nLab
electric charge

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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 .

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 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 (92.68.97.89)