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

• Aharonov-Bohm effect

References

Section 5 of

• Theodore Frankel, The Geometry of Physics - An Introduction

• geometry of physics, The electromagnetic field strength