Physical Interpretation of EM Field Lagrangian

[bolbteppa]

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 [itex]F \wedge * F[/itex], where [itex]F[/itex] is the exterior derivative of a 4-potential [itex]A[/itex]. Another way to say this is that [itex]F[/itex] is the four-dimensional curl of a 4-potential [itex]A[/itex], i.e. the http://users.wfu.edu/parslerj/math733/lecture%20notes%201-4.pdf [itex]A[/itex], and since we can physically interpret the curl of a vector field as the instantaneous rotation of the elements of volume that [itex]A[/itex] acts on, it seems as though we can interpret varying [itex]F \wedge * F[/itex] 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 [itex]F_{ij}F^{ij}[/itex] and so a similar interpretation must exist... If I interpret [itex]F_{ab}[/itex] as I've interpreted [itex]F[/itex] above, i.e. a 4-d curl, and [itex]F^{cd}[/itex] similarly just in the dual space, then in order to get a scalar from these I have to take http://physicspages.com/2013/03/15/electromagnetic-field-tensor-contractions-with-metric-tensor/ [itex]F_{ab}F^{cd}[/itex], 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 [itex]A_idx^i[/itex] 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 http://users.wfu.edu/parslerj/math733/lecture%20notes%201-4.pdf 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 [itex]F_{ij}[/itex] 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 [itex]\mathcal{L} = T - V[/itex] 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.

 

[vanhees71]

I'm not so used to the Cartan calculus. So I stick to the Ricci index calculus.

The basic principles of physics are based on group theoretical analyses of the symmetries of physical laws. In the case of electrodynamics (a better name would be electromagnetics) these are the space-time symmetry (Poincare symmetry) of special relativity and the gauge invariance for the description of massless vector fields with only descrete intrinsic (polarization) degrees of freedom that follow from the representation theory of the Poincare group. In addition we know that the electromagnetic interaction also respects space-reflection (parity) symmetry.

Now we have the vector field [itex]A^{\mu}[/itex] as a fundamental building block. Now we want to construct a Lagrange density obeying the symmetries and consisting of at most first-order derivatives of the field. The only building blocks for the Lagrangian thus are [itex]A^{\mu}[/itex] the pseudometric [itex]\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)[/itex], and the partial derivatives [itex]\partial_{\mu}[/itex] of the field. Then there's the gauge-invariant combination
[tex]F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu},[/tex]
the Faraday tensor. From this we can build the invariant
[tex]\mathcal{L}_1=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}.[/tex]
Under the proper orthochronous Lorentz group we could also use the Levi-Civita tensor as an additional building block defining the Hodge dual of the Faraday tensor
[tex]^{\dagger}F^{\mu \nu} = \frac{1}{2}\epsilon^{\mu \nu \rho \sigma} F_{\rho \sigma}.[/tex]
From this we'd have the additional invariant
[tex]\mathcal{L}_2=\alpha F_{\mu \nu} ^{\dagger}F^{\mu \nu},[/tex]
but this is not invariant under space reflections, and thus we have to set [itex]\alpha=0[/itex], and the unique free-field Lagrangian for the electromagnetic field is [itex]\mathcal{L}_1[/itex].

 

[bolbteppa]

Thanks for the response, as far as I can see my post would just offer some geometric motivations and interpretations of some of the steps in your post, seemingly making things such as the necessity of first order partial derivatives obvious rather than things we have to stipulate, and furthermore explains exactly why we choose one invariant as opposed to another invariant.

As far as I can see, the motivations for first order partial derivatives are the principle of superposition, which is tantamount to saying 'because it works', and Maxwell's equations, which are experimentally verified, and that's great but I think my little explanation just gives a more primitive geometric motivation for why they are first order partial derivatives in the first place, let's you geometrically interpret varying the action, and then let's you geometrically interpret the covariant form of Maxwell's equations as an obvious consequence of varying the action and as nothing but a condition to obviously be satisfied...

Gauge invariance seems to me to be completely tautological when we realize our concern is to preserve volumes of instantaneous rotation of the field, not preserving gauge invariance seems to be equivalent to saying that we are no longer working with a 4-D curl, i.e. no longer working with the instantaneous rotation of elements of volume of the em potential, so destroying gauge invariance is just ridiculous. Stipulating that we musn't destroy gauge invariance seems as though it's like saying that we can't just add pi to a dot product, or an elephant to one side of 3 + 2 = 5 in order to preserve 'equality invariance', because it destroys the meaning of what we were doing in the first place.

Having thought about all this a little I think it shows that defining the EM field tensor the way Landau or Jackson would do it implicitly sneaks in the equation of continuity, because you've shifted the emphasis from volumes to scalar quantities (where the scalar quantity is constructed out of the volume contained within the 4-D curl) whereas my little geometric motivation just seems to be deriving all of this from the equation of continuity in the first place.

As far as I can see, the missing ingredient in Landau's or Jackson's construction of the form of the EM field tensor is basically just that they are not using differential forms and have to go to scalar's immediately, whereas differential forms gives you the freedom to think in terms of volumes. Thinking in terms of scalars then makes the equation of continuity a theorem to be proven, but I think it contains all the information about the whole process in the first place.

I haven't the time to think about the relations to the other elements of the Poincare group etc... until the summer, but I bet this will shed light on that, assuming it's more or less correct. Even if what I've said it wrong, I'm sure it could be cleaned up, I'd love to know if there's a reference somewhere describing things this way.

 

[marmoset]

You might find some help in Burke's book Applied Differential Geometry. He doesn't discuss the electromagnetic field lagrangian explicitly but he has a whole chapter on electromagnetism with differential forms and a section (six pages) on lagrangian field theory. The whole book will probably interest you if you're looking for simple geometric representations of physics/maths in general.

 

[PJC01]

As a scientist, it is important to always seek a physical interpretation for mathematical equations and concepts. In the case of the electromagnetic Lagrangian, it is possible to provide a geometric and physical motivation for its form by using differential forms and their picture interpretations.

Firstly, the Lagrangian for the electromagnetic field without current sources can be written in terms of differential forms as F \wedge * F where F is the exterior derivative of a 4-potential A. This can also be interpreted as the four-dimensional curl of the 4-potential A, representing the instantaneous rotation of the elements of volume that A acts on. Therefore, varying F \wedge * F can be seen as trying to minimize the instantaneous four-dimensional volume of rotation of the electromagnetic field.

Moreover, using F_{ij}F^{ij} in the Lagrangian also has a physical interpretation. By interpreting F_{ab} as a 4-dimensional curl and F^{cd} as a 4-dimensional divergence, the contraction of these two terms gives the divergence of the volume of rotation. Thus, minimizing the action is equivalent to minimizing the flow of rotation per unit volume, providing a physical motivation for this form of the Lagrangian.

In terms of the A_idx^i term in the Lagrangian, further interpretation can be obtained by considering the Lorentz force law and the other Maxwell equations. By interpreting this term in terms of current, we can see that it relates to the flow of current and how it affects the electromagnetic field.

It is also worth noting that these interpretations can have a global interpretation in terms of fiber bundles, which could provide further insight into the physical meaning of the Lagrangian.

In conclusion, while the mathematical derivation of the electromagnetic Lagrangian is well-understood, it is important to also seek a physical interpretation for its form. Using differential forms and their picture interpretations, we can provide a geometric and physical motivation for the form of the Lagrangian, shedding light on the underlying physical principles at play in electromagnetism.