nLab Aharonov-Bohm effect

Idea

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.

Details

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

$$x = r cos(\phi)\,,\;\;\; y = r sin(\phi) \,.$$

Accordingly we have the differential 1-forms

$$\mathbf{d}x = cos(\phi)\mathbf{d}r - r sin(\phi) \mathbf{d}\phi$$

$$\mathbf{d}y = sin(\phi)\mathbf{d}r + r cos(\phi) \mathbf{d}\phi$$

hence

$$\begin{aligned} \mathbf{d}\phi & = \frac{1}{r}cos(\phi)\mathbf{d}y - \frac{1}{r}sin(\phi) \mathbf{d}x \\ & = \frac{1}{r^2} x \mathbf{d}y - \frac{1}{r^2} y \mathbf{d}x \end{aligned} \,.$$

Here the expression on the right extends smoothly also to the $x$-axis and this extension we call

$$\theta \coloneqq \frac{1}{r^2} x \mathbf{d}y - \frac{1}{r^2} y \mathbf{d}x \;\; \in \Omega^1(\mathbb{R}^2 - \{0\}) \,.$$

From the way this is constructed it is clear that $\theta$ is a closed differential form

$$\mathbf{d}\theta = 0 \,.$$

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

$$A \coloneqq \theta$$

then:

1. the field strength vanishes $F = \mathbf{d}A = 0$;

2. 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.

Related concepts

• fiber bundles in physics

• Dirac charge quantization, magnetic monopole

• cosmic topology

• anyonic braid group statistics as Aharanov-Bohm effect for a fictitious gauge field

References

The effect was first predicted by

• W. Ehrenberg, R. E. Siday, The Refractive Index in Electron Optics and the Principles of Dynamics, Proceedings of the Physical Society. Section B, 62 8 (1949) 1 [doi:10.1088/0370-1301/62/1/303]

It is named after:

• Yakir Aharonov, David Bohm, Significance of Electromagnetic Potentials in the Quantum Theory, Phys. Rev. 115 (1959) 485 [doi:10.1103/PhysRev.115.485, pdf]

Early discussion with emphasis of the role of connections on fiber bundles in physics and generalization to non-abelian Yang-Mills theory:

• Tai Tsun Wu, Chen Ning Yang, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D 12 (1975) 3845 [doi:10.1103/PhysRevD.12.3845]

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