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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

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

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

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

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

Related concepts

• Gauss law

• hypercharge

• conserved current, charge

• electric charge, magnetic charge

• Ohm's law

• color charge

• flux

• dyon

• Hodge-Maxwell theorem

• higher electric background charge coupling