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\centerline {\bf The Maxwell-Dirac equations : Asymptotic completeness and
the infrared problem}
\gsaut
\centerline {\bf Mosh\'e FLATO$^1$, Jacques SIMON$^1$ and Erik TAFLIN$^{1,2}$}
\saut
\hskip2.2cm $^1$ D\'epartement de Math\'ematiques, Universit\'e de Bourgogne,
\hskip2.5cm B.P. 138, F-21004 DIJON Cedex, FRANCE.
\hskip2.2cm $^2$ Permanent address : The New Technology Division, Union des
\hskip2.5cm Assurances de Paris,
\hskip2.5cm 20 ter, rue de Bezons, F-92411 COURBEVOIE Cedex, FRANCE.
\gsaut
\hskip2.5cm {\it Dedicated to Elliott Lieb with our affection.}
\gsaut
\gsaut
\saut
{\bf Abstract :} In this article we present an announcement of results concerning :
a) A solution to the Cauchy problem for the M-D equations, namely global existence,
for small initial data at $t = 0,$ of solutions for the M-D equations.
b) Arguments from which asymptotic completeness for the M-D equations follows.
c) Cohomological interpretation of the results in the spirit of nonlinear
representation theory and its connection to the infrared tail of the electron in
M-D classical field theory.
The full detailed results will be published elsewhere.
\gsaut
\saut
\sevenrm
[In press in Reviews in Mathematical Physics, special issue
dedicated to Elliott Lieb]
\tenrm
\vfill\eject
{\bf 1. Introduction : statement of the problem.}
\psaut
The classical Maxwell-Dirac (M-D) equations in $1+3$ dimensions read in conventional
notations (electron charge $e=1$ ; Dirac matrices $\gamma^\mu, \hskip.1cm 0 \leq
\mu \leq 3$ ; metric tensor $g^{\mu \nu}, \hskip.1cm g^{0 0} = 1, \hskip.1cm g^{ii} =
- 1$ for $1 \leq i \leq 3$ and $g^{\mu \nu} = 0$ for $\mu \neq \nu,$ ; $\gamma^\mu
\hskip.1cm \gamma^\nu + \gamma^\nu \hskip.1cm \gamma^\mu = 2 g^{\mu \nu}$ ; $\carre =
\partial_\mu \hskip.1cm \partial^\mu)$ :
$$\carre \hskip.1cm A_\mu = \bar{\psi} \hskip.1cm \gamma_\mu \hskip.1cm \psi,
\hskip.1cm 0 \leq \mu \leq 3, \eqno{(1.1 \hbox{a})}$$
$$(i \gamma^\mu \hskip.1cm \partial_\mu + m) \hskip.1cm \psi = A_\mu \hskip.1cm
\gamma^\mu \hskip.1cm \psi, \hskip.1cm m > 0, \eqno{(1.1 \hbox{b})}$$
$$\partial_\mu \hskip.1cm A^\mu = 0, \eqno{(1.1 \hbox{c})}$$
where $\bar{\psi} = \psi^+ \hskip.1cm \gamma^0, \hskip.1cm \psi^+$ being the
Hermitian conjugate of the Dirac spinor $\psi.$
Let us introduce some notations :
Let $E$ be a Hilbert space of vectors $(f, \dot{f}, \alpha)$
where the functions $f : \Rrm^3 \fl \Rrm^4, \hskip.1cm \dot{f} : \Rrm^3 \fl \Rrm^4,
\hskip.1cm \alpha : \Rrm^3 \fl \hskip.1cm \Crm^4$ belong to spaces to be specified
in paragraph 2. M (resp. D) is the space of vectors $(f, \dot{f})$ (resp. $\alpha$)
for $(f, \dot{f}, \alpha) \in E.$ As will be explained in detail in paragraph 2
there is a linear strongly continuous representation $U^1$ of the Poincar\'e group
in $E,$ which is defined by the linear part of equations (1.1a) and (1.1b) and
$U^1 = U^{M1} \oplus U^{D1}$ where $U^{M1}$ acts on M and $U^{D1}$ on D. $E_\infty$
denotes the space of differentiable vectors in $E$ for the representation $U^1.$
It contains vectors $(f, \dot{f}, \alpha),$ such that $f$ is a long range potential
decreasing as $(1 + \vert x \vert)^{\rho - 3/2}, \hskip.1cm x \in \Rrm^3$ and
$\dot{f}$ decreasing as $(1 + \vert x \vert)^{\rho - 5/2}$ for some $1/2 < \rho < 1.$
We shall consider solutions of equations (1.1a), (1.1b) and (1.1c) which are in
$E_\infty$ in the sense that $(A(t), \dot{A}(t), \psi (t)) \in E_\infty$ at time
$t,$ where $\dot{A}_\mu (t) = {d \over dt} \hskip.1cm A_\mu (t).$
It is well-known that the construction of the observables on the Fock space of
QED (Quantum Electrodynamics) requires infrared corrections to eliminate the
infrared divergencies in the perturbative expression of the quantum scattering
operator. These corrections are introduced by hand with the purpose to give {\it a
posteriori} a finite theory. In this paper we shall announce rigorous results which
we have obtained concerning the infrared problem for the (classical) M-D equations.
Our belief is that such results can {\it a priori} be of interest for QED, especially
for the infrared regime and combined with the deformation-quantization approach
[1], [2], [3].
On the classical level the infrared problem consists of determining to which extent
the long range interaction created by the coupling $A^\mu \hskip.1cm j_\mu$
between the electromagnetic potential $A_\mu$ and the electric current $j_\mu =
\bar{\psi} \hskip.1cm \gamma_\mu \hskip.1cm \psi$ is an obstruction for the separation,
when $\vert t \vert \fl \infty,$ of the nonlinear relativistic system into two
asymptotic isolated relativistic systems, one for the electromagnetic potential
$A_\mu$ and one for the Dirac field $\psi.$ It will be seen in this article that
there is such an obstruction, which in particular implies that asymptotic in and
out states do not transform according to a linear representation of the Poincar\'e
group. This constitutes a serious problem for the second quantization of the
asymptotic (in and outgoing) fields, since the particle interpretation usually requires
free relativistic fields, i.e. at least a linear representation of the Poincar\'e
group ($U^1$ in our case). Therefore we have to introduce nonlinear representations
$U^{as}$ of the Poincar\'e group ${\cal P}_0,$ in the sense of [4], which can give
the transformation of the asymptotic in and out states under ${\cal P}_0$ and
which can permit a particle interpretation.
There are two reasons which permit to
determine the class of asymptotic representations $U^{as}.$ First, the classical
observables, $4$-current density, $4$-momentum and $4$-angular momentum are invariant
under gauge transformations. Second, if the evolution equations of the asymptotic
fields become linear after a gauge transformation one can use the freedom of
gauge in the second quantization of the fields. It is therefore reasonable
to postulate that asymptotic representations $g \mapsto U^{as}_g$ are linear modulo
a nonlinear gauge transformation depending on $g$ and respecting the Lorentz gauge
condition. Since we compare solutions of the M-D system and the asymptotic
system only for large time we can from a technical point of view, using stationary
phase methods, define the asymptotic representations in the following way:
$$U^{as}_g (u) = (U^{as M}_g (u), U^{as D}_g (u)), \hskip.1cm U^{as M} = U^{M1},
\eqno{(1.2 \hbox{a})}$$
$$(U^{as D}_g (u))^\wedge \hskip.1cm (k)
= \sum_{\varepsilon = \pm} \hskip.1cm e^{i \varphi^\varepsilon_g (u,k)} \hskip.1cm
P_\varepsilon (k) \hskip.1cm (U^{D1}_g \hskip.1cm \alpha)^\wedge \hskip.1cm (k),
\hskip.1cm g \in {\cal P}_0, \eqno{(1.2 \hbox{b})}$$
where $u = (f, \dot{f}, \alpha) \in E, \hskip.1cm P_\varepsilon$ is the orthogonal
projector in $D$ on the subspace of initial of data with energy sign $\varepsilon = \pm,$
the symbol $\wedge$ denotes Fourier transform and the function
$(g,u,k) \mapsto \varphi^\varepsilon_g (u,k)$
from ${\cal P}_0 \times E_\infty \times \Rrm^3$ to $\Rrm$ is $C^\infty$ .
