Contents

# Contents

## Idea

In electromagnetism the electromagnetic field is modeled by a degree 2 differential cocycle $\hat F \in H(X, \mathbb{Z}(2)_D^\infty)$ (see Deligne cohomology) with curvature characteristic 2-form $F \in \Omega^2(X)$.

With $\star$ denoting the Hodge star operator with respect to the corresponding pseudo-Riemannian metric on $X$, the right hand of

$d \star F = j_{el} \in \Omega^3(X)$

is the conserved current called the electric current on $X$. Conversely, with $j_{el}$ prescribed this equation is one half of Maxwell's equations for $F$.

If $X$ is globally hyperbolic and $\Sigma \subset X$ is any spacelike hyperslice, then

$Q_{el} := \int_\Sigma j_{el}$

is the charge of this current: the electric charge encoded by this configuration of the electromagnetic field.

Notice that due to the above equation $d j_{el} = 0$, so that $Q$ is independent of the choice of $\Sigma$. When unwrapped into separate space and time components, the expression $d j_{el} = 0$ may be expressed as

$div j + \frac{\partial\rho}{\partial t} = 0$

which is a statement of the physical phenomenon of charge conservation .

## Remarks

• While electric current is modeled by just a differential form, magnetic charge has a more subtle model. See magnetic charge .

• The above has a straightforward generalization to higher abelian gauge fields such as the Kalb-Ramond field and the supergravity C-field: for a field modeled by a degree $n$ Deligne cocycle $\hat F$ the electric current $j_{el}$ is the right hand of

$d \star F = f_{el} \in \Omega^{n+1}(X) \,.$

Last revised on May 12, 2022 at 10:32:44. See the history of this page for a list of all contributions to it.