MAXWELL - DIRAC EQUATIONS 9
argument (l,—£k/uj(k)) instead of (u;(fc), ek), which corresponds to the choice in [15].
The phase function si ' in (1.17c), which determines fi+, is defined by formula (1.18) and
the choice of A ( + ) = A ( + ) 1 -j- A ( + ) 2 given by (1.22a). With the choice Xo = 1 t n e function
hg introduced after (1.17b) is given by (hg{u))(y) = ^(B^1^ + a) - B^l(a),y) and
satisfies Dhg(u) = 0 if (/, / ) e E°f.
1.3.b Statements. We state the main results of this article for the case where t +oo.
There are analog results for t oo.
Theorem I. Let 1/2 p 1. If n 4 then U^:V0 x ££ p -* £ ° p is a continuous
nonlinear representation ofVo in E„p and, in addition, the function U^:Vo x E%£ £££
is C°°. Moreover U^ is not equivalent by a
C2
map to a linear representation on E%£.
If U[ and U2 are defined via (1.23a) by two choices of the function Xo, they are
equivalent.
Theorem I partially sums up Theorems 3.12-3.14.
Theorem II. Let 1/2 p 1. There exist an open neighbourhood UQQ (resp. Ooo ) of zero
in V^ (resp. E%g), a diffeomorphism £7+: Odo —+ IA00 and a C°° function U:Vo x Uoo
£^00; defining a nonlinear representation ofVo, such that:
i) jtUe*P(tx){u) = Tx{Uexp{tX){u)), X e p,t e R,u e Woo,
u) n
+
o u^ = ugon+
iii) Hm (||t^
(tft)
(n
+
(ti)) - Ue^{tPo)(fJ)\\MP
+ l|t^p(*ft)(n+(*)) - £ ^i+)(tt,t,-^ft(-w)tC(ti=b)ttllD) - °
e=±
foru = (f,f,a)eO&).
This theorem (see Theorem 6.19) solves in particular the Cauchy problem for small
initial data and proves asymptotic completeness. By the construction of the wave operator
f2+ in chapter 6, the solution (A(t7 -),A(t, -)^(t, •)) = t/exp(tp0)('u) °f the Cauchy problem
satisfies
sup3 ((1 + |x| + t ) 3 / 2 - ' | A
M
( t , z ) | + (1 + \x\ + t)|ft,AM(t,x)| + (1 + \x\+tff2\^{t,x)\)
00.
to
1.3.C Cohomological interpretation. These results and conditions (1.14) and (1.17c)
have a natural cohomological interpretation. We only consider the case where t —• 00, and
since the representations U^ defined for different Xo v * a (l-23a) are equivalent, we only
consider the case where Xo = 1- A necessary condition for £/(+) and £7+ to be a solution
of equation (1.14) is that the formal power series development of Ug , Ug and Q+ in the
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