[Click here for a PDF of this post with nicer formatting]

# Reading.

Covering chapter 4 material from the text [1].

Covering lecture notes pp.103-113: variational principle for the electromagnetic field and the relevant boundary conditions (103-105); the second set of Maxwell’s equations from the variational principle (106-108); Maxwell’s equations in vacuum and the wave equation in the non-relativistic Coulomb gauge (109-111)

# Review. Our action.

Our dynamics variables are

A = 1, \cdots, N$} \\ A^i(x) & \quad \mbox{$A = 1, \cdots, N$}\end{array}\right.\end{aligned} \hspace{\stretch{1}}(2.1)$

We saw that the interaction term could also be written in terms of a delta function current, with

and

Variation with respect to gave us

Note that it’s easy to get the sign mixed up here. With our metric tensor, if the second index is the summation index, we have a positive sign.

Only the and depend on .

# The field action variation.

\paragraph{Today:} We’ll find the EOM for . The dynamical degrees of freedom are

Here are treated as “sources”.

We demand that

We need to impose two conditions.

\begin{itemize}

\item At spatial , i.e. at , we’ll impose the condition

This is sensible, because fields are created by charges, and charges are assumed to be localized in a bounded region. The field outside charges will at . Later we will treat the integration range as finite, and bounded, then later allow the boundary to go to infinity.

\item at and we’ll imagine that the values of are fixed.

This is analogous to and in particle mechanics.

Since is given, and equivalent to the initial and final field configurations, our extremes at the boundary is zero

\end{itemize}

PICTURE: a cylinder in spacetime, with an attempt to depict the boundary.

# Computing the variation.

Looking first at the variation of just the bit we have

Our variation is now reduced to

We can integrate this first term by parts

The first term is a four dimensional divergence, with the contraction of the four gradient with a four vector .

Prof. Poppitz chose split of to illustrate that this can be viewed as regular old spatial three vector divergences. It is probably more rigorous to mandate that the four volume element is oriented , and then utilize the 4D version of the divergence theorem (or its Stokes Theorem equivalent). The completely antisymmetric tensor should do most of the work required to express the oriented boundary volume.

Because we have specified that is zero on the boundary, so is , so these boundary terms are killed off. We are left with

This gives us

# Unpacking these.

Recall that the Bianchi identity

gave us

How about the EOM that we have found by varying the action? One of those equations is

since .

Because

we have

The messier one to deal with is

Splitting out the spatial and time indexes for the four gradient we have

The spatial index tensor element is

so the sum becomes

This gives us

or in vector form

Summarizing what we know so far, we have

or in vector form

# Speed of light

\paragraph{Claim}: “” is the speed of EM waves in vacuum.

Study equations in vacuum (no sources, so ) for .

where

In terms of potentials

Since we also have

some rearrangement gives

The remaining equation , in terms of potentials is

We can make a gauge transformation that completely eliminates 6.28, and reduces 6.27 to a wave equation.

with

Can choose to make ()

Can also find a transformation that also allows

\paragraph{Q:} What would that second transformation be explicitly?

\paragraph{A:} To be revisited next lecture, when this is covered in full detail.

This is the Coulomb gauge

From 6.27, we then have

which is the wave equation for the propagation of the vector potential through space at velocity , confirming that is the speed of electromagnetic propagation (the speed of light).

# References

[1] L.D. Landau and E.M. Lifshitz. *The classical theory of fields*. Butterworth-Heinemann, 1980.