We shall call such nonlinear representations "gauge projectively linear".
The nonlinear relativistic system is defined by a nonlinear representation $T$
of the Poincar\'e Lie algebra ${\bf p}$ (in the sense of [4]) on a manifold $V_\infty
\subset E_\infty$ of initial conditions satisfying the Lorentz gauge condition,
which in the case of the M-D system (1.1a), (1.1b) and (1.1c) is in fact a reformulation
of the relativistic covariance of the system. If $g \mapsto U_g$ is a nonlinear
representation of the Poincar\'e group ${\cal P}_0,$ with the corresponding Lie
algebra representation $T$ we can conclude that the classical infrared problem
consists of determining gauge projectively linear representations $U^{(+)}$ and
$U^{(-)}$ equivalent to $U$ by the (modified) wave operators $\Omega_+$ and $\Omega_-,$
i.e.
$$U^{(\varepsilon)}_g = \Omega^{-1}_\varepsilon \hskip.1cm \circ \hskip.1cm U_g
\hskip.1cm \circ \hskip.1cm \Omega_\varepsilon, \hskip.1cm g \in {\cal P}_0,
\hskip.1cm \varepsilon = \pm, \eqno{(1.3 \hbox{a})}$$
where $U_{exp (t P_0)} \hskip.1cm (\Omega_\varepsilon (u)),$ $P_0$ being the time
evolution generator in ${\cal P}_0,$ converges in a sense that we shall specify
to $U^{(\varepsilon)}_{exp (t P_0)}$ $(u)$
as $\varepsilon t \fl \infty, \hskip.1cm \varepsilon = \pm.$ What is
important is that the limit takes place for the observables,
i.e. at least modulo gauge transformations (even not respecting the Lorentz condition).
Therefore we impose that, for $u = (f, \dot{f}, \alpha) \in E_\infty$ and satisfying
the Lorentz condition for the linear part of the M-D equations :
$$\displaystyle{\lim_{\eta t \fl \infty}} \hskip.1cm (\Vert U^M_{exp (t P_0)}
\hskip.1cm (\Omega_\eta (u)) - U^{M1}_{exp (t P_0)} \hskip.1cm (f, \dot{f}) \Vert_M
\eqno{(1.3 \hbox{b})}$$
$$+ \Vert U^D_{exp (t P_0)} \hskip.1cm (\Omega_\eta (u)) - \sum_{\varepsilon = \pm}
\hskip.1cm e^{i s^{(\eta)}_\varepsilon \hskip.1cm (u,t,-i \partial)} \hskip.1cm
P_\varepsilon \hskip.1cm (-i \partial) \hskip.1cm U^{(\eta)D}_{exp (t p_0)} (u)
\Vert_D) = 0,$$
$\eta = \pm,$ where $(u,t,k) \mapsto s^{(\eta)}_\varepsilon \hskip.1cm (u,t,k)$
is a real $C^\infty$ function and $s^{(\eta)}_\varepsilon \hskip.1cm (u,t,-i \partial)$
is the operator defined by the inverse Fourier transform of the multiplication
operator $k \mapsto s^{(\eta)}_\varepsilon \hskip.1cm (u,t,k).$ We also impose growth
conditions on the phase $s^{(\eta)}_\varepsilon,$ so that the major contribution
comes from $U^{(\eta)D}$ :
$$\vert s^{(\eta)}_\varepsilon \hskip.1cm (u,t,k) \vert \leq C \hskip.1cm w(k)
\hskip.1cm (1 + \vert t \vert)^\delta, \hskip.1cm \vert {d \over dt} \hskip.1cm
s^{(\eta)}_\varepsilon \hskip.1cm (u,t,k) \vert \leq C \hskip.1cm w(k) \hskip.1cm
(1+ \vert t \vert)^{\delta - 1}, \eqno{(1.3 \hbox{c})}$$
where $w(k) = (m^2 + \vert k \vert^2)^{1/2}$ and $C$ is a constant depending on
$u.$ We note that the equivalence (1.3a) requires the existence of $\Omega_\eta$
and its inverse, i.e. asymptotic completeness. For future reference let us denote
$$W^{(\eta)}_t (u) = \sum_{\varepsilon = \pm}
\hskip.1cm e^{i \hskip.1cm s^{(\eta)}_\varepsilon
\hskip.1cm (u,t,-i \partial)} \hskip.1cm P_\varepsilon (-i \partial).
\eqno{(1.3 \hbox{d})}$$
Next we shall study the cohomological interpretation of conditions (1.3a) and
(1.3b). We only consider the case $t \fl \infty.$ A necessary condition for $U^{(+)}$
and $\Omega_+$ to be a solution of equation (1.3a) is that the formal power series
development of $U^{(+)}_g,$ $U_g$ and $\Omega_+$ in the initial conditions satisfy
the cohomological equations defined in [4] and [5]. In particular (after trivial
transformations) the second order terms $U^{(+)2}_g, \hskip.1cm U^2_g$ and $\Omega^2_+$
shall satisfy
$$\delta \hskip.1cm \Omega^2_+ = C^2, \hskip.1cm \delta \hskip.1cm R^{(+)2} = 0
\eqno{(1.4)}$$
where $\delta$ is the coboundary operator defined by the representation $g \mapsto
U^1_g \hskip.1cm A^2 \hskip.1cm (\otimes^2 \hskip.1cm U^1_{g^{-1}})$ of the Poincar\'e
group ${\cal P}_0$ on bilinear maps, $C^2 = R^{(+)2} - R^2$ is the cocycle defined
by $R^2_g = U^2_g \hskip.1cm (U^1_{g^{-1}} \hskip.1cm \otimes \hskip.1cm U^1_{g^{-1}})$
and $R^{(+)2}_g = U^{(+)2}_g \hskip.1cm (U^1_{g^{-1}} \hskip.1cm \otimes \hskip.1cm
U^1_{g^{-1}}), \hskip.1cm g \in {\cal P}_0.$ Equation (1.4) shows that the cochain
$R^{(+)2}$ has to be a cocycle and then that the cocycle $C^2$ has to be a coboundary.
