6 INTRODUCTION

1.2 T H E INFRARE D PROBLE M

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 [ /

9

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)

JR3

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

e

H 3 ) a ) A ( k ) = Pe(k)a(k) = - ( j +

e

( - £

7

V

f c i

+ m7°)-;(fc)-1)d(k), (1.16)

3 = 1