Let us describe the world-line of the electron in spacetime by the equation
Za= Za(s), (2-1)
where za(s) is a function of the proper-time s, and dzo/ds > 0. The electromagnetic
potential at the point Xa satisfies the Maxwell's equations
Ad,
A = 0, (2-2)
OA, = 47rJ, (2-3)
where Ja is the charge-current density vector. With our present model of the
electron, Ja vanishes everywhere except on the world-line of the electron, where it
is infinite
/dz
Ja e Cj 6(xo zo)6(xi zi)6(x2 z2)6(x3 z3)ds (2-4)
for an electron of charge e. The electromagnetic field tensor Fabcan be derived from
the potential Aa
Fab aAb abA (2-5)
Eqs (2-2) and (2-3) have many solutions and thus do not fix the field uniquely.
One may use a solution provided by the well-known retarded potentials of Li6nard
and Wiechert. We call the field derived from these potentials Fai. One can obtain
other solutions by adding to this one any solution of Eq. (2-2) and
Aa, = 0, (2-6)
representing a field of radiation. Then, the actual field F$,b for our one-electron
problem will be the superposition of the field from the retarded potentials and the
field from the solutions of Eq. (2-6) that represent the incoming electromagnetic
waves incident on our electron
Fab Fab + ab.
act ret in
(2-7)