nLab electromagnetic potential

Definition

The electromagnetic field on a spacetime $X$ is mathematically modeled by a circle bundle with connection $\nabla$ on $X$. If the underlying bundle is trivial, or else on local coordinate patches $\mathbb{R}^n \hookrightarrow X$ over which it is so, this connection is equivalently a differential 1-form $A \in \Omega^1(\mathbb{R}^n)$.

This is then called the electromagnetic potential of the electromagnetic field (sometimes: “vector potential” or “gauge potential of the electromagnetic field”).

Its de Rham differential

$$F \coloneqq \mathbf{d}A$$

is the actual field strength of the electromagnetic field.

On a 4-dimensiona Minkowski spacetime with its canonical coordinates $\{t,x^1, x^2, x^3\}$, the electromagnetic potential $A$ is naturally expanded into corredinate components, traditionally written as

$$A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3 \,.$$

Here

• $\phi$ is the electric potential

• $\vec A = [A_1, A_2, A_3]$ is the magnetic potential

(for this choice of coordinates).

Related concepts

• gauge potential

• Aharonov-Bohm effect

References

The identification of gauge potentials with connections on fiber bundles is due to:

• 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 at fiber bundles in physics.

History:

(The notion of electromagnetic potential has roots in the somewhat vague notion of the “electrotonic state” of Michael Faraday.)

• A. C. T. Wu, Chen Ning Yang, Evolution of the concept of vector potential in the description of the fundamental interactions, International Journal of Modern Physics A 21 16 (2006) 3235-3277 [doi:10.1142/S0217751X06033143]

• Chen Ning Yang: Vector Potentials and Connections, talk at 13th Annual Geometry Festival Connections in Modern Mathematics and Physics, Stony Brook University (April 1998) [video:YT]

• Chen Ning Yang, The conceptual origins of Maxwell’s equations and gauge theory, Phyics Today 67 11 (2014) [doi:10.1063/PT.3.2585, pdf]

Review:

• Theodore Frankel, Section 5 of: The Geometry of Physics - An Introduction

• geometry of physics, The electromagnetic field strength

• Wikipedia, Magnetic vector potential