Electromagnetic momentum in the Aharonov-Bohm quantum interference experiment from a physical perspective

TL;DR

The paper addresses the Aharonov–Bohm effect by proposing a classical electromagnetic momentum mechanism: an external charge in the vector potential of a current loop acquires an EM momentum $\mathbf{p}_e=\frac{Q\mathbf{A}}{c}$, which, through either density-based or force-based arguments, can be distributed across the sources of the vector potential. The authors show that this momentum is nonlocal and resides in the composite charge–solenoid system, producing equal and opposite momenta for symmetric charges and yielding the observed AB phase shift via $\Delta \varphi=\frac{Q}{c\hbar}\oint \mathbf{A}\cdot d\mathbf{x}=\frac{Q\Phi}{c\hbar}$. They provide both a direct current-loop calculation and an alternative force-flux derivation, demonstrating consistent results and offering a classical electromagnetism account for the AB momentum without invoking topological quantum explanations. Overall, the work clarifies the physical origin and localization (in the system as a whole) of the AB momentum, reinforcing the physical relevance of the vector potential and linking the quantum phase shift to classical momentum accounting in a nonlocal, gauge-invariant framework.

Abstract

In the Aharonov-Bohm setup, a double-slit experiment, when a long but thin solenoid of current is introduced between the two coherent beams of electrons behind the slits, an extra phase difference between the interfering beams appears, as shown by a shift in the interference pattern. This mysterious effect, purportedly arises owing to an electromagnetic momentum, attributed to the presence of a vector potential at the location of either beam, due to the solenoid of current even when the magnetic field is zero outside the solenoid. It has remained a puzzle, how mere potential, thought to be just a mathematical tool for calculating electromagnetic field, can give rise to electromagnetic momentum in a system. Experimentally the effect has been amply verified, with hardly any doubts that the observed effect is real. A satisfactory physical explanation of the existence of momentum, at least under the aegis of classical electromagnetism, is still missing since inception of the idea more than half a century back. We show here the presence of electromagnetic momentum in the product of the drift velocities of the current-carrying charges within the solenoid and the mass equivalent of their potential energies in the electric field of the external charges.

Figures (3)

• Figure 1: A schematic of the AB experimental setup. Coherent beams of electrons, passing through two slits, $S1$ and $S2$, separated along the y-axis, form an interference pattern on the screen. When a long but thin solenoid of electric current is introduced behind the two slits, midway between the two beams, an extra phase between the two electron beams appears, as shown by a shift in the interference pattern. In the region of electron beams, the magnetic field is nil ($\mathbf {B}=0$) though the vector potential is finite ($\mathbf {A}\ne 0$).
• Figure 2: A charge $Q_1$ lies a distance $R_1$ from a small current loop $ABCD$, carrying a uniform current $I$. Two equal current elements, ${\rm d}{l}$, separated by distance $2r$, on two opposite sides of the current loop, are shown.
• Figure 3: A current loop $ABCD$ is carrying current $I$. There is an electric field ${\mathbf E}$, uniform over the current loop. The electric field ${\mathbf E}$ does positive work, $I {\mathbf E} \cdot {\rm d}{\mathbf l}$ on a current element ${\rm d}{\mathbf l}$ in the arc $ABC$ of the current loop and an equal and opposite work on a similar current element ${\rm d}{\mathbf l}$ in the arc $CDA$, the two current elements separated by a distance $S$, on two opposite sides of the current loop. This implies transfer of an infinitesimal EM energy, between two current elements across the shaded area ${\rm d}{\mathbf a}$ of the circuit, which represents an element of EM momentum in the system.