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
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