This is equivalent to the existence of a solution $U^{(+)}, \hskip.1cm \Omega_+$
modulo terms of order at least three. There are similar equations for higher order
terms
$$\delta \hskip.1cm \Omega^n_+ = C^n, \hskip.1cm \delta \hskip.1cm R^{(+)n} = 0,
\eqno{(1.5)}$$
where $C^n$ and $R^{(+)n}$ are functions of $\Omega^2_+,...,\Omega^{n-1}_+$ and
$U^{(+)2},...,U^{(+)n}.$ In a previous paper [6] we proved that there exists a
modified wave operator and global solutions of the M-D equations for a set of
scattering data $(f, \dot{f}, \alpha)$ satisfying
$\hat{f}_\mu (k) = \hat{\dot{f}}_\mu (k) = 0$
for $k$ in a neighbourhood of zero, i.e. $f_\mu$ and $\dot{f}_\mu$ have no low
frequencies component. It follows from that paper that the usual wave operator
(i.e. $U^{(+)} = U^1$ and $s^{(+)}_\varepsilon = 0$ in (1.3b)) does not exist,
even in this case where there are no low frequencies. In fact there is an obstruction
to the existence of such a solution of equation (1.5) for $n = 3$ due to the self-coupling
of $\psi$ with the electromagnetic potential created by the current $\bar{\psi}
\hskip.1cm \gamma_\mu \hskip.1cm \psi.$ However, as was proved in [6] there exists
a modified wave operator $\Omega_+$ satisfying (1.5) for $n \geq 2,$ with $U^{(+)} = U^1.$
Moreover it was proved that the phase functions $s^{(+)}_\varepsilon$ are given by
(see formulas (1.5), (3.28) and (3.33) of [6])
$$s^{(+)}_\varepsilon \hskip.1cm (u,t,k) = S_\varepsilon (u,t,k) - \varepsilon \hskip.1cm
w(k) \hskip.1cm t, \eqno{(1.6)}$$
where $S_\varepsilon$ is a certain approximate solution of the Hamilton-Jacobi
equation for a relativistic electron in an external electromagnetic potential :
$$({\partial \over \partial t} \hskip.1cm {\cal S}_\varepsilon (u,t,k) + A_0 \hskip.1cm
(t, - \nabla_k \hskip.1cm {\cal S}_\varepsilon (u,t,k)))^2
- \sum^3_{i=1} \hskip.1cm (k_i + A_i (t, - \nabla_k \hskip.1cm {\cal S}_\varepsilon
(u,t,k)))^2 = m^2. \eqno{(1.7)}$$
With the spaces defined in [6] it was possible to choose (formula (3.33a) of [6])
$$S_\varepsilon (u,t,k) = \varepsilon t w(k) - \theta (A^{(+)}, t, -
\varepsilon t k w (k)^{-1}), \eqno{(1.8)}$$
where $w (k) = (m^2 + \vert k \vert^2)^{1/2}, \hbox{ and } \theta
(A^{(+)}, y) = \int_{L(y)} \hskip.1cm A^{(+)}_\mu \hskip.1cm (z) \hskip.1cm dz^\mu,$
$$L(y) = \{ z \in \Rrm^4 \vert z = sy, \hskip.1cm 0 \leq s \leq 1 \},$$
$A^{(+)}$ being a certain approximate solution of equation (1.1a) containing the
long range part of $A.$ Thus in the absence of low frequencies we can choose $\Omega_+$
such that it intertwines the linear and the nonlinear representation of ${\cal P}_0.$
Now if $f_\mu, \dot{f}_\mu$ have a non-zero low frequency part, as is the
case of the Coulomb potential $(\hat{f}_\mu (k) \sim \vert k \vert^{-2})$ then
there is a cohomological obstruction already for $n=2$ if $U^{(+)2} = 0.$ The
essential point now is that the cocycle $R^2$ can be split into a trivializable
part $- C^2$ (a coboundary) and into a nontrivial part $R^{(+)2}$ which defines
$U^{(+)2},$ and therefore the whole representation $U^{(+)}.$ We shall call this
nontrivial part the infrared cocycle. As we shall see in paragraph 2, $U^{(+)}_g$
is trivial on the Lorentz subgroup $SL (2, \hskip.1cm \Crm)$
(i.e. $\varphi^\varepsilon_g = 0$
for $g \in SL (2, \hskip.1cm \Crm)$ in (1.2b)). Moreover denoting by $P_\mu, \hskip.1cm
0 \leq \mu \leq 3,$ the translation generators in ${\bf p} =
\Rrm^4 + {\bf sl}(2, \hskip.1cm \Crm)$
and denoting $\tilde{T}^{(+)D}_{P_\mu} (u) = {d \over ds} \hskip.1cm U^{D1}_{exp (-s
P_\mu)} \hskip.1cm U^{(+)D}_{exp (s P_\mu)} \hskip.1cm (u)
\vert_{s=0},$ we have for $u = (f, \dot{f}, \alpha)$ :
$$(\tilde{T}^{(+)D}_{P_\mu} (u))^\wedge (k) = - i \hskip.1cm \sum_{\varepsilon = \pm}
\hskip.1cm \theta^\infty \hskip.1cm (\chi' \partial_\mu \hskip.1cm
B, \hskip.1cm l_\varepsilon) \hskip.1cm P_\varepsilon (k) \hskip.1cm \hat{\alpha} (k),
\eqno{(1.9 \hbox{a})}$$
where
$$\theta^\infty \hskip.1cm (H,y) = \int^\infty_0 \hskip.1cm y^\nu \hskip.1cm
H_\nu (\tau y) \hskip.1cm d \tau, \hskip.1cm y \in \Rrm^4, \hskip.1cm H = (H_0,
H_1,H_2,H_3), \eqno{(1.9 \hbox{b})}$$
where $B_\nu$ is the free field with data $f_\nu, \dot{f}_\nu,$ $l_\varepsilon =
(w(k), - \varepsilon k), \hskip.1cm k \in \Rrm^3$ is the on-shell four-momentum
and $z \mapsto \chi' (z)$ a cutoff function equal to zero for $(z^2_0 - (z^2_1 +
z^2_2 + z^2_3))^{1/2} \leq 1$ and equal to one for $(z^2_0 - (z^2_1 + z^2_2 + z^2_3))^{1/2}
\geq 2.$
Let $S_\varepsilon, \hskip.1cm \varepsilon = \pm$ be the functions defined by
$$e^{i S_\varepsilon (u,t, -i \partial)} \hskip.1cm P_\varepsilon (-i \partial) \alpha =
e^{i s^{(+)}_\varepsilon \hskip.1cm (u,t, -i \partial)} \hskip.1cm P_\varepsilon
\hskip.1cm (-i \partial) \hskip.1cm U^{(+)D}_{exp (t P_0)} (u), \eqno{(1.10)}$$
where $u = (f, \dot{f}, u).$ Then $S_\varepsilon$ is given by (1.8). When the
limit $\theta (\chi' B, t, - \varepsilon k \hskip.1cm w(k)^{-1} t)
\fl \theta^\infty \hskip.1cm (\chi' B, l_\varepsilon), \hskip.1cm l_\varepsilon =
(w(k), - \varepsilon k)$ exists for $t \fl \infty,$ then the infrared cocycle is
in fact a coboundary. In particular, this is the case when $\hat{f}_\mu, \hat{\dot{f}}_\mu$
are equal to zero in a neighbourhood of zero.
In paragraph 2 we shall outline the proof of the facts mentioned previously as
well as that of the existence of the nonlinear representation $g \mapsto U_g$
of the Poincar\'e group on an invariant domain of initial data ${\cal O}_0,$ that
of the existence of modified wave operators $\Omega_\pm : {\cal O}_\pm \fl {\cal O}_0$
and that of asymptotic completeness. To prove these facts we construct as in [6], [9],
[10], [11] an approximate solution of the nonlinear evolution equation for
a given scattering data $(f, \dot{f}, \alpha).$ The enveloping algebra techniques
in [11] and new energy estimates and $L^2 - L^\infty$ estimates for the Dirac equation
permit to prove that the rest term (difference between what will be the solution
and the approximate solution) exists and is a decreasing function of time in $L^2$
spaces. We obtain a solution of the M-D equations with scattering data
$u_+ = (f, \dot{f}, \alpha).$
The corresponding initial data will be denoted by $u.$ The seminorms of $u$ considered
as a $C^n$-vector of $U^1$ are proved to be bounded by the norms of $T_Y (u), \hskip.1cm
Y \in {\cal U} ({\bf p}),$ where $T_Y (u)$ is the extension of $T_X (u),
\hskip.1cm X \in {\bf p}$
to the enveloping algebra ${\cal U} ({\bf p})$ defined in [11]. We can now conclude that
modified wave operators $\Omega_\varepsilon : {\cal O}'_\varepsilon \fl {\cal O}'_0,
\hskip.1cm \varepsilon = \pm$ exist on small open neighbourhoods ${\cal O}'_\varepsilon,
\hskip.1cm \varepsilon = \pm$ of the scattering data space which is here the Fr\'echet
space of $C^\infty$-vectors of $U^1.$ The maps $\Omega_\varepsilon$ are proved to
satisfy the hypothesis of the implicit functions theorem for Fr\'echet spaces. This
proves that ${\cal O}'_\varepsilon, \hskip.1cm \varepsilon = \pm$ and ${\cal O}_0$
can be chosen such that $\Omega_\varepsilon$ are invertible. This gives the
asymptotic completeness and the existence of $U_g$ on a Poincar\'e invariant
domain. The explicit expression of $U^{(+)}$ is found by a suitable choice of
$W^{(+)} (u)$ in (1.3d).
