Topology of the Aharonov-Bohm effect in different reference frames

TL;DR

The paper frames the Aharonov-Bohm (AB) effect as a spacetime flux phenomenon, deriving a covariant AB phase expression and highlighting its topological nature. It then analyzes magnetic and electric AB configurations in different inertial frames, showing that while the total AB phase is Lorentz-invariant, the individual electric and magnetic flux contributions are frame-dependent, with cases where one contribution vanishes and the phase is carried entirely by the other. This leads to frames where a so-called magnetic (or electric) AB effect is observed entirely through the electric (or magnetic) flux, underscoring that the nomenclature is frame-relative. The results emphasize that electromagnetic AB effects are fundamentally topological and covariant, with implications for interpreting AB-type experiments and relativistic formulations of quantum interference.

Abstract

Recent works showed that the Aharonov-Bohm (AB) phase difference for a quantum charged particle can be written in terms of electric and magnetic fluxes in a spacetime surface whose boundaries are the possible particle worldlines in the interferometer, relative to the possible paths. After presenting this result in a more detailed way, reinforcing its topological nature, we study the magnetic and electric versions of the AB effect in different inertial reference frames. We find a particular reference frame for a magnetic AB effect where the magnetic flux has a null contribution for the AB phase difference, which is entirely due to an electric flux. Also, we find a particular reference frame for an electric AB effect where the electric flux has a null contribution for the AB phase difference, which is entirely due to a magnetic flux. In this sense, the nomenclatures 'magnetic AB effect' and 'electric AB effect' lose their meaning. We have electromagnetic AB effects.

Figures (6)

• Figure 1: General AB interferometer. The quantum charged particle has two possible paths in the interferometer, labeled $a$ and $b$. $\bm{x}_{0}$ is the position where the incident particle wave function is split for both paths and $\bm{x}_{f}$ the position where the wave functions are recombined. The quantum particle always propagate in regions with null electromagnetic fields, but that may have nonzero potentials, such that we may have a nonzero AB phase difference between the paths.
• Figure 2: Spacetime diagram of a quantum particle in the interferometer depicted in Fig. \ref{['fig:ilustracao']}. The interferometer is considered to be in the $xy$ plane and time is represented in the vertical direction. The two possible particle worldlines in the interferometer, $\mathbf{x}_a(t)$ and $\mathbf{x}_b(t)$, are represented in black. One possible spacetime surface $\Omega$ whose boundaries are the possible particle worldlines is represented in blue. A curve connecting paths $a$ and $b$ through the surface $\Omega$ at a specific time is in red, with a differential displacement $d\bm{x}$ indicated. The origin of the system of coordinates was defined at $\bm{x}_{0}$.
• Figure 3: Magnetic AB scheme with a square interferometer. The interferometer paths, in blue, enclose an infinite cylindrical solenoid, in red, which produces a uniform magnetic field inside it and a null field outside.
• Figure 4: Spacetime diagrams for the magnetic AB scheme of Fig. \ref{['fig:casomag']} in different reference frames, with the same representations as figure \ref{['fig:ilustracaodeomega']}. In both reference frames the solenoid position in the $xy$ plane is represented in red. (a) In the interferometer rest frame $S$, there is no electric field and the magnetic flux is the responsible for the AB phase of Eq. (\ref{['faseABcampos-vetor']}). (b) In this special reference frame $S'$, the particle velocity is null in the regions where it was propagating in the $x$ direction in frame $S$. On this way, the magnetic flux in the represented spacetime surface $\Omega'$ is null, since the area element is null in the superposition of this spacetime surface with the region where the magnetic field is nonzero, such that the electric flux is the responsible for the AB phase of Eq. (\ref{["faseABmagneticoS'"]}).
• Figure 5: (a) Electric AB effect in the laboratory frame $S$. The potential difference between the interferometer paths (in blue) is created by a parallel-plate capacitor (in red), which is charged at time $t_I$ producing a uniform electric field $\bm{E}$ inside it while the quantum particle is in a superposition state at the positions $\bm{x}_a(t_I)$ and $\bm{x}_b(t_I)$. The capacitor is charged for a short period $T$, such that the particle movement can be disregarded during this period. $\bm{L}$ represents the portion of the straight line that connects $\bm{x}_a(t_I)$ to $\bm{x}_b(t_I)$ inside the capacitor. The velocity $\bm{v}$ between the reference frame $S'$ and the interferometer rest frame $S$ is also represented. (b) Spacetime diagram of a quantum particle in the interferometer depicted in panel (a), with the same representations as figure \ref{['fig:ilustracaodeomega']}. The spacetime region with a nonzero electric field is represented in green. The vector $\bm{L}$ is represented as the red arrow.