models and simulations for Aharonov-Bohm effect

Li, Yuzhe

The Aharonov-Bohm (AB) effect demonstrates that charged particles can acquire a phase shift from a magnetic vector potential, even when propagating through regions devoid of magnetic fields. This dissertation develops time-domain numerical simulators to systematically investigate this effect across 2D and 3D coordinate systems. In 2D models, the Schrodinger equation is integrated using the Crank-Nicolson scheme. The Cartesian simulation reproduces flux-dependent fringe shifts, establishes grid resolution criteria, and demonstrates the necessity of the Coulomb gauge to avoid singular gauge artifacts. A 2D cylindrical model then verifies the coordinate-independence of the simulated AB phase shift. For 3D systems, a Strang-splitting scheme is introduced. Applied to a finite-length solenoid with elliptic-integral vector potentials, it utilizes FFT-based azimuthal diagonalization to isolate the pure AB phase inversion by initializing the wavepacket at the equatorial plane under small magnetic flux. Under fringing fields, the scheme captures the transition to classical Lorentz deflection, illustrating the quantum-classical correspondence. Finally, a spherical-coordinate model featuring a spherical-shell barrier employs a regularized wavefunction 𝑢 = 𝑟𝜓 to efficiently capture pre-slit phase inversions. To summarize, these platforms demonstrate the AB effect and provide a versatile framework for investigating quantum interference in realistic 3D geometries.

•Jan 1, 2026•0 citations

Magnetic forces in the absence of a classical magnetic field

Ismael L. Paiva, Yakir Aharonov, Jeff Tollaksen +1 more

It is shown that, in some cases, the effect of discrete distributions of flux lines in quantum mechanics can be associated with the effect of continuous distributions of magnetic fields with special symmetries. In particular, flux lines with an arbitrary value of magnetic flux can be used to create energetic barriers, which can be used to confine quantum systems in specially designed configurations. This generalizes a previous work where such energy barriers arose from flux lines with half-integer fluxons. Furthermore, it is shown how the Landau levels can be obtained from a two-dimensional grid of flux lines. These results suggest that the classical magnetic force can be seen as emerging entirely from the Aharonov-Bohm effect. Finally, the basic elements of a semiclassical theory that models the emergence of classical magnetic forces from fields with special symmetries are introduced.

arXiv:2001.10172•Physical Review A•Apr 14, 2020•4 citations

Principal Bundles and the Aharonov-Bohm Effect

Victoria Radzikowska

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.

•Jan 1, 2026•0 citations
This is a dissertation in two parts. In the first one, the Aharonov-Bohm effect is investigated. It is shown that solenoids (or flux lines) can be seen as barriers for quantum charges. In particular, a charge can be trapped in a sector of a long cavity by two flux lines. Also, grids of flux lines can approximate the force associated with continuous two-dimensional distributions of magnetic fields. More, if it is assumed that the lines can be as close to each other as desirable, it is explained how the classical magnetic force can emerge from the Aharonov-Bohm effect. Continuing, the quantization of the source of the magnetic field, and not just of the degrees of freedom of the particle interacting with it, is considered. Special attention is given to the cases where the source has a relatively small spreading and is post-selected. As it will be discussed, in those cases, the weak value plays a role in the determination of the effective vector potential "experienced" by the particle. In the second part of this work, notions from functional analysis are extended to Banach algebras and completions of Grassmann algebras. A notion of analyticity is given to the functions of a single Banach algebra variable. This notion allows the introduction of holomorphic polynomials, power series, and rational functions. With that, the analogous of Hilbert spaces of power series are also considered. Finally, closures of Grassmann algebras with respect to the 1 and the 2-norms are explored. The analogous of the complex analysis in the open disk or a half-plane (usually referred to as Schur analysis) is presented in the 1-norm closure. Also, a Wiener-like algebra, interpolation problems, and a process known as the Schur algorithm are studied in this setting. Now, an inner product between two elements can be introduced in the 2-norm closure, revealing similarities between this space and the non-commutative Fock-Bargamann-Segal space. It is, then, defined a class of stochastic processes. To conclude, the derivatives of these processes are analyzed in an analogous of the space of stochastic distributions.
We give a complete quantum analysis of the Aharonov-Bohm (AB) magnetic phase shift involving three entities: the electron, the charges constituting the solenoid current, and the vector potential. The usual calculation supposes that the solenoid's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized electron moving in this vector potential. Recently, Vaidman presented a semiclassical calculation [L. Vaidman, Phys. Rev. A 86, 040101 (2012)], later confirmed by a fully quantum calculation of Pearle and Rizzi [preceding paper, Phys. Rev. A 95, 052123 (2017)], where it is supposed that the electron's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized solenoid charges moving in this vector potential. Here we present a third calculation, which supposes that the electron and solenoid currents may be well approximated as classical sources. The AB phase shift is then shown to be acquired by the quantized vector potential. We next show these are three equivalent alternative ways of calculating the AB shift. We consider the exact problem where all three entities are quantized. We approximate the wave function as the product of three wave functions: a vector potential wave function, an electron wave function, and a solenoid wave function. We apply the variational principle for the exact Schr\"odinger equation to this approximate form of solution. This leads to three Schr\"odinger equations, one each for vector potential, electron, and solenoid, each with classical sources for the other two entities. However, each Schr\"odinger equation contains an additional real ccc-number term, the time derivative of an extra phase. We show that these extra phases are such that the phase of the total wave function produces the AB shift. Since none of the three entities requires different treatment from any of the others, this leads to three alternative views of the physical cause of the AB magnetic effect.
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.
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 Aharonov-Bohm (AB) effect demonstrates that charged particles can acquire a phase shift from a magnetic vector potential, even when propagating through regions devoid of magnetic fields. This dissertation develops time-domain numerical simulators to systematically investigate this effect across 2D and 3D coordinate systems. In 2D models, the Schrodinger equation is integrated using the Crank-Nicolson scheme. The Cartesian simulation reproduces flux-dependent fringe shifts, establishes grid resolution criteria, and demonstrates the necessity of the Coulomb gauge to avoid singular gauge artifacts. A 2D cylindrical model then verifies the coordinate-independence of the simulated AB phase shift. For 3D systems, a Strang-splitting scheme is introduced. Applied to a finite-length solenoid with elliptic-integral vector potentials, it utilizes FFT-based azimuthal diagonalization to isolate the pure AB phase inversion by initializing the wavepacket at the equatorial plane under small magnetic flux. Under fringing fields, the scheme captures the transition to classical Lorentz deflection, illustrating the quantum-classical correspondence. Finally, a spherical-coordinate model featuring a spherical-shell barrier employs a regularized wavefunction 𝑢 = 𝑟𝜓 to efficiently capture pre-slit phase inversions. To summarize, these platforms demonstrate the AB effect and provide a versatile framework for investigating quantum interference in realistic 3D geometries.
It is shown that, in some cases, the effect of discrete distributions of flux lines in quantum mechanics can be associated with the effect of continuous distributions of magnetic fields with special symmetries. In particular, flux lines with an arbitrary value of magnetic flux can be used to create energetic barriers, which can be used to confine quantum systems in specially designed configurations. This generalizes a previous work where such energy barriers arose from flux lines with half-integer fluxons. Furthermore, it is shown how the Landau levels can be obtained from a two-dimensional grid of flux lines. These results suggest that the classical magnetic force can be seen as emerging entirely from the Aharonov-Bohm effect. Finally, the basic elements of a semiclassical theory that models the emergence of classical magnetic forces from fields with special symmetries are introduced.
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.