MAXWELL - DIRAC EQUATIONS 17

iii) A necessary condition for the 1-cohomology with values in an irreducible representation

to be nonzero is that the center of the enveloping algebra be trivially represented. In the

general case it is sufficient to have at least one invertible operator in the representation of

the center of the enveloping algebra for the corresponding 1-cohomology to vanish.

The linear unitary representation V1 of the Poincare group canonically associated

with the Klein-Gordon equation in 1 + I dimensions, I 1, has the property that the

representation g »-» VgA(g)2V*1) defines a representation of the enveloping algebra, which

has an invertible element, namely the (mass)2-operator. This was the basic idea in [26] to

prove for the first time that the quadratic NLKG equation in 1 + 3 dimensions has global

solutions and a wave operator.

In the case of systems of NLKG equations with several nonzero masses, the (mass)2-

operator is in general not one to one. Such equations and the corresponding non-linear

Lie algebra representations are formally linearizable since the cohomology is trivial [31].

Moreover the present methods can be adapted [27] to include such cases. In fact the

methods shown there permit to include quadratic interactions in 1+3 dimensions, that

had not been treated before.

Combination of the idea of the invertibility of the

(mass)2-operator

from [26] and

new enveloping algebra "majoration" techniques permitted, in [28], to solve the small data

Cauchy problem and to prove asymptotic completeness for the NLKG equation in 1 + I

dimensions, / 2, with arbitrary C°° nonlinearity in the field and in its first space-time

derivatives. Moreover, when the equation is Poincare covariant, then the wave operators

intertwine the linear and nonlinear representations of the Poincare group, cf. §1.5 of this

introduction.

1.7 ORGANIZATION OF THE MONOGRAPH

In chapter 2, we prove that the sequence E& of Hilbert spaces admits a family of

smoothing operators (Theorem 2.5) in the sense of [25], in order to use an implicit function

theorem in the Frechet space E^.

In chapter 3, we prove that U^ is a nonlinear representation (Theorem 3.12) which

is nonlinearizable (Theorem 3.13).

In chapter 4, we construct approximate solutions (An,/n), n 0, (see formulas

(4.135a)-(4.135b)) of the M-D system. They are obtained, essentially by using stationary

phase methods and by iterating formulas (4.137b)-(4.137c). Their decrease properties are

given by Theorem 4.9 and Theorem 4.10. They are approximate solutions in the sense

that a certain remainder term (A^jf, A^) (see (4.140a)-(4.140b)) satisfies Theorem 4.11.

In particular the remainder term for the electromagnetic potential A^f is short-range (it

belongs to

L2(R3,R4)

for each fixed time).

In chapter 5, we prove equal time weighted

L2-L2

and

L2-L°°

estimates for the linear

inhomogeneous Dirac equation (i^d^ + m — 7MGM)/i = g, with external field G (Theorem

5.5 and Theorem 5.8). Combination of these estimates with an energy estimate (Corollary

5.2) leads to existence of h and to estimates on /i, adapted to the nonlinear problems

treated in chapter 6.

In chapter 6, we end the construction of an approximate solution (A*,/*) (formulas

(6.1a)-(6.2b) and (6.30)) satisfying the Lorentz gauge condition (Proposition 6.2). The

existence of the rest term (K, $) (see formulas (6.30-(6.31c)) follows by the construction of