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The Aharonov-Bohm effect is a configuration of the electromagnetic field which has vanishing electric/magnetic field strength (vanishing Faraday tensor $F = 0$) and but is nevertheless non-trivial, in that the vector potential $A$ is non-trivial. Since the vector potential affects the quantum mechanical phase on the wavefunction of electrons moving in an electromagnetic field, in such a configuration classical physics sees no effect, but the phase of quantum particles, which may be observed as a interference pattern on some screen, does.
More technically, a configuration of the electromagnetic field is generally given by a circle-principal connection and an Aharonov-Bohm configuration is one coming from a flat connection, whose curvature/field strength hence vanishes, but which is itself globally non-trivial. This is only possible on spaces (spacetimes) which have a non-trivial fundamental group, hence for instance it doesn’t happen on Minkowski spacetime.
In practice one imagines an idealized electric current-carrying solenoid in Euclidean space. Away from the solenoid itself the magnetic field produced by it gives such a configuration.
Let $\mathbb{R}^2 - \{0\}$ be the plane with the origin removed, and consider the space $(\mathbb{R}^2 - \{0\}) \times \mathbb{R}$ (thought of as 3d Cartesian space with the z-axis removed) and spacetime $(\mathbb{R}^2 - \{0\}) \times \mathbb{R}^2$ (thought of as the previous configuration statically moving in time).
For the following argument only the topological structure of the space matters, and nothing needs to explicitly depend on the $z$-coordinate and the time-coordinate, so for notational simplicity we may suppress these and consider just $\mathbb{R}^2 - \{0\}$.
On this space minus the x-axis consider the polar coordinates $(\phi,r)$ with
Accordingly we have the differential 1-forms
hence
Here the expression on the right extends smoothly also to the $x$-axis and this extension we call
From the way this is constructed it is clear that $\theta$ is a closed differential form
However, on $\mathbb{R}^2 - \{0\}$ this is not an exact form. In other words, if one regards $\theta$ as the vector potential being the configuration of an electromagnetic field
then:
the field strength vanishes $F = \mathbf{d}A = 0$;
but there is no gauge transformation relating $A$ to the trivial field configuration.
This is possible because $\mathbb{R}^2 - \{0\}$ is not simply connected and hence the Poincaré lemma does not apply.
anyonic braid group statistics as Aharanov-Bohm effect for a fictitious gauge field
The effect was first predicted by
It is named after:
Early discussion with emphasis of the role of connections on fiber bundles in physics and generalization to non-abelian Yang-Mills theory:
See also:
L. Mangiarotti, Gennadi Sardanashvily, section 6.6 of Connections in Classical and Quantum Field Theory, World Scientific, 2000
Mikio Nakahara, Section 10.5.3 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)
Wikipedia, Aharonov-Bohm effect
Last revised on June 7, 2024 at 09:55:55. See the history of this page for a list of all contributions to it.