1. Introduction

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 diver-

gencies in the perturbative expression of the quantum scattering operator. These correc-

tions are introduced by hand with the purpose to give a posteriori a finite theory. In this

monograph we shall prove rigorous results which we have obtained concerning the infrared

problem for the (classical) Maxwell-Dirac equations. Our belief is that such results can

a priori be of interest for QED, especially for the infrared regime and combined with the

deformation-quantization approach [1], [2], [7]. Our results show in particular that, also in

the classical case one obtains infrared divergencies, if one requires free asymptotic fields as

it is needed in QED. But before continuing the physical motivation of this work, we shall

describe the mathematical context.

1.1 T H E MATHEMATICAL FRAMEWOR K

1.1.a The Equations. We shall use conventional notations: electron charge e = 1; Dirac

matrices 7^, 0 \x 3; metric tensor g*v, g00 = 1, gu = - 1 for 1 i 3 and g*v = 0

for fi^u] 7^7" + YY = W \ dM = d/(dy^); D = d ^ (with the Einstein summation

convention and the index raising convention). Then the classical Maxwell-Dirac (M-D)

equations read:

DAIA = ffatitl, 0 / x 3 , (1.1a)

( i

7

^

M

+ m)^ = A

M 7

^ , m 0, (1.1b)

dpA" = 0, (1.1c)

where ip = I/J+7O, ^ + being the Hermitian conjugate of the Dirac spinor ip. We write

equations (1.1a) and (1.1b) as an evolution equation:

^(A^A^i)) = (i

M

(t), A ^ ( t ) ) + ( 0 , ^ ( t )

7

^ ( t ) ) , (1.2a)

j^(t) = V^(i) - iA^rm, (l-2b)

where V = - £ *

= 1

7 V d j + ^ V A = £ *

= 1

9|, t e R and where A^t), A^t): E 3 -+ R,

ip(t):R3 — C 4 . The Lorentz gauge condition (1.1c) takes on initial conditions A^fo),

Apito), 0 /i 3, and r/j(t0) at t = t0 the form (cf. [17], [14])

3

i°(to) + £ft^(to) = 0, (1.3a)

2 = 1

3

AA°(t0) + mt0)\2+x^(*o) = ° (L3b)

i=l

where A* =g»vAu.

1