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
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