18

INTRODUCTION

a contraction mapping (formula (6.33)) in a Banach space !FN (Corollary 6.8). In particular

K(t) e

L2(R3,M4).

The existence of solutions of the M-D equations (Theorem 6.10) and

the existence of a modified wave operator, denoted temporarily by Oi (formula (6.172),

Theorem 6.12) are then proved. The existence of

Cl^1

is shown (Theorem 6.13) by using

an implicit functions theorem in the Frechet space E^. The disadvantage of Oi is that it

gives a nonexplicit expression for the the asymptotic representation because it requires the

use of the solutions of the M-D systems. To overcome this problem, we introduce the final

modified wave operator tt^ (formulas (6.227a), (6.227b) and (6.276)) and its extension

Q+ to a Poincare invariant domain. fi+ has the advantage of giving a simple expression

(see (1.17a) and (1.17b)) for the asymptotic representation £/(+).

This monograph (at least in so far as the M-D equations are concerned) is largely

self-contained. A few technical results are borrowed from quoted references. It may be

read with little or no knowledge of nonlinear group representation theory, but the latter

is crucial to a real understanding of the methods and choice of spaces. It can be helpful

to consult the overview papers [13] and [30]. A short condensed version of the present

monograph has been published in [15].

1.8 FINAL 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^ijj of a c-number free electromagnetic potential A^ 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

ipa(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 [/(+), which is "gauge projectively linear", converge when t — oo

to the usual free field observables. Therefore there should be no observable phenomena

which distinguish U^ from U1, at least as far as these observables are concerned. Since

the representation U is equivalent to U^ by Q+, we shall call U a "gauge projectively

linearizable" nonlinear representation. Of course if the Dirac field or the electromagnetic

potential itself was an observable, then this would no longer be true, i.e. it would then be

possible to distinguish U^ and

U1

by the observables.

iii) One should note that the phase factor (3.33a of [14]), which looks as a very familiar

factor in abelian and non abelian gauge theories, was obtained in [14] 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^ field decreasing slower than r - 1 / 2 and suitable

initial data for j^A^, one would obtain phase factors with higher powers of AM.

iv) The same methods can be used for nonabelian gauge theories (of the Yang-Mills type)

coupled with fermions. The aim here is to separate asymptotically the linear (modulo an

infrared problem that can be a lot worse in the nonabelian case) equation for the spinors

from the pure Yang-Mills equation (the A^ part). The next step would then be to linearize

analytically the pure Yang-Mills equation (that is known [12] to be formally linearizable),

and then to combine all this with the deformation-quantization approach to deal rigorously