On the classical level the infrared problem consists of determining to which extent the
long-range interaction created by the coupling A^j^ between the electromagnetic potential
A^ and the current j ^ = ^7/iV7 is a n obstruction for the separation, when \t\ oo, of the
nonlinear relativistic system into two asymptotic isolated relativistic systems, one for the
electromagnetic potential A^ and one for the Dirac field i/. It will be proved here 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 Poincare group. This constitutes
a serious problem for the second quantization of the asymptotic (in and out going) fields,
since the particle interpretation usually requires free relativistic fields, i.e. at least a linear
representation of the Poincare group (U1 in our case). Therefore we have to introduce
nonlinear representations U^ and U^ of Vo, in the sense of [10], which can give the
transformation of the asymptotic in and out states under Vo and which can permit a
particle interpretation.
There are two reasons which permit to determine the class of asymptotic represen-
tations U^e\ e = ±. First, the classical observables, 4-current density, 4-momentum and
4-angular momentum are invariant under gauge transformations. Second, if the evolution
equations 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 the
asymptotic representations g «— Ug are linear modulo a nonlinear gauge transformation
depending on g and respecting the Lorentz gauge condition. We shall make precise the
meaning of this statement at the end of this introduction.
Moreover to determine the class of admissible modified wave operators it is reasonable
to postulate that solutions of the M-D equations (l.la)-(l.lc) should converge when t
zboo in Ep to free solutions (i.e. solutions of equations (l.la)-(l.lc) but with vanishing
right-hand side) modulo a gauge transformation, not even respecting the Lorentz gauge
condition; such transformations are admissible since they leave invariant the observables.
In mathematical terms the infrared problem of the M-D equations then consists of
determining two diffeomorphisms Cl£:Ooo —• Uoo, £ = ±, the modified wave operators,
where 0c© is an open neighbourhood of zero in E%£ and where Uoo is a neighbourhood of
zero in V&, satisfying
C 7 ^ = f i
- 1 o [ /
o n
, g€V0,e = ±, (1.14)
where the asymptotic representations are C°° functions U&:P0 x ^ - ^ E%. In order
to satisfy the two preceding postulates, we impose supplementary conditions on U^ and
f2£, which we shall justify at the end of this introduction. Let the Fourier transformation
/ i—• / be defined by
f(k) = (2TT)- 3 / 2 / e-ikxf(x)dx. (1.15)
The orthogonal projections P£(—id) in D on initial data with energy sign e, e = ±, for the
Dirac equation are given by:
1 3
( P
H 3 ) a ) A ( k ) = Pe(k)a(k) = - ( j +
( - £
f c i
+ m7°)-;(fc)-1)d(k), (1.16)
3 = 1
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