MAXWELL  DIRAC EQUATIONS 13
where u = (/, / , a), then we say that Q is a gaugeprojective map.
To show that the asymptotic representation g H Ug is equal to
U1
modulo a gauge
pro jective map depending on g, we write the asymptotic condition (1.17c), with Ug (u)
instead of u:
ll^*ft)(n+(^+)(«)))

^xVo)(^1M(//))IU,
(i37)
+ \\u?MtPo)(n+(u^(u))) 
^^+,^^^p.[^)u^tBi)ui^D(u)\\D
 o,
£
when t — oo. Definition (1.23a) of the function ipg defining Ug by (1.17a) and (1.17b),
shows that
pg(u, ek)  i ? ( #
) !
(  + a)  £
(+)1
(), (t, etk/uj{k))) • 0, (1.38)
when t — oo, for g =
exp(aMPM).
We recall that
ipgiu)
= 0 for g € 5L(2,C) and we note
that Q?(#, •) = 0 for H(y) =
B^x(y
+ a) 
#(+)1(?/).
It follows from (1.37) and (1.38)
that, if A = #(#,), then
\\Ue^p{tPo)((^(U^u))) 
U^{tPo){UlM{fJ))\\MP
(1.39)
+ l l ^ ( t i ^
when t — oo, where #(£,fc)= A(£, —tk/u(k)). The definition of
A(+)
and ^(w, (t, x)) give
that
4+)(tfj+)M**0
= 4
+ )
(^(u),£ , &)• Since A = #(A,), where AM = 9MA, it follows
from (1.39) that
\W^{tPo){{^{U^u))) 
Ul^tPo)UlM{fJ)\\MP
(1.40)
+
II^P(.PO)((^(^
+ )
^)))
 Y,*u™{Q{ulu)^  °.
when t — oo, where G is the gauge transformation given by A (as in (1.35) with c = 0).
This shows that Qg = UgU^x is a gaugeprojective transformation according to (1.36).
1.5 RELATED WORK ON RELATIVISTICALLY COVARIANT LONG RANGE SCATTERING
1.5.1 Concerning covariance of wave operators, a fundamental result was proved in
[24], for the nonlinear KleinGordon equation (NLKG) with a (at least) cubic nonlinear
term. It states that there exist wave operators for small initial data, satisfying asymptotic
completeness. It also proved that, for some nonlinear terms, the limitation to small initial
data is no more necessary and that the wave operators intertwine the corresponding linear
and nonlinear representations of Poincare group.
The existence of the wave operators and the property of asymptotic completeness
for small initial data was proved in [28] for (NLKG), in 1 + n dimensions, n 2, with
an arbitrary C°° nonlinear term (in the field and its first spacetime derivatives), van
ishing together with its first derivatives at the origin. Moreover, when the equation is