Before stating the results and outlining the proofs in paragraph 2 we make the
following remarks :
i) It can be natural to think of the underlying classical theory of QED not as
the M-D equations with $c$-number spinor components but as a theory with anticommuting
spinor components. Since the second order terms in the spinor component
of the wave operator originates in the coupling $A_\mu \hskip.1cm \psi$ of a
$c$-number free electromagnetic potential $A_\mu$ and a free Dirac-field, we
believe that the infrared cocycle will remain the same for a theory with
anticommuting spinor components. Further for higher order terms, the nilpotency
property $\psi_\alpha (x)^2 = 0$ can ameliorate the infrared problems arising from
the self-coupling.
ii) The observables, $4$-current, $4$-momentum and $4$-angular momentum, defined
for the asymptotic representation $U^{(+)},$ which is "gauge projectively linear",
converge when $t \fl \infty$ to the usual free field observables. Therefore
there should be no observable phenomenas which distinguish $U^{(+)}$ from $U^1,$
at least as far as these observables are concerned. Since the representation $U$ is
equivalent to $U^{(+)}$ by $\Omega_+,$ we shall call $U$ a "gauge projectively
linearizable" nonlinear representation. Of course if the Dirac field itself was
an observable, then this would no longer be true,
i.e. it would then be possible to distinguish
$U^{(+)}$ and $U^1$ by the observables.
iii) One should note that the phase-factor (3.33a of [6]), which
looks as a very familiar factor in abelian and nonabelian gauge
theories, was obtained in [6] in the different context of the
Hamilton-Jacobi equation associated with the full Maxwell-Dirac
equations. Would one have taken initial data for the $ A_{\mu}$
field decreasing slower than $r^{-1/2}$ and suitable initial
data for ${d \over dt}A_\mu $, one would obtain phase factors with
higher powers of $A_\mu$.
\saut
{\bf 2. Statement of results and outline of proofs.}
\psaut
Let $M^\rho, \hskip.1cm - 1/2 < \rho < \infty$ be the completion of $S(\Rrm^3,
\Rrm^4) \hskip.1cm \oplus \hskip.1cm S(\Rrm^3, \Rrm^4)$ with respect to the norm
$$\Vert (f, \dot{f}) \Vert_{M^\rho} = (\Vert \vert \nabla \vert^\rho f
\Vert^2_{L^2(\Rrm^3, \Rrm^4)} +
\Vert \vert \nabla \vert^{\rho - 1} \dot{f} \Vert^2_{L^2 (\Rrm^3, \Rrm^4)})^{1/2}.
\eqno{(2.1 \hbox{a})}$$
$M^\rho$ is a Hilbert space. Let $D = L^2 (\Rrm^3, \hskip.1cm \Crm^4)$ and let
$E^\rho = M^\rho \hskip.1cm \oplus \hskip.1cm D, \hskip.1cm - 1/2 < \rho < \infty,$
be the Hilbert space with norm
$$\Vert (f, \dot{f}, \alpha) \Vert_{E^\rho} = (\Vert (f, \dot{f}) \Vert^2_{M^\rho} +
\Vert \alpha \Vert^2_D)^{1/2}. \eqno{(2.1 \hbox{b})}$$
When there is no possibility of confusion we write $E$ (resp. $M$) instead of $E^\rho$
(resp. $M^\rho$) for $1/2 < \rho < 1.$ $M^{{\sevenrm o} \rho}$ is the closed subspace
of elements $(f, \dot{f}) \in M^\rho, \hskip.1cm - 1/2 < \rho < \infty$
such that
$$\dot{f}_0 = \sum_{1 \leq i \leq 3} \hskip.1cm \partial_i \hskip.1cm f_i, \hskip.1cm
f_0 = - \sum_{ 1 \leq i \leq 3} \hskip.1cm \vert \nabla \vert^{-2} \hskip.1cm
\partial_i \hskip.1cm \dot{f}_i. \eqno{(2.2 \hbox{c})}$$
A solution $B_\mu$ of $\carre \hskip.1cm B_\mu = 0, \hskip.1cm 0 \leq \mu \leq 3$
with initial conditions $(f, \dot{f}) \in M^\rho$ satisfies the gauge condition
$\partial_\mu \hskip.1cm B^\mu = 0$ if and only if
$(f, \dot{f}) \in M^{{\sevenrm o} \rho}.$
Let $\Pi = \{ P_\mu, \hskip.1cm M_{\alpha \beta} \vert 0 \leq \mu \leq 3, \hskip.1cm
0 \leq \alpha < \beta \leq 3 \}$ be a standard basis of the Poincar\'e Lie algebra
${\bf p} = \Rrm^4 + {\bf sl}(2, \hskip.1cm \Crm).$ $P_0$ is the time translation
generator, $P_i, \hskip.1cm 1 \leq i \leq 3$ the space translation generators,
$M_{ij}, \hskip.1cm 1 \leq i < j \leq 3$ the space rotation generators and $M_{oj},
\hskip.1cm 1 \leq j \leq 3$ the boost generators. We define $M_{\alpha \beta} =
- M_{\beta \alpha}$ for $0 \leq \beta \leq \alpha \leq 3.$ There is a linear
(strongly) continuous representation $U^1$ of ${\cal P}_0 = \Rrm^4 \hskip.1cm
. \hskip.1cm SL (2, \hskip.1cm \Crm)$ in $E^\rho, \hskip.1cm - 1/2 < \rho <
\infty$ with differentiable vectors $E^\rho_\infty$ such that its differential is
the following linear representation $T^1$ of ${\bf p}$ on $E^\rho_\infty$ :
$$T^1_{P_0} \hskip.1cm (f, \dot{f}, \alpha) = (\dot{f}, \Delta f, {\cal D} \alpha),
\Delta = \sum_{1 \leq i \leq 3} \partial^2_i, \hskip.1cm {\cal D} =
- \sum_{1 \leq j \leq 3}
\gamma^0 \gamma^j \partial_j + i \gamma^0 \hskip.1cm m \hskip.1cm ; \eqno{(2.3 \hbox{a})}$$
$$T^1_{P_i} (f, \dot{f}, \alpha) = \partial_i (f, \dot{f}, \alpha),
\hskip.1cm 1 \leq i \leq 3 \hskip.1cm ; \hskip1cm \eqno{(2.3 \hbox{b})}$$
$$T^1_{M_{ij}} (f, \dot{f}, \alpha) \hskip.1cm (x) = (x_i \partial_j - x_j
\partial_i) \hskip.1cm (f, \dot{f}, \alpha) \hskip.1cm (x) + (n_{ij} f, \hskip.1cm
n_{ij} \dot{f}, \hskip.1cm \sigma_{ij} \hskip.1cm \alpha) \hskip.1cm (x),
\eqno{(2.3 \hbox{c})}$$
$$1 \leq i < j \leq 3, \hskip.1cm \sigma_{ij} = - 1/2 \hskip.1cm \gamma_i \gamma_j
\in {\bf su}(2), \hskip.1cm n_{ij} \in {\bf so}(3) \hskip.1cm ;$$
$$T^1_{M_{oi}} (f, \dot{f}, \alpha) \hskip.1cm (x) = (x_i \hskip.1cm
\dot{f} (x), \hskip.1cm \sum^3_{j=1} \hskip.1cm \partial_j x_i \hskip.1cm
\partial_j f(x), \hskip.1cm x_i {\cal D} \alpha (x)) + (n_{oi} \hskip.1cm f, \hskip.1cm
n_{oi} \hskip.1cm \dot{f}, \hskip.1cm \sigma_{oi} \alpha) (x), \eqno{(2.3 \hbox{d})}$$
$$1\leq i \leq 3, \hskip.1cm
\sigma_{oi} = {1 \over 2} \hskip.1cm \gamma_0 \hskip.1cm \gamma_i \in
{\bf sl}(2, \hskip.1cm \Crm),
\hskip.1cm n_{oi} \in {\bf so}(3,1).$$
The explicit form of $n_{\alpha \beta}, \hskip.1cm 0 \leq \alpha < \beta \leq 3$
defining a vector representation of ${\bf p}$ on $\Rrm^4$ is not important in this
paper. We note that $U^1$ leaves $E^{{\sevenrm o} \rho} =
M^{{\sevenrm o} \rho} \oplus D$ invariant.
