nLab
electric charge

Idea

In electromagnetism the electromagnetic field is modeled by a degree 2 differential cocycle $\hat{F}\in H(X,\mathbb{Z}(2{)}_D^{\mathrm{\infty }})$ (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

$$d\star F=j_{\mathrm{el}}\in {\Omega }^3(X)$$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 $\hat{F}$ the electric current $j_{\mathrm{el}}$ is the right hand of

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

Related concepts

• conserved current, charge

• electric charge, magnetic charge

• flux

• higher electric background charge coupling