Lecture 8.

Today we discussed solutions to the wave equation, which arises for
A^\mu in the Lorentz gauge with no sources, as well as many other
physical contexts.

We went through the rationale of separation of variables for such a
partial differential equation. We found solutions A(x,y,z,t) =
T(t)X(x)Y(y)Z(z), where each function T(t), X(x), etc., solves an
ordinary simple harmonic oscillator equation. The arguments of the
resulting sinusoidal solutions for X, Y, Z, and T were k_x x, k_y y,
k_z z, and c|\vec{k}| t, respectively. We multiplied them to get
solutions of the general form 

A_{\vec{k}} = a_{\vec{k}} exp{i (\omega t - \vec{k}\dot\vec{x}) } 
		+ complex conjugate

where \vec{k} is any spatial vector, \omega = c|\vec{k}|, and
a_\vec{k} is an arbitrary constant coefficient.

We discussed why these solutions describe waves: instantaneously,
A_{\vec{k}} has a sinusoidal waveform. A point at a particular point
on that waveform has phase \phi = \omega t - kx (for \vec{k} taken in
the x-direction). Thus the phase stays constant if the point moves
forward with velocity \omega/k = c. Thus as we watch time advance the
waveform shape moves forward at a velocity c.

We then constructed the most general solution to the wave
equation. Because the wave equation is linear, the principle of
superposition holds --- sums of solutions are solutions. In fact the
A_\vec{k} form a complete, orthogonal basis of solutions: any solution
of the wave equation can be written as a sum of A_\vec{k}, with
particular coefficients a_\vec{k}. A snapshot of that sum at a
particular time just gives the Fourier decomposition of the
instantaneous field; then the phase (\omega t - \vec{k} \dot \vec{x})
just prescribes how each Fourier component evolves in time, to obey
the wave equation.

We finally wrote these solutions A_\vec{k} in covariant form. We
claimed that, since a particle whose field obeys a wave equation is
massless, the particle's energy is just |\vec{p}|c (the rest mass
contribution vanishes). Thus we take k^\mu = (|\vec{k}|, \vec{k}) to
be the momentum 4-vector of a photon with momentum \vec{k}. We lower
this to find the covector k_\mu = = (|\vec{k}|, - \vec{k}).  We can
then read off how these components appear in the phase of our
solutions as

A_{\vec{k}} = a_{\vec{k}} exp{i k_\mu x^\mu }  + complex conjugate ,

a manifestly Lorentz covariant form.

---
KB