electric charge



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

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

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

is the conserved current called the electric current on XX. Conversely, with j elj_{el} prescribed this equation is one half of Maxwell's equations for FF.

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

Q el:= Σj 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 dj el=0d j_{el} = 0, so that QQ is independent of the choice of Σ\Sigma. When unwrapped into separate space and time components, the expression dj el=0d j_{el} = 0 may be expressed as

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

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


  • 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 nn Deligne cocycle F^\hat F the electric current j elj_{el} is the right hand of

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

Revised on May 21, 2014 01:57:14 by Anonymous Coward (