Historical research guide · 0857 · 1998
Magnetic force exerted by the Aharonov-Bohm line
A. L Shelankov
In one minute
The Aharonov-Bohm line is an idealised thread of magnetic flux: outside it the magnetic field is exactly zero, the vector potential is not, and a passing electron registers the difference in its phase. Alexander Shelankov, at Umeå University, settles a long argument about what else the electron feels. The scattering pattern from such a line is perfectly symmetric left to right, which had persuaded many physicists that no sideways force could exist. Shelankov solves the scattering problem again in the paraxial approximation — the small-angle method borrowed from optics — and finds that a beam of finite width is deflected bodily to one side, exactly as a Lorentz force would deflect it. The asymmetry hides in the forward direction, where the outgoing wave cannot be separated into an incident part and a scattered part, so the usual cross-section bookkeeping misses an interference term. Shrink the flux and Planck’s constant drops out, leaving the classical Lorentz force. The same calculation confirms the Iordanskii force on a superfluid vortex.
Connection
Chapter 10 rests on the vector potential being physical and something you can act with, and this paper says what it does to a real beam rather than to an idealised plane wave: it steers it. Batelaan’s group later built that experiment and measured the predicted asymmetry, /library/stm-81bc78c3b1, while the time-of-flight result at /library/stm-d405e2fac7 shows there is no longitudinal force to go with the transverse one. For chapter 5 the interest is the last paragraph: the same mathematics describes a vortex line in superfluid helium, which is the site’s standing analogy for a structured vacuum.
From the source
The magnetic field around an Aharonov-Bohm line is zero while the gauge vector potential is finite, generated by the magnetic flux concentrated in the line; the construction now serves in many contexts, and the superflow around a vortex line in superfluid helium acts on a normal excitation much as the vector potential of such a line acts on a charge.Introduction, opening paragraph
Whether the line exerts a Lorentz-like transverse force on a moving charge had remained genuinely open: the Aharonov-Bohm differential cross-section is left-right symmetric, which appears to give zero net transverse momentum transfer, while summing the partial waves gives a finite transverse cross-section proportional to the sine of twice the flux in units of the flux quantum. The two routes disagree, and the paper sets out to resolve which is right.Introduction, Eqs. (1) and (2)
A new solution to the scattering problem is obtained in the paraxial, or parabolic, approximation, valid for small scattering angles; it reproduces the known Aharonov-Bohm solution when the incident wave is an infinite plane wave, and it removes the forward-direction singularity as soon as the incident wave is given a finite width.Eqs. (3) to (7); footnote on the Cornu spiral
In a gedanken experiment where a beam-like wave meets the line, the beam is deflected as a whole: the transverse momentum transfer per collision is proportional to the sine of twice the reduced flux and to the intensity of the incoming wave at the line, so the deflection is of the order of the beam’s own angular width, and a finite overlap of the incoming wave with the line is a prerequisite.Eq. (8) and the paragraph following it
For an arbitrary incoming wave the effective transverse force is h-bar times the sine of twice the reduced flux, multiplied by the cross product of the incident current density at the position of the line with the unit vector along the line. When the flux is much smaller than the flux quantum, Planck’s constant disappears from the expression and the classical Lorentz force density is recovered.Eq. (9) and the paragraph following it
The cross-section argument fails because in the forward direction the outgoing wave cannot be split into incident and scattered pieces, so the momentum transfer cannot be expressed through the differential cross-section; the S-matrix treatment shows the missing piece is a parity-odd interference between the transmitted and the scattered wave, which acts only within the angular width of the incident beam. Integrated over phonons, the same force is the Iordanskii force acting on a vortex line in helium-4 from the normal component.S-matrix discussion, Eqs. (10) and (11); concluding remark on the Iordanskii force
The way in
https://doi.org/10.1209/epl/i1998-00408-4How to cite it
A. L Shelankov (1998) Magnetic force exerted by the Aharonov-Bohm line doi:10.1209/epl/i1998-00408-4