# Contents

## Idea

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

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

$d\star F={j}_{\mathrm{el}}\in {\Omega }^{3}\left(X\right)$d \star F = j_{el} \in \Omega^3(X)

is the conserved current called the electric current on $X$. Conversely, with ${j}_{\mathrm{el}}$ 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

${Q}_{\mathrm{el}}:={\int }_{\Sigma }{j}_{\mathrm{el}}$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 $d{j}_{\mathrm{el}}=0$, so that $Q$ is independent of the choice of $\Sigma$. When unwrapped into separate space and time components, the expression $d{j}_{\mathrm{el}}=0$ may be expressed as

$\mathrm{div}j+\frac{\partial \rho }{\partial t}=0$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 $\stackrel{^}{F}$ the electric current ${j}_{\mathrm{el}}$ is the right hand of

$d\star F={f}_{\mathrm{el}}\in {\Omega }^{n+1}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$d \star F = f_{el} \in \Omega^{n+1}(X) \,.

Revised on December 21, 2011 01:49:42 by Urs Schreiber (83.91.122.110)