We have not quite finished the job of building a proper relativistic field
theory of electromagnetism. That is because we would like to be able to
obtain all of the equations of motion (that is, physics) describing the system
from a covariant action principle. We have done that for the particles *in* the fields, but what about the fields themselves? In fact, since the
particles produce (and hence modify) the fields, we do not even have the
correct solutions for the particles alone, yet. Let us see if we can develop
a suitable Lagrangian for the fields that leads, ideally, to Maxwell's
equations.

The **Rules** for building a field theory Lagrangian are of interest in and
of themselves, since they are quite general. The rules are:

- Take the position and velocity coordinates for continuous space time and replace them with field variables.
- Label the field variables with discrete (coordinate direction) labels and with continuous (position) variables.
- Replace the ``velocity'' with the 4-gradient.
- Require the action to be stationary w.r.t. variations in the field variables themselves and their gradients.

q_i & &rarr#to;& &phis#phi;_k(x)

q_i & &rarr#to;& &part#partial;^&alpha#alpha;&phis#phi;_k(x)

L = &sum#sum;_i L_i(q_i,q_i) & &rarr#to;& &int#int;L(&phis#phi;_k,&part#partial;^&alpha#alpha; &phis#phi;_k) d^3x

d dt( Lq_i ) = Lq_i & &rarr#to;& &part#partial;^&beta#beta;L&part#partial;^&beta#beta; &phis#phi;_k = L&phis#phi;_k .

When we make an action integral, we integrate
over time, making the
total integral four dimensional. We therefore call
the *Lagrangian density* in four dimensions. Note that the action will be
covariant provided the Lagrangian *density* is a 4-scalar. This is what
I have meant whenever I have inadvertantly called the ``Lagrangian'' a scalar.
Good, clean, relativistic theories with or without particles are made out of
scalar Lagrangian densities, not Lagrangians per se:
A = &int#int;&int#int;L d^3x dt = &int#int;L d^4 x .

We now do the usual dance. We know that for the fields must be a scalar. We also know that Maxwell's equations relate the fields to the currents that produce them, and also link the electric and magnetic fields. We thus need to build a theory out of . Various ways we can do this include

and still messier pieces like

The first two terms are invariant under the transformations of the full Lorentz group. The third is not a scalar under inversion, but a pseudoscalar (odd under inversion). We reject it. The last is a mess. We reject it. We want a term quadratic in the 4-gradients of the fields. This is the first term. We want a source term to couple the fields and the particles. The second term does that.

So, we try a Lagrangian density with just these two terms, with unknown constants and that have to be determined so that they correctly reconstruct Maxwell's equations in whatever system of units we like: L = - Q F_&alpha#alpha;&beta#beta; F^&alpha#alpha;&beta#beta; - R J_&alpha#alpha;A^&alpha#alpha;.

We need to take derivatives of with respect to , so it is useful to write this Lagrangian in the form: L = - Q g_&lambda#lambda;&mu#mu; g_&nu#nu;&sigma#sigma; ( &part#partial;^&mu#mu;A^&sigma#sigma;- &part#partial;^&sigma#sigma;A^&mu#mu;)(&part#partial;^&lambda#lambda;A^&nu#nu;- &part#partial;^&nu#nu;A^&lambda#lambda;) - R J_&alpha#alpha;A^&alpha#alpha;. When we form we get delta functions whenever and are equal to a pair of the indices above. We therefore get four terms: L(&part#partial;^&beta#beta;A^&alpha#alpha;) = - Q g_&lambda#lambda;&mu#mu; g_&nu#nu;&sigma#sigma; { &delta#delta;_&beta#beta;^&mu#mu;&delta#delta;_&alpha#alpha;^&sigma#sigma; F^&lambda#lambda;&nu#nu; - &delta#delta;_&beta#beta;^&sigma#sigma;&delta#delta;_&alpha#alpha;^&mu#mu;F^&lambda#lambda;&nu#nu; + &delta#delta;_&beta#beta;^&lambda#lambda;&delta#delta;_&alpha#alpha;^&nu#nu;F^&mu#mu;&sigma#sigma; - &delta#delta;_&beta#beta;^&nu#nu; &delta#delta;_&alpha#alpha;^&lambda#lambda;F^&mu#mu;&sigma#sigma; } where the first two terms come from delta functions formed from the first term and the second two terms come from delta functions formed from the second term.

is symmetric (in fact, diagonal). The is antisymmetric. When we do the sums against the -functions, the four terms make identical contributions: L(&part#partial;^&beta#beta;A^&alpha#alpha;) = - 4Q F_&beta#beta;&alpha#alpha; = 4 Q F_&alpha#alpha;&beta#beta; . The other part of the E-L equation (which corresponds in position space to the ``potential'', or ``force'' term) is LA^&alpha#alpha; = - R J_&alpha#alpha;. Therefore the equations of motion for the electromagnetic field can be written 4Q &part#partial;^&beta#beta;F_&beta#beta;&alpha#alpha; = R J_&alpha#alpha;.

