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 ${ℝ}^n\hookrightarrow X$ over which it is so, this connection is equivalently a differential 1-form $A\in {\Omega }^1({ℝ}^n)$.

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

$$F≔dA$$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

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

Here

• $\varphi$ is the electric potential

• $\stackrel{\rightharpoonup }{A}=[A_1,A_2,A_3]$ is the magnetic potential

(for this choice of coordinates).

References

Section 5 of

• Theodore Frankel, The Geometry of Physics - An Introduction

• geometry of physics, The electromagnetic field strength