Following closely [11] we introduce the graded sequence of Hilbert spaces $E^\rho_i,$
$i \geq 0, \hskip.1cm E^\rho_0 = E^\rho,$ where $E^\rho_\infty \subset E^\rho_j \subset E^\rho_i$
for $i \leq j.$ $E^\rho_i$ is the space of $C^i$-vectors of the representation
$U^1.$ Suppose given an ordering $X_1 < X_2 < ... < X_{10}$ on $\Pi.$ Then, in
the universal enveloping algebra ${\cal U} ({\bf p})$ of ${\bf p},$
the subset of all products
$X^{\alpha_1}_1 ... X^{\alpha_{10}}_{10}, \hskip.1cm 0 \leq \alpha_i, \hskip.1cm
1 \leq i \leq 10$ is a basis $\Pi'$ of ${\cal U} ({\bf p}).$ If $Y = X^{\alpha_1}_1 ...
X^{\alpha_{10}}_{10} \in \Pi',$ we define $\vert Y \vert = \vert \alpha \vert =
\sum_{0 \leq i \leq 10} \hskip.1cm \alpha_i.$ Let $E^\rho_i, \hskip.1cm i \in \Nrm$
be the completion of $E^\rho_\infty$ with respect to the norm
$$\Vert u \Vert_{E^\rho_i} = (\sum_{Y \in \Pi', \hskip.1cm \vert Y \vert \leq i}
\hskip.1cm \Vert T^1_Y (u) \Vert^2_{E^\rho})^{1/2}, \eqno{(2.4)}$$
where $T^1_Y, \hskip.1cm Y \in {\cal U} ({\bf p})$ is defined by the canonical extension
of $T^1$ to the enveloping algebra ${\cal U} ({\bf p})$ of ${\bf p}.$
Let $M^{{\sevenrm o} \rho}_i =
M^\rho_i \cap M^{{\sevenrm o} \rho}, \hskip.1cm E^{{\sevenrm o} \rho}_i =
E^\rho_i \cap E^{{\sevenrm o} \rho}$ and similarly for $E^{{\sevenrm o} \rho}_\infty,
\hskip.1cm M^{{\sevenrm o} \rho}_\infty.$
To understand better what the elements of the spaces $E^\rho_i$ are we introduce
the following seminorms $q_n, \hskip.1cm n \geq 0$, on $E^\rho_\infty$ :
$$q_n (u) = (q^M_n (v)^2 + q^D_n (\alpha)^2)^{1/2}, \hskip.1cm u = (v, \alpha) \in
E^\rho_\infty,$$
$$v \in M^\rho_\infty, \hskip.1cm \alpha \in D_\infty,$$
$$q^M_n (v) = (\sum_{\vert \mu \vert \leq \vert \nu \vert \leq n} \hskip.1cm \Vert
M_\mu \hskip.1cm \partial^\nu v \Vert^2_{M^\rho})^{1/2} \eqno{(2.5 \hbox{a})}$$
and
$$q^D_n (\alpha) = (\sum_{\vert \mu \vert \leq n, \hskip.1cm \vert \nu \vert \leq n}
\hskip.1cm \Vert M_\mu \hskip.1cm \partial^\nu \hskip.1cm \alpha \Vert^2_D)^{1/2},
\eqno{(2.5 \hbox{b})}$$
where $\mu = (\mu_1, \mu_2, \mu_3), \hskip.1cm \nu = (\nu_1, \nu_2, \nu_3)$ are
multi-indices and $M_\mu (x) = x^\mu = x^{\mu_1}_1 \hskip.1cm x^{\mu_2}_2 \hskip.1cm
x^{\mu_3}_3.$
We have the following result, which is obtained by a close study of the explicit expression
for $T^1_Y, \hskip.1cm Y \in {\cal U} ({\bf p})$ :
\saut
{\bf Proposition 2.1 :} \it There are constants $C_n > 0$ such that \rm
$$C^{-1}_n \hskip.1cm \Vert u \Vert_{E^\rho_n} \leq q_n (u) \leq C_n \hskip.1cm
\Vert u \Vert_{E^\rho_n}, \hskip.1cm n \geq 0.$$
The decrease properties of elements $u = (f, \dot{f}, \alpha) \in E^\rho_\infty$
follow from this proposition; $\alpha \in S(\Rrm^3, \hskip.1cm \Crm^4)$ and
for $1/2 < \rho < 1$ we obtain using Sobolev estimates :
\saut
{\bf Proposition 2.2 :} \it If $(f, \dot{f}) \in M^\rho_\infty,
\hskip.1cm 1/2 < \rho < 1$ then
i) $(1+ \vert x \vert)^{3/2 - \rho} \hskip.1cm \vert f(x) \vert \leq C \hskip.1cm
\Vert (f,0) \Vert_{M^\rho_1},$
ii) $(1+\vert x \vert)^{5/2 - \rho + \vert \nu \vert} \hskip.1cm (\vert \partial^\nu
\partial_i \hskip.1cm f(x) \vert + \vert \partial^\nu \dot{f} (x) \vert)
\leq C_{\vert \nu \vert} \hskip.1cm \Vert (f, \dot{f})
\Vert_{M^\rho_{\vert \nu \vert + 2}},
\vert \nu \vert \geq 0,$
iii) If in addition $x^\alpha \partial^\beta \partial_i f \in L^p$
and $x^\alpha \partial^\beta
\dot{f} \in L^p, \hskip.1cm p = 6(5-2 \rho)^{-1},$ $\vert \alpha \vert \leq \vert
\beta \vert \leq n,$
\noindent
$1 \leq i \leq 3$ then \rm
$$\Vert (f, \dot{f}) \Vert_{M^\rho_n} \leq C_n \hskip.1cm \sum_{0 \leq \vert \alpha \vert
\leq \vert \beta \vert \leq n} \hskip.1cm (\sum_{0 \leq i \leq 3} \hskip.1cm \Vert
x^\alpha \partial^\beta \partial_i f) \Vert_{L^p} + \Vert x^\alpha \partial^\beta
\dot{f} \Vert_{L^p}).$$
This proposition shows in particular that long range potentials satisfying
$$\vert \partial^\nu \hskip.1cm f(x) \vert \leq C_\nu \hskip.1cm
(1+\vert x \vert)^{\rho - 3/2 -
\vert \nu \vert - \varepsilon}, \hskip.1cm \vert \partial^\nu \hskip.1cm \dot{f} (x) \vert
\leq C_\nu \hskip.1cm (1+\vert x \vert)^{\rho - 5/2 - \vert \nu \vert - \varepsilon}
\eqno{(2.6)}$$
for all $\vert \nu \vert \geq 0$ and some $\varepsilon > 0,$ are in $M^\rho_\infty,
\hskip.1cm 1/2 < \rho < 1.$
A nonlinear representation $T$ of ${\bf p}$ on $E^\rho_\infty,$ in the sense of [4],
is obtained by the fact that the M-D equations are manifestly covariant :
$$T_X = T^1_X + T^2_X, \hskip.1cm X \in {\bf p}, \eqno{(2.7)}$$
where $T^1$ is given by (2.3a) - (2.3d) and for
$u = (f, \dot{f}, \alpha) \in E^\rho_\infty$ :
$$T^2_{P_0} (u) = (0, \bar{\alpha} \hskip.1cm \gamma \hskip.1cm \alpha, \hskip.1cm
-i \hskip.1cm f_\mu \hskip.1cm \gamma^0 \hskip.1cm \gamma^\mu \hskip.1cm \alpha),
\hskip.1cm \gamma = (\gamma_0, \gamma_1, \gamma_2, \gamma_3) ;
\hskip1cm \eqno{(2.8 \hbox{a})}$$
$$T^2_{P_i} = 0, \hskip.1cm 1 \leq i \leq 3 \hskip.1cm ; \hskip.1cm T^2_{M_{ij}} = 0,
\hskip.1cm 1 \leq i < j \leq 3 \hskip.1cm ; \hskip1cm \eqno{(2.8 \hbox{b})}$$
$$T^2_{M_{oj}} (u) = x_j \hskip.1cm T^2_{P_0} (u), \hskip.1cm 1 \leq i \leq 3.
