A (t(x) 47 G"ret(x, z) Ja'(z)d4z, (2-117)
adv(x) 4 Gad(, z ()d4z, (2-118)
by which the advanced and retarded fields of the particle are written
re ret A Aret (2-119)
b b;a a;b,
Fadv Aadv Aadv (2-120)
b -a b;a a;b
The total field may be expressed in the forms
Fab F + Fret out adv. (2-121)
Fab Fab + ab ab + Fab (2
Alternatively, one may express the total field in the form
Fab fab + Fab, (2-122)
where
Fab 2 Ft+ F2adv) (2-123)
fab Fa + Fout
S l1" rad Rout 1 rad (-124)
iab + 2Fab ab ab (2-124)
2 2
F rad Fret _adv (2-125)
ab ab Fa- .(2-125)
Similarly to Eqs. (2-119) and (2-120), the fields Fab and Fard may be expressed in
terms of potentials Aa and Arad which are defined by the integral expressions of
the form (2-117) and (2-118), involving the functions Gaa' and Grada,, respectively.
The various fields defined thus satisfy the equations
gl/2Fretab;b gl/2Fadvab;b gl/2Fab;b 4= 4
(2-126)