Lecture 9.

Today we put our Lorentz-covariant Maxwell equations for A^\mu into a
form that is gauge-invariant as well. We did this by introducing the
field strength tensor F^{\mu\nu}, whose components are the physical E
and B fields, and are invariant under gauge transformations.

We first reviewed last lecture's conclusion about the general form of
solutions to the wave equation. We applied this to the electromagnetic
field A^\mu, which obeys the wave equation in Lorentz gauge where
\partial_\mu A^\mu = 0. This means the general solution for A^\mu has
the same basis of solutions exp{ik_\mu x^\mu} as before, but with
coefficients to each term being four-vectors, a_\vec{k}^\mu . The
Lorentz condition induces a constraint on the coefficient vector:
k^\mu a_\vec{k}^\mu = 0, which means that the coefficient vector must
be perpendicular (in 4-space) to the momentum 4-vector.

We then introduced F^{\mu\nu} as a four-dimensional generalization of
curl A. We showed F^{\mu\nu} was gauge-invariant. As a 4-dimensional
antisymmetric tensor, F^{\mu\nu} has 16 entries - 4 vanishing diagonal
entries = 12 nonzero entries, half (6) of which are independent. We
calculated these components, getting F^{0i} = - E^i and F^{ij} =
-\epsilon^{ij}_k B^k, where Latin indices are spatial and
\epsilon_{ijk} is the totally antisymmetric tensor introduced in
class: it gives zero if any of the indices (ijk) coincide; +1 if (ijk)
is an even permutation of the cycle (123), and -1 if (ijk) is an odd
permutation.

We were then able to recognize Maxwell's equations for A^\mu (in full
generality, with no gauge choice necessary) as 

\partial_\mu F^{\mu\nu} = j^\nu /c

a form which is not only Lorentz-covariant (since it has a 4-vector on
each side of the equation), but is also gauge-invariant (both sides
are unchanged by gauge transformations). We showed how the
time-component of this equation yielded Gauss' law, and the space
components Ampere's law.

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KB