\hskip2cm \eqno{(2.8 \hbox{c})}$$
The gauge condition (1.1c) takes on initial data $u = (f, \dot{f}, \alpha)$ the
form (cf. [7])
$$\dot{f}_0 = \sum^3_{i=1} \hskip.1cm \partial_i f_i, \hskip.1cm \Delta
f_0 = \sum^3_{i=1} \hskip.1cm \partial_i f_i - \vert \alpha \vert^2.
\eqno{(2.8 \hbox{d})}$$
Let $V^\rho_N$ be the subset of elements in $E^\rho_N$ satisfying (2.8d). It is
easy to establish the following result :
\saut
{\bf Proposition 2.3 :} \it There is a continuous polynomial of degree two $F : E^\rho_N
\fl E^\rho_N, \hskip.1cm 1/2 < \rho < 1, \hskip.1cm N \geq 1,$ which is a diffeomorphism
from $E^\rho_N$ onto $E^\rho_N$ and which is such that $V^\rho_N = F^{-1} \hskip.1cm
(E^{{\sevenrm o} \rho}_N).$ \rm
Continuing to follow [11], we extend the linear map $X \mapsto T_X$ from ${\bf p}$
to the vector space of all differentiable maps from $E_\infty$ to $E_\infty,$ to
the enveloping algebra ${\cal U} ({\bf p})$ by defining inductively $T_{\bf 1} = I,$ where
${\bf 1}$ is the identity in ${\cal U} ({\bf p}),$ and $T_{YX}$ by
$$T_{YX} = DT_Y.T_X, \hskip.1cm \hbox{ for} \hskip.1cm Y \in {\cal U} ({\bf p}),
\hskip.1cm X \in {\bf p}, \eqno{(2.9)}$$
where $(DA.B) (f)$ is the Fr\'echet derivative of $A$ at the point $f$ in the
direction $B(f).$ It was proved in [11] that this defines a linear map from
${\cal U} ({\bf p})$ to the vector space of
differentiable maps from $E_\infty$ to $E_\infty$
(see formula (1.10) of [11] and the sequel) and that
$${d \over dt} \hskip.1cm T_{Y(t)} \hskip.1cm (u(t)) = T_{P_0 Y(t)}
\hskip.1cm (u(t)), \hskip.1cm Y \in {\cal U} ({\bf p}), \eqno{(2.10 \hbox{a})}$$
where
$$Y(t) = exp (t P_0) \hskip.1cm Y \hskip.1cm exp (-t P_0),
\eqno{(2.10 \hbox{b})}$$
if
$${d \over dt} \hskip.1cm u(t) = T_{P_0} \hskip.1cm (u(t)). \eqno{(2.10 \hbox{c})}$$
Next we introduce the nonlinear representations $U^{(+)}.$ Let $u_+ = (f, \dot{f},
\alpha) \in E^{{\sevenrm o} \rho}_\infty$ and let
$$J^{(+)}_\mu \hskip.1cm (t,x) = ({m \over t})^3 \hskip.1cm ({t \over \delta (t,x)})^5
\times$$
$$\sum_{\varepsilon = \pm} \hskip.1cm ((P_\varepsilon \hskip.1cm (-i \partial)
\alpha)^\wedge \hskip.1cm (- { \varepsilon \hskip.1cm m \hskip.1cm x \over \delta
\hskip.1cm (t,x)}))^+ \hskip.1cm \gamma^0 \hskip.1cm \gamma_\mu \hskip.1cm (P_\varepsilon
(-i \partial) \alpha)^\wedge \hskip.1cm (- { \varepsilon \hskip.1cm m \hskip.1cm x
\over \delta \hskip.1cm (t,x)}), \eqno{(2.11)}$$
where $\delta (t,x) = (t^2 - \vert x \vert^2)^{1/2} > 0, \hskip.1cm t > 0$ and
$$P_\varepsilon (k) = {1 \over 2} \hskip.1cm (I + \varepsilon \hskip.1cm w(k)^{-1}
\hskip.1cm (- \sum^3_{j=1} \hskip.1cm \gamma^0 \hskip.1cm \gamma^j \hskip.1cm k_j +
m \hskip.1cm \gamma^0)) \eqno{(2.12)}$$
defines the orthogonal projectors in $D$ on the subspace of energy sign $\varepsilon.$
The Fourier transform is defined by
$$\hat{f} (k) = (2 \pi)^{- 3/2} \hskip.1cm \int_{\Rrm^3} \hskip.1cm e^{i k x}
\hskip.1cm f(x) \hskip.1cm dx,$$
and by $(P_\varepsilon (-i \partial) \alpha)^\wedge (- { \varepsilon mx \over \delta
(t,x)})$ we mean the value of the function $(P_\varepsilon (-i \partial) \alpha)^\wedge$
for the argument $k = - { \varepsilon mx \over \delta (t,x)}$.
Since $\alpha \in S (\Rrm^3, \hskip.1cm \Crm^4),$ (2.11) defines for a given $t>0$
a function $J^{(+)}_\mu (t,.)$ in $S (\Rrm^3, \Rrm),$ vanishing for $\vert x \vert \geq t.$
$J^{(+)}_\mu$ is the leading contribution obtained from the stationary phase method
to $\sum_\varepsilon \hskip.1cm (e^{{\cal D}t} \hskip.1cm P_\varepsilon (-i \partial)
\alpha)^+ \hskip.1cm \gamma^0 \hskip.1cm \gamma_\mu \hskip.1cm (e^{{\cal D}t} \hskip.1cm
P_\varepsilon (-i \partial) \alpha),$ when $t \fl \infty.$ Choose $\chi \in C^\infty
(\Rrm),
\chi (\tau) = 0$ for $\tau \leq 1, \hskip.1cm \chi (\tau) = 1$ for
$\tau \geq 2$ and $0 \leq \chi (\tau) \leq 1$ for $\tau \in \Rrm.$ We define
$$A^{(+)}_\mu \hskip.1cm (t,x) = \chi (\delta (t,x)) \hskip.1cm B^{(+)}_\mu (t,x),
\hskip.1cm \hbox{ for} \hskip.1cm \delta (t,x) \geq 1,
\hskip.1cm t \geq 0, \eqno{(2.13 \hbox{a})}$$
$\hskip2.6cm A^{(+)}_\mu \hskip.1cm (t,x) = 0$ otherwise,
$$B^{(+)}_\mu \hskip.1cm (t,.) = B^{(+)1}_\mu (t,.)
