Semiclassical asymptotics of the Aharonov-Bohm interference process

TL;DR

This work addresses the discontinuities that appear in semiclassical descriptions of the Aharonov-Bohm effect with a magnetic string by introducing a novel semiclassical limit that converts the discrete sum over canonical angular momenta into a continuous integral. The resulting analysis shows that the backward direction recovers the Van Vleck/Gutzwiller semiclassical propagator, while the forward direction yields a split-wave interference with two topologically distinct paths acquiring Dirac's magnetic phase, with the interference pattern scaling universally with the angular part of Hamilton's principal function. The study also connects these asymptotics to Berry's whirling-wave representation, revealing that only a subset of whirls survive in the semiclassical limit and that half-integer flux yields exact forward-propagator behavior. Together, these results clarify the origin of phase discontinuities and provide a principled bridge between exact AB propagators, semiclassical theory, and whirling-wave decompositions, with implications for interpreting AB-type interference in quantum systems.

Abstract

In order to determine the origin of discontinuities which arise when the semiclassical propagator is employed to describe an infinitely long and infinitesimally thin solenoid carrying magnetic flux, we give a systematic derivation of the semiclassical limit of the motion of an otherwise free charged particle. Our limit establishes the connection of the quantum mechanical canonical angular momentum to its classical counterpart. Moreover, we show how a picture of Aharonov-Bohm interference of two half-waves acquiring Dirac's magnetic phase when passing on either side of the solenoid emerges from the quantum propagator, and that the typical scale of the resulting interference pattern is fully determined by the ratio of the angular part of Hamilton's principal function to Planck's constant. The semiclassical propagator is recovered in the limit when this ratio diverges. We discuss the relation of our results to the whirling-wave representation of the exact propagator.

Figures (9)

• Figure 1: Elementary manifestation AHARONOVBOHMEHRENBERGSAKURAIFEYNMANII of the Aharonov-Bohm effect in a double-slit thought experiment in which charged particles, emanating from the source $i$ to reach the screen $s$, pass either side of a thin solenoid placed behind the slit-system, between the two openings. The probability amplitudes associated with the two topologically distinct paths $\gamma_I$ and $\gamma_{II}$ acquire Dirac's magnetic phase factor, giving rise to a phase difference which amounts to the flux $\Phi$ enclosed by the curve which ensues after a reversal of either path.
• Figure 2: Charged particle propagating from $\vec{x}'$ to $\vec{x}"$ in the presence of a magnetic string carrying magnetic flux $\Phi$, and in the vicinity of an infinitely extended wall. (a) As long the two possible trajectories, $\gamma_I$ unperturbed, and $\gamma_{II}$ once reflected at the wall, enclose the magnetic string, the semiclassical propagator (\ref{['eq:SemicalssicalPropagator']}) generates a probability amplitude similar to (\ref{['eq:AbElementary']}) as encountered in the elementary manifestation of the Aharonov-Bohm effect. (b) If the endpoint $\vec{x}"$ is displaced such that the closed curve ensuing after the reversal of either path no longer contains the magnetic string, the amplitude (\ref{['eq:AbElementary']}) abruptly changes into (\ref{['eq:AbElementary2']}) which is no longer a function of the flux $\Phi$. This renders (\ref{['eq:AbElementary']}) discontinuous whenever a classical trajectory passes directly through the magnetic string.
• Figure 3: Charged particle propagating from $\vec{x}'$ to $\vec{x}"$, in the presence of an (infinitely massive) Coulomb attractor $Z$, as well as of a magnetic string carrying magnetic flux $\Phi$. The Coulomb potential gives rise to at least two trajectories MCGPVIII from $\vec{x}'$ to $\vec{x}"$, $\gamma_I$ connecting both points in a rather direct way, and $\gamma_{II}$ passing in a sharp turn behind the center of attraction. The classical trajectories are invariant under rotations of $\vec{x}'$ and $\vec{x}"$ about $Z$ such that the process in (a, adapted from DISS), enclosing the flux $\Phi$, can be transformed into the process depicted in (b), in which the trajectories no longer run around the magnetic string. Thus also in this scenario the semiclassical propagator (\ref{['eq:SemicalssicalPropagator']}) features discontinuities whenever a classical trajectory passes directly through the flux.
• Figure 4: (Adapted from DISS) Coordinate system in which the magnetic string, carrying the flux $\Phi$, pierces the origin of the two-dimensional plane, and $\vec{x}'$ is aligned along the positive $x$-axis. For processes from $\vec{x}'$ to $\vec{x}"$, small angles around $\varphi_b=0$ define the backward direction, and the same applies for $\varphi_f=0$ for processes in forward direction.
• Figure 5: Real part of the half-waves $K_0^+$ and $K_0^-$ bending around the origin in a counterclockwise (a) and clockwise (b) rotational sense, respectively, divided by the free propagator factor $K_0$, where $x = z \cos \varphi_b$, $y = z \sin \varphi_b$. Amended accordingly by Dirac's magnetic phase, the superposition of these half-waves describes the forward Aharonov-Bohm interference process (\ref{['eq:Kforward']},\ref{['eq:KOP']}) in the asymptotic limit. The sum $K_0^+ + K_0^-$ amounts to the free-particle propagator $K_0$.