Changes

We start off with Maxwell's Equation in the Lorentz gauge:

We start off with Maxwell's Equation in the Lorentz gauge:

:<math>\square^2A^\mu(\mathbf{r},t) = \square^2A^\mu (r)=\mu j^\mu (r)</math><br><br>

:<math>\square^2A^\mu(\mathbf{r},t) = \square^2A^\mu (r)=\mu j^\mu (r)</math><br>

where we use the metric signature (+,+,+,-) and<br><br>

where we use the metric signature (+,+,+,-) and<br>

:<math>A^\mu = (\mathbf{A},\frac{\Phi} {c})</math>,

:<math>A^\mu = (\mathbf{A},\frac{\Phi} {c})</math>

:<math>\square^2=\part_\mu \part^\mu = \nabla^2 - \frac{1}{c^2} \frac{\part^2}{\part t^2}</math>

:<math>\square^2=\part_\mu \part^\mu = \nabla^2 - \frac{1}{c^2} \frac{\part^2}{\part t^2}</math>

:<math>j^\mu = (\mathbf{j},c\rho), \part_\mu= (\mathbf{\nabla}, \frac{1}{c} \frac{\part}{\part t})</math><br><br>

:<math>j^\mu = (\mathbf{j},c\rho), \part_\mu= (\mathbf{\nabla}, \frac{1}{c} \frac{\part}{\part t})</math><br>

Lorentz Gauge: <math>\part_\mu A^\mu = 0 \rArr \mathbf{\nabla} \cdot \mathbf{A}-\frac{1}{c^2} \frac{\part\Phi}{\part t}=0</math><br><br>

The gauge condition for the Lorentz gauge is

Introduce Green's function at<math> (\mathbf{r},t)=r \quad</math> from some impulse source at<math> r'=(\mathbf{r}',t') \quad</math><br><br>

:<math>\part_\mu A^\mu = 0 \rArr \mathbf{\nabla} \cdot \mathbf{A}-\frac{1}{c^2} \frac{\part\Phi}{\part t}=0</math><br>

Introduce the Green's function at <math> (\mathbf{r},t)=r \quad</math> from some impulse source at<math> r'=(\mathbf{r}',t') \quad</math><br><br>

<math>\square^2_rG(r,r')=\delta^4(r-r')</math><br><br>

<math>\square^2_rG(r,r')=\delta^4(r-r')</math><br><br>

Let <math> \tilde{G} (q) = \frac{1}{(2\pi)^2} \int d^4r e^{-iq\cdot r} G(r,0)</math><br><br>

Let <math> \tilde{G} (q) = \frac{1}{(2\pi)^2} \int d^4r e^{-iq\cdot r} G(r,0)</math><br><br>