- \int^\infty_t \vert \nabla \vert^{-1} \hbox{ sin } (\vert \nabla \vert
(t-s)) \hskip.1cm J^{(+)}_\mu (s,.) \hskip.1cm ds, \eqno{(2.13 \hbox{b})}$$
where $t > 0$, and
$$B^{(+)1}_\mu (t,.) = \hbox{ cos } (\vert \nabla \vert t) \hskip.1cm f_\mu +
\vert \nabla \vert^{-1} \hskip.1cm \hbox{ sin } (\vert \nabla \vert t) \hskip.1cm
\dot{f}_\mu.$$
The cutoff function $\chi'$ : $(t,x) \mapsto \chi (\delta (t,x))$ has been introduced to
exclude, in a Lorentz invariant way the points $(0,x), \hskip.1cm x \in \Rrm^3$
from the support of $A^{(+)}_\mu.$
Then we define the nonlinear representation $U^{(+)}$ of ${\cal P}_0$ on
$E^{{\sevenrm o} \rho}_\infty$ by
$$U^{(+)}_g \hskip.1cm (u_+) = (I \oplus I \oplus e_g \hskip.1cm (f, \dot{f})) \hskip.1cm
U^1_g \hskip.1cm u_+, \hskip.1cm g \in {\cal P}_0, u_+ \in E^{{\sevenrm o} \rho}_\infty,
\eqno{(2.14)}$$
where $e_g (f, \dot{f})$ is the linear operator from $D$ to $D$ given by
$$e_g (f, \dot{f}) = I \hbox{ for } g \in SL(2, \hskip.1cm \Crm) \eqno{(2.15 \hbox{a})}$$
and
$$(e_g (f, \dot{f}) \hskip.1cm \alpha)^\wedge \hskip.1cm (k) = exp (i \sum_{\varepsilon
= \pm} \hskip.1cm \theta^\infty \hskip.1cm (\chi' (B^{(+)1} \hskip.1cm ({\bf .} +
a) - B^{(+)1} ({\bf .})), \hskip.1cm l_\varepsilon (k)) \eqno{(2.15 \hbox{b})}$$
for $g = exp (a^\mu \hskip.1cm P_\mu), \hskip.1cm l_\varepsilon (k) = (w(k), -
\varepsilon k), \hskip.1cm k \in \Rrm^3.$
Finally we introduce, for given $u_+ = (f, \dot{f}, \alpha) \in
E^{{\sevenrm o} \rho}_\infty,$
the operator $W^{(+)}_t \hskip.1cm (u_+)$ satisfying (1.3a) - (1.3d) by
$$(W^{(+)}_t \hskip.1cm (u_+) \beta)^\wedge (k) = \sum_{\varepsilon = \pm} \hskip.1cm
e^{i s^{(+)}_\varepsilon (u_+,t,k)} P_\varepsilon (k) \hskip.1cm \hat{\beta} (k),
\eqno{(2.16)}$$
where $s^{(+)}_\varepsilon (u_+,t,k) = - \theta (A^{(+)}, t, - \varepsilon t k
w(k)^{-1}) - \theta^\infty (\chi' (B^{(+)} (. + b(t)) - B^{(+)} (.), \hskip.1cm
l_\varepsilon (k))$, and $b(t) = (t,0,0,0).$
\saut
{\bf Proposition 2.4 :} \it If $n \geq 4$ then $U^{(+)}$ is a continuous nonlinear
(analytic) representation of ${\cal P}_0$ on $E^{{\sevenrm o} \rho}_n,
\hskip.1cm 1/2 < \rho < 1$ and it is a $C^\infty$ nonlinear representation of ${\cal P}_0$
on $E^{{\sevenrm o} \rho}_\infty.$ \rm
We now state the main results. There is an invariant neighbourhood in $V^\rho_\infty$
on which $U_g, \hskip.1cm g \in {\cal P}_0$ exists :
\saut
{\bf Theorem 2.5 :} \it There exists a neighbourhood ${\cal O}_0$ of zero in $V^\rho_\infty,
\hskip.1cm 1/2 < \rho < 1$ such that $T$ is the differential of a unique $C^\infty$ nonlinear
representation $U$ of ${\cal P}_0$ on ${\cal O}_0.$
In particular the M-D equations have global solutions for initial data in ${\cal O}_0.$ \rm
\psaut
Theorem 2.5 follows from the fact that the modified wave operators $\Omega_\pm$
exist and are invertible. We state the result only for the case of $\Omega_+.$
\saut
{\bf Theorem 2.6 :} \it Let $1/2 < \rho < 1.$
Then there exists a neighbourhood ${\cal O}_0$ of zero in
$V^\rho_\infty,$ a neighbourhood ${\cal O}_+$ of zero in $E^{{\sevenrm o} \rho}_\infty$
and a $C^\infty$-diffeomorphism $\Omega_+ : {\cal O}_+ \fl {\cal O}_0$ satisfying :
i) $U_g \hskip.1cm \circ \hskip.1cm \Omega_+ =
\Omega_+ \hskip.1cm \circ \hskip.1cm U^{(+)}_g,$
for $g \in {\cal P}_0,$
ii) If $h(t) = \hbox{ exp } (t \hskip.1cm P_0),$ then for $u_+ \in {\cal O}_+ :$ \rm
$$\displaystyle{\lim_{t \fl \infty}} \hskip.1cm \Vert U_{h(t)} \hskip.1cm (\Omega_+
(u_+)) - W^{(+)}_t (u_+) \hskip.1cm U^{(+)}_{h(t)} \hskip.1cm
(u_+) \Vert_E = 0.$$
We shall outline the crucial steps of the proof of Theorem 2.6. As in the case
of the Klein-Gordon equation [11], the linear operators $T^1_Y, \hskip.1cm Y \in
{\cal U} ({\bf p})$ are bounded by the nonlinear operators
$T_Y, \hskip.1cm Y \in {\cal U} ({\bf p}).$
Namely, with the notation (cf. (2.38) of [11])
$${\cal P}_N (a) = (\sum_{Y \in \Pi', \hskip.1cm \vert Y \vert \leq N} \hskip.1cm
\Vert a_Y \Vert^2_E)^{1/2}, \hskip.1cm N \geq 0, \eqno{(2.17)}$$
where $Y \mapsto a_Y$ is a linear map from ${\cal U} ({\bf p})$ to $E$
we have the following result :
\saut
{\bf Proposition 2.7 :} \it There exists a neighbourhood
${\cal O}$ of zero in $E$ such that for $u \in E_N
\cap {\cal O}$ :
$${\cal P}_N \hskip.1cm (T(u)) \leq C_N \hskip.1cm \Vert u \Vert_{E_N} \leq F_N
\hskip.1cm ({\cal P}_1 (u)) \hskip.1cm {\cal P}_N (u), \hskip.1cm N \geq 0,$$
where $C_N$ is independent of $u,$ $F_N$ is a nondecreasing function on $\Rrm^+$
and for $N = 0,1,$ $F_N$ is a constant. \rm
The next step in the proof consists of establishing approximate solutions of the
M-D equations for given scattering data $u_+ = (f, \dot{f}, \alpha) \in
E^{{\sevenrm o} \rho}_\infty,
\hskip.1cm 1/2 < \rho < 1.$ This is done by iterating
$$A_{n+1, \mu} (t) = A_{0,\mu} (t) - \int^\infty_t \hskip.1cm \vert \nabla \vert^{-1}
\hskip.1cm sin (\vert \nabla \vert \hskip.1cm (t-s)) \hskip.1cm \sum_{\varepsilon = \pm}
\hskip.1cm (\phi'^{\varepsilon}_n)^+ \hskip.1cm \gamma^0 \hskip.1cm \gamma_\mu
\hskip.1cm \phi'^\varepsilon_n (s) \hskip.1cm ds, \eqno{(2.18 \hbox{a})}$$
$$\phi'^\varepsilon_{n+1} (t) = \phi'^\varepsilon_0 (t) + i \int^\infty_t \hskip.1cm
e^{(t-s) {\cal D}} \hskip.1cm (A_{n,\mu} + B_{n, \mu}) \hskip.1cm \gamma^0 \hskip.1cm
\gamma^\mu \hskip.1cm \phi'^\varepsilon_n (s) \hskip.1cm ds,
\hskip.1cm t > 0 \eqno{(2.18 \hbox{b})}$$
where $B_{n,\mu} (y) = - \partial_\mu \theta (A_n, y), \hskip.1cm y
\in \Rrm^4, \hskip.1cm \phi'^\varepsilon_0 (t) = P_\varepsilon (-i \partial)
e^{ {\cal D} t} \hskip.1cm \alpha,$ $A_{0,\mu} (t) = \hbox{ cos } (\vert \nabla \vert t)
\hskip.1cm f_\mu + \vert \nabla \vert^{-1} \hskip.1cm \hbox{ sin } (\vert \nabla \vert t)
\hskip.1cm \dot{f}_\mu.$ By using stationary phase methods it follows that for
$n$ sufficiently large and $1/2 < \rho < 1, \hskip.1cm 0 \leq \rho' \leq 1$ :
$$(1+t)^{\rho' - \rho+1} \hskip.1cm \Vert (A_{n+1} (t) - A_n (t),
\hskip.1cm \dot{A}_{n+1} (t) -
\dot{A}_n (t)) \Vert_{M^{\rho'}} \eqno{(2.19)}$$
$$+ (1+t)^{3/2 - \rho} \hskip.1cm \Vert \phi'_{n+1} (t) - \phi'_n \Vert_D +$$
$$+ (1+t + \vert x \vert)^{5/2 - \rho + \vert \nu \vert - \varepsilon} \hskip.1cm
(1+t)^\varepsilon \hskip.1cm \vert {\partial^{\nu_0} \over \partial t^{\nu_0}}
\hskip.1cm \partial^{\nu_1}_1 \hskip.1cm \partial^{\nu_2}_2 \hskip.1cm
\partial^{\nu_3}_3 \hskip.1cm (A_{n+1} (t,x) - A_n (t,x)) \vert$$
$$+ (1+t+\vert x \vert)^{3 - \rho} \hskip.1cm \vert \phi'_{n+1} (t,x) - \phi'_n
(t,x) \vert \leq C_\varepsilon \hskip.1cm \Vert u \Vert_{E_{N_0}},$$
for some $N_0$ depending on $\rho$, the constants $C_\varepsilon$ depending on
$\rho$ and $\Vert u \Vert_{E_{N_0}}.$ Similar estimates are satisfied for the
derivatives of $A_n, \dot{A}_n, \phi'_n$ with respect to the scattering data and the
terms obtained by the action of the enveloping algebra ${\cal U} ({\bf p})$ on these
derivatives.
