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