45.2 Curvature, electric and magnetic fields

45.2 Curvature, electric and magnetic fields

While the connection or vector potential $A$ is the fundamental geometrical quantity needed to construct theories with gauge symmetry, one often wants to work instead with certain quantities derived from A that are invariant under gauge transformations. To a mathematician this is the curvature of a connection, to a physicist these are the electric and magnetic field strengths derived from a vector potential. The definition is:

Definition (Curvature of a connection, electromagnetic field strengths)

The curvature $of$ a connection $A_{\mu }$ is given by

$$F_{\mu \nu }=\frac{i}{e}[D_{\mu }^A,D_{\nu }^A]$$

which can more explicitly be written

$$F_{\mu \nu }=\frac{\partial A_{\nu }}{\partial x_{\mu }}-\frac{\partial A_{\mu }}{\partial x_{\nu }}$$

Note that while $D_{\mu }^A$ is a diferential operator, $[D_{\mu }^A,D_{\nu }^A]$ and thus the curvature is just a multiplication operator.

The electromagnetic field strengths break up into those components with a time index and those without:

Definition (Electric and magnetic fields)

The electric and magnetic fields are two functions from ${\mathbf{R}}^4$ to ${\mathbf{R}}^3$ , with components given $by$

$$E_j=F_{j0}=-\frac{\partial A_j}{\partial t}+\frac{\partial A_0}{\partial x_j}$$ $$B_j=\frac{1}{2}{ϵ}_{jkl}{F}_{kl}={ϵ}_{jkl}\frac{\partial A_l}{\partial x_k}$$

or, in vector notation

$$\mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t}+\nabla A_0,\mathbf{B}=\nabla \times \mathbf{A}$$

$E$ is called the electric field, $B$ the magnetic field.

These are invariant under gauge transformations since

$$\begin{array}{rl}\mathbf{E}\to \mathbf{E}-\frac{\partial }{\partial t}\nabla \phi +\nabla \frac{\partial \phi }{\partial t}& =\mathbf{E}\\ \mathbf{B}\to \mathbf{B}+\nabla \times \nabla \phi & =\mathbf{B}\end{array}$$

Here we use the fact that

$$\nabla \times \nabla f=0\text{\hspace{1em}(45.2)}$$

for any function $f.$

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来源版本：2025-10-20

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