Let $M$ be such that (2.19) is satisfied with $n = M - 1$ and let
$$K_\mu (t) = A_\mu (t) - A_{M,\mu} (t), \hskip.1cm \Phi (t) = e^{i \theta (A,t)}
\hskip.1cm \psi (t) - \phi'_M (t), \eqno{(2.20)}$$
$t \geq 0,$ where $A_\mu$ and $\psi$ are the unknown variables in the M-D equations.
To prove the existence of a solution of the equation for the new variables $(K,\Phi)$
we shall need a new energy estimate. If $G_\mu, \partial_\nu G_\mu \in
L^\infty (\Rrm^+ \times \Rrm^3, \Rrm^4)
\cap C^0(\Rrm^+, (1 - \Delta )^{-1/2} M^1),
0 \leq \mu, \nu \leq 3,$ it then follows from [8] that the
equation $(i \gamma^\mu \partial_\mu + m) \hskip.1cm h = \gamma^\mu
G_\mu h$ defines a unitary evolution operator
family $w (t,s), \hskip.1cm t,s \geq 0,$ strongly continuous
on $D$ and continuously differentiable on
$W^{1,2} \hskip.1cm (\Rrm^3, \hskip.1cm \Crm^4).$ Therefore the solution in $D$
of the equation
$$(i \gamma^\mu \partial_\mu + m) \hskip.1cm h - G_\mu
\gamma^\mu h = F_\mu \gamma^\mu r (\in D) \eqno{(2.21 \hbox{a})}$$
is obtained by integration with $w(t,s).$ Suppose that $r$ satisfies the equation
$$(i \gamma^\mu \partial_\mu + m) \hskip.1cm r = q .\eqno{(2.21 \hbox{b})}$$
\psaut
{\bf Proposition 2.8 :} \it Let $Q (t,x) = t \hskip.1cm F_0 (t,x) +
\sum_{1 \leq i \leq 3} \hskip.1cm x_i
\hskip.1cm F_i (t,x),$ $\xi_{M_{oi}} = x_i {\partial \over \partial t}
+ t \partial_i.$ If $0 \leq t \leq t'$ and $0 \leq a \leq 1$ then
$$\vert \hskip.1cm \Vert h(t) \Vert_D - \Vert h(t') \Vert_D
\hskip.1cm \vert \leq {2 \over m} \hskip.1cm
\sup_{t \leq s \leq t'} \hskip.1cm \Vert \gamma^\mu \hskip.1cm F_\mu (s) \hskip.1cm
r(s) \Vert_D$$
$$+ {1 \over m} \hskip.1cm \int^{t'}_t \hskip.1cm (\Vert r(s) \hskip.1cm \partial_\mu
F^\mu (s) + {1 \over 2} \hskip.1cm r(s) \hskip.1cm \gamma^\mu
\gamma^\nu \hskip.1cm (\partial_\mu F_\nu (s) - \partial_\nu F_\mu (s))$$
$$+ i \gamma^\mu \gamma^\nu \hskip.1cm G_\mu (s) \hskip.1cm F_\nu (s) +
i \gamma^\mu F_\mu (s) \hskip.1cm q(s) \Vert_D$$
$$+ 2 \hskip.1cm \Vert (1+s)^{a-1} \hskip.1cm \vert (Q(s) + F_0 (s)) \hskip.1cm
{d \over ds} \hskip.1cm r(s) - \sum_{1 \leq i \leq 3} \hskip.1cm F_i (s) \hskip.1cm
(\xi_{M_{oi}} \hskip.1cm r) (s) \vert^{1-a}$$
$$\times \vert F^\mu (s) \hskip.1cm \partial_\mu \hskip.1cm r(s)
\vert^a \Vert_D) \hskip.1cm ds,$$
where the summation convention is used for the repeated upper and lower indices
$\mu$ and $\nu.$ \rm
This energy estimate permits to prove that the rest term and its derivatives exist
in $L^2$ for each $t \geq 0.$
Using the fact that there exists a smothing operator on $E^\rho_\infty$ we can
now prove that the wave operator exists : $\Omega_+ (u_+) = u(0) = (A(0), \dot{A} (0),
\psi (0)).$
\saut
{\bf Theorem 2.9 :} \it Let $1/2 < \rho < 1.$ There exists $N_0 \geq 0$
and a neighbourhood ${\cal O}_{N_0}$
of zero in $E^{{\sevenrm o} \rho}_{N_0}$ such that $\Omega_+ : {\cal O}_{N_0}
\cap E^{{\sevenrm o} \rho}_\infty \fl E^{{\sevenrm o} \rho}_\infty$
is a one-to-one $C^\infty$ map satisfying
$$\Vert (D^l \hskip.1cm \Omega_+) \hskip.1cm (u ; u_1,...,u_l) \Vert_{E_L} \leq C_{L,l}
\hskip.1cm \Vert u \Vert_{E_{N_0+l+L}} \hskip.1cm \Vert u_1 \Vert_{E_{N_0}}...
\Vert u_l \Vert_{E_L}$$
$$+ C'_{L,l} \hskip.1cm \Vert u_1 \Vert_{E_{N_0+l+L}} \hskip.1cm \Vert u_2
\Vert_{E_{N_0}}...
\Vert u_l \Vert_{E_{N_0}}
+...+ C'_{L,l} \hskip.1cm \Vert u_1 \Vert_{E_{N_0}}...\Vert u_{l-1} \Vert_{E_{N_0}}
\hskip.1cm \Vert u_l \Vert_{E_{N_0+l+L}},$$
for each $L \geq 0, \hskip.1cm l \geq 0, \hskip.1cm u_1,...,u_l
\in E^{{\sevenrm o} \rho}_\infty,$
where $C_{L,l}$ and $C'_{L,l}$ are constants depending only of $\Vert u \Vert_{E_{N_0}}.$
Moreover $\Omega_+ ({\cal O}_{N_0} \cap E^{{\sevenrm o} \rho}_\infty)
\subset V^\rho_\infty.$ \rm
\psaut
$D^l$ denotes here the derivative of order $l.$
Finally, using once more the energy estimate in Proposition 2.8 one proves that
the derivative $D \hskip.1cm \Omega_+$ has a right inverse satisfying an inequality
similar to that in Theorem 2.9. The existence of the inverse $\Omega^{-1}_+$ now
follows from the implicit functions theorem in a Fr\'echet space.
\vfill\eject
{\bf REFERENCES.}
\psaut
[1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer : "Deformation
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61-110 (1978).
\psaut
[2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer : "Deformation
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\psaut
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\psaut
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\psaut
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\psaut
[6] M. Flato, J. Simon, E. Taflin : "On Global Solutions of the Maxwell-Dirac
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\psaut
[7] L. Gross. : "The Cauchy problem for the coupled Maxwell and Dirac equations". Commun.
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\psaut
[8] T. Kato : "Quasilinear equations of evolution, with application to partial
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\psaut
[9] J.C.H. Simon : "A wave operator for a non-linear Klein-Gordon equation". Lett.
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\psaut
[10] J.C.H. Simon, E. Taflin : "Wave operators and analytic solutions for systems of
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\psaut
[11] J.C.H. Simon, E. Taflin : "The Cauchy problem for non-linear Klein-Gordon Equations".
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\end
ENDBODY