Just when you thought it was safe to go back into the classroom, along comes
Jaws himself. Green's functions are your *friends!*

The inhomogeneous Maxwell equations are now compactly written as

(17.87) |

(17.88) |

(17.89) |

To solve this inhomogeneous differential equation, we construct simultaneously
a Green's function

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(17.91) |

Next week we will concentrate on the integral equation solutions themselves.
Now let us see how to construct the appropriate (covariant) Green's function.
As usual, the principle part of the Green's function can involve only the
absolute distance between the points. Thus if
we seek solutions to

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There are several ways we could go about solving this equation. They are all equivalent at some level or another. For example, we have already solved this equation for a single fourier component in Chapter 9. We could transform this result and obtain a four dimensional result. However, a more general procedure is to construct the solution from scratch.

The four dimensional fourier transform of the desired Green's function is
defined by

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The integrand in this expression is singular when .
Recall that the presence of singularities means that we have to decide
how to treat them to get a well-defined result. There are several ways
to do this, and each has a physical interpretation. If we integrate
over the ``time'' component first, we get

(17.97) |

Let's do the integral carefully (in case your contour integration is bit
rusty). Note that the poles of this integral are both real. This means
that the integral is ambiguous - it can be assigned any of several
possible values depending on how we choose to evaluation it. It is
beyond the scope of these notes to evaluate the consequences of making
and physically interpreting each of these choices. Instead we will
choose to include *both* poles completely within a standard contour
closed in the upper or lower half plane respectively, and then take
limits such that the poles return to the real axis after the integral
because this particular choice leads us very simply to the advanced and
retarded forms of the Green's function that we already obtained when
discussing the fourier transform of the incoming or outgoing spherical
Green's functions for the Helmholtz equation.

First we have to decide which way to close the contour. Examining the
integrand, we note that if
the integrand vanishes
on a lower-half contour like in the figure above. We displace the
poles down slightly so that they lie inside the contour :
. Finally, let
be a
complex variable such that the real axis is .

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We can then write the Green's function as

where is the spatial separation of the points and .

Using a trig identity (or if you prefer expanding the 's in terms
of exponentials and multiplying out, then changing variables and
exploiting the fact that only even terms survive) to extend the integral
to we can write this as:

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These remaining integrals are just one dimensional Dirac delta
functions. Evaluating, we get:

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If we had chosen the other contour, identical arguments would have led us to
the *advanced* Green's function:

(17.103) |

For what it is worth, the Green's functions can be put in covariant form. One
almost never uses them in that form, and it isn't pretty, so I won't bother
writing it down. We can now easily write down formal solutions to the wave
equation for *arbitrary* currents (*not* just harmonic ones):

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It is a worthwhile exercise to meditate upon what might be a suitable
form for the inhomogeneous terms if one considerst the integration
four-volume to be *infinite* (with no inhomogeneous term at all)
and then split the infinite volume up into the interior and exterior of
a finite four-volume, as we did with incoming and outgoing waves before,
especially when there are many charges and they are permitted to
interact.

Dirac noted that choosing a ``retarded'' Green's function, just as
choosing an ``outgoing wave'' Green's function before, results in a
somewhat misleading picture given that the actual physics is completely
time-reversal symmetric (indeed, independent of using a *mixed*
version of the Green's functions in either case). He therefore
introduced the ``radiation field'' as the *difference* between the
``outgoing'' and the ``incoming'' inhomogenous terms given the
contraint that the actual vector potential is the same regardless of
the choice of Green's function used::

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