Sidney Coleman wrote a report in 1961 that derived the LAD equation for a classical electron using what was then 'modern theory':
I've tried to copy his diagram below which formats OK using classic Google groups but not in the newer, current version. On page 35 he writes:
If we remember that in Maxwellian dynamics, the radiation field is given by the difference of the advanced and retarded fields, it is easy to see that the electron radiates.
\ |t /*
\ | /*
\ | /.
\ | /
\ | / *
\ \ | / /
_______________\__\ /__* ___/__________x
\ / \ * /
/ | \ /
/ \| \ */
/ | \/.
/ |\ /\
/ | \/ \*
/ | /\P \*
/ | \
The figure shows the world line of an electron undergoing hyperbolic acceleration. The past light cone from the point P does not interact with the electron worldline, so the retarded potential vanishes; the future light cone intersects it once, so the advanced potential doesn't. Thus there is a radiation field at P."
I don't agree with this. The past light cone of the observation point P not intersecting with the electron's world line means the retarded field at P is undefined rather than being zero, and hence the radiation field there is also undefined.
Thanks in advance for any comments.