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Using differential forms and their picture interpretations, I wonder if it's possible to give a nice geometric & physical motivation for the form of the Electromagnetic Lagrangian density?

The Lagrangian for the electromagnetic field without current sources in terms of differential forms is $F \wedge * F$, where $F$ is the exterior derivative of a 4-potential $A$. Another way to say this is that $F$ is the four-dimensional curl of a 4-potential $A$, i.e. the anti-symmetric part of the flow of the determinant of the Jacobian of a vector field $A$, and since we can physically interpret the curl of a vector field as the instantaneous rotation of the elements of volume that $A$ acts on, it seems as though we can interpret varying $F \wedge * F$ as saying that we are trying to minimize the instantaneous four-dimensional volume of rotation of the electromagnetic field (since the Hodge dual on 2-forms gives 2-forms 'perpendicular' to our original ones, wedging a form with it's dual gives us a 4-d volume, so here we are getting the rotation of a volume element in spacetime).

Is that correct?

There is also the issue of defining the same action just in different spaces, using $F_{ij}F^{ij}$ and so a similar interpretation must exist... If I interpret $F_{ab}$ as I've interpreted $F$ above, i.e. a 4-d curl, and $F^{cd}$ similarly just in the dual space, then in order to get a scalar from these I have to take the trace of the matrix product $F_{ab}F^{cd}$, which seems to me as though it can be interpreted as the divergence of the volume of rotation, thus minimizing the action seems to be saying that we are minimizing the flow of rotation per unit volume.

Is this correct?

If these interpretations are in any way valid, can anyone suggest a similar interpretation for the $A_idx^i$ term in the Lagrangian, either when we're getting the Lorentz force law or the other Maxwell equations? Vaguely thinking about interpreting this term in terms of current and getting Maxwell's equations hints at what I've written above to have at least some validity!

Interestingly, if correct I would imagine all of this has a fantastic global interpretation in terms of fiber bundles, if anybody sees a relationship that would be interesting.

(Page 9 of this pdf are where I'm getting this interpretation of divergence and curl via the Jacobian, and I'm mixing it with the geometric interpretation of differential forms ala MTW's Gravitation)

I understand Landau's mathematical derivation of the $F_{ij}$ field tensor, Lorentz invariant scalar w.r.t. to the Minkowski inner product, linearity of the EOM, and eliminating direct dependence on the potentials, but physical motivation for it's form is lacking. Since one can loosely interpret minimizing $\mathcal{L} = T - V$ as minimizing the excess of kinetic over potential energy over the path of a particle, and for a free particle as simply minimizing the energy, I don't see why a loose interpretation of the EM Lagrangian can't be given. Any thoughts are welcome.

References:

  1. Math 733: Vector Fields, Differential Forms, and Cohomology, Lecture notes, R. Jason Parsley
  2. Warnick, Selfridge, Arnold - Teaching Electromagnetic Field Theory Using Differential Forms
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As a guess, I'd say it's something on the lines of the total center of energy of the electromagnetic system, including all sources, moving along a geodesic. –  Larry Harson May 5 at 1:19
    
There is no specific difference in physical interpretation when one uses differential forms. As for your question of fiber bundles: check out G-principal bundles, then check out U(1)-principal bundles. Volume 2 of Dubrovin's 'Modern Geometry' has some great material on this topic. –  dj_mummy May 7 at 17:01

1 Answer 1

Ok will attempt an answer to this one, although i state that the language of differential geometry and differential forms (especially in relativistic 4-D space) is not sth i use every-day. However i totally agree that a physical picture is important (especially in physics :))

Well the process of "zero-ing" a derivative (or differential form) is not only for extremum values (minimum/maximum) It is also related to "exactness"

So what does this mean in less "symbolic" and more "physical" terms.

Since the differential and the exactness/closedness relates to a kernel of a transformation, in simple terms this means that there is a "stable-point of (dynamic) balance" at that area (nothing is "lost" or "added").

When sources are added, the form is no longer "exact". Sources are not zero, so the kernel is not zero and there are interactions.

If you want a "more geometric" picture (eg in terms of "rotation"), this is related to the relativistic concept of "time" (which i disagree at some points).

Nevertheless one can still assume a geometric picture like this:

The A potential (which is "un-observable") is used as a "phase". This A potential can be seen as a configuration which has a stable form but can change position.

Think of it as a DNA molecule which is formed into a loop. The information on the DNA molecule is not changed but its relative permutation/rotation/position CAN change and the difderential form just states that this should be irrelevant ("gauge invariance")

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