The Effect of a Magnetic Flux Line in Quantum Theory

The nonlocal exchange of the conserved, gauge invariant quantity $e^{\frac{i}{\hslash }(p_k-\frac{e}{c}{A}_k)L^k}$, $L^k=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, $k=1,2$ between the charged particle and the magnetic flux line (in the $k=3$ direction) is responsible for the Aharonov-Bohm effect. This exchange occurs at a definite time, before the wave packets are brought together to interfere, and can be verified experimentally.

Figure 1

Conservation law of modular momentum ${\pi }_1^{\prime }=\cos (2\pi \frac{p_1^{\prime }}{p_0})$, ${\pi }_2^{\prime }=\cos (2\pi \frac{p_2^{\prime }}{p_0})$, where $p_0$ is a constant. $p_1^{\prime }$, $p_2^{\prime }$ are the components of the momentum in any given direction after a collision. Because of a collision, the system moves from one point of the ellipse to another, for example, from $a$ to $b$.

Figure 2

Charged particle in a superposition of three wave packets moving in the $x$ direction. Only the distributions of the velocities in the directions $AB$ and $AC$ change when the respective lines cross the solenoid, while the distribution of the velocity in the direction $BC$ does not change during the motion.

Figure 3

View from the reference frame where the center of mass of the charged particle is at rest: As the solenoid crosses the circle line of radius $r$, the modular angular velocity about the $z$ axis of the charged particle changes.