If one checks back in one's notes, one sees that this is indeed the covariant form of the inhomogeneous Maxwell's equations if and : &part#partial;^&beta#beta;F_&beta#beta;&alpha#alpha; = &mu#mu;_0 J_&alpha#alpha; follows from the Lagrangian density: L = - 14 F_&alpha#alpha;&beta#beta; F^&alpha#alpha;&beta#beta; - &mu#mu;_0 J_&alpha#alpha;A^&alpha#alpha;.

Therefore the Lagrangian we have constructed yields the inhomogeneous Maxwell
equations, but not the homogeneous ones. That is okay, though, because we
have constructed the
in terms of the
in such a
way that the homogeneous ones are satisfied *automatically*! To observe
that this miracle is true, we recall the covariant form of the homogeneous
equations:
&part#partial;_&alpha#alpha;F^&alpha#alpha;&beta#beta; = 0.
Also,
F^&alpha#alpha;&beta#beta; = 12 &epsi#epsilon;^&alpha#alpha;&beta#beta;&gamma#gamma;
&delta#delta; F_&gamma#gamma;&delta#delta; .
Thus
&part#partial;_&alpha#alpha;F^&alpha#alpha;&beta#beta; & = & 12 &part#partial;_&alpha#alpha;
&epsi#epsilon;^&alpha#alpha;&beta#beta;&gamma#gamma;&delta#delta; F_&gamma#gamma;&delta#delta;

& = & &part#partial;_&alpha#alpha;&epsi#epsilon;^&alpha#alpha;&beta#beta;&gamma#gamma;&delta#delta;
&part#partial;_&gamma#gamma;A_&delta#delta;

& = & &epsi#epsilon;^&alpha#alpha;&beta#beta;&gamma#gamma;&delta#delta; &part#partial;_&alpha#alpha;
&part#partial;_&gamma#gamma;A_&delta#delta;
is the first term. But
is symmetric, while
is antisymmetric in the same two
indices, so the contraction on the two indices vanishes (work it out term by
term if you doubt it).

Thus the homogeneous equations are satisfied by our *definition* of
quite independent of any dynamics. In four dimensions, all
of the inhomogeneous source terms must appear in equations with the form of
the inhomogeneous equation above, and only one of these equations can result
from the action principle. The similarity transformation to the fields we
observe is thus the ``natural'' form of the ME's, and in four dimensions the
homogeneous equations are really not independent as far as the action
principle is concerned. Note that this is fundamentally because our field
strength tensor derives from the definition of the magnetic field as the curl
of the vector field
(which is divergenceless) which is built into
the definition.

As a final quixotic note, observe that if we take the 4-divergence of both
sides of the inhomogeneous Maxwell equations:
&part#partial;^&alpha#alpha;&part#partial;^&beta#beta;F_&beta#beta;&alpha#alpha; =
&mu#mu;_0 &part#partial;^&alpha#alpha;J_&alpha#alpha;
the left hand side vanishes because again, a *symmetric*
differential operator is contracted with a *completely
antisymmetric* field strength tensor. Thus
&part#partial;^&alpha#alpha;J_&alpha#alpha;= 0 ,
which, by some strange coincidence, is the charge-current conservation
equation in four dimensions. Do you get the feeling that something very deep
is going on? This is what I love about physics. Beautiful things are really
*beautiful!*

We will now blow off the ``proca'' Lagrangian, which would be appropriate if the photon had a mass. It doesn't, but if it did you would need to read this chapter. It might, of course, so you should probably read the chapter anyway, but it currently (bad pun) doesn't so I'm going to make light of it (worse pun) and continue.

If we had one more month, we would now study the covariant forms of the stress
tensor. It is important, but it is also quite difficult, and necessitates a
broader discussion than we can now afford. To treat the subject properly, we
would need to treat parts of chapter 17 simultaneously, and we would need to
do a *lot* of algebra. This would mean that we would miss (in all
probability) being able to learn the Liénard-Wiechart potential, which is
far more important. We will therefore content ourselves with merely defining
the stress tensor, remarking on some of its properties without proof, and
moving on. You are responsible for working your way through this chapter,
according to your needs, inclinations, and abilities, on your own.