MAXWELL - DIRAC EQUATIONS 13
where u = (/, / , a), then we say that Q is a gauge-projective 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,
(i-37)
+ \\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 gauge-projective 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 non-linear Klein-Gordon equation (NLKG) with a (at least) cubic non-linear
term. It states that there exist wave operators for small initial data, satisfying asymptotic
completeness. It also proved that, for some non-linear terms, the limitation to small initial
data is no more necessary and that the wave operators intertwine the corresponding linear
and non-linear 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°° non-linear term (in the field and its first space-time derivatives), van-
ishing together with its first derivatives at the origin. Moreover, when the equation is
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