ABSTRACT
As we have seen before, the general driven wave equation for the electromagnetic four-potential is
(5.1)
One of the most fundamental properties of the wave equation is the fact that it is linear: if we know the solutions for two different four-current sources,
and
(5.2)
then, the solution for a linear superposition of sources is also known;
(5.3)
Now, we can express any four-current as a linear superposition of deltafunctions:
(5.4)
Therefore, the solution to the wave equation driven by a delta-function is of particular importance, as we can write, in the Lorentz gauge,
(5.5)
Aµ µ0 jµ+ ∂ µ ∂νAν( ).=
µ,
A1 µ µ0 j1µ+ ∂ µ ∂νA1ν( ),=
A2 µ µ0 j2µ+ ∂ µ ∂νA2ν( ),=
λ1A1µ λ2A2µ+( ) µ0 λ1 j1µ λ2 j2µ+( )+ ∂ µ ∂ν λ1A1µ λ2A2µ+( )[ ].=
jµ xλ( ) jµ xλ( )δ4 xλ xλ-( ) d4x.∫∫∫∫=
G xλ xλ-( ) µ0δ4 xλ xλ-( )+ 0,=
and the general solution to the wave equation, with an arbitrary four-current source is
(5.6)
The solution is known as the Green function or propagator for the linear wave equation under consideration.
