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)