Master Dissertation

Preface
Overview
This dissertation is about the causal approach to finite quantum electrodynamics
(QED). The word causal refers to the main assumption on which the
method we will introduce is based. In short it is the statement that what
happens at time t > s doesn’t influence what happens at earlier times t < s.
The word finite means that there do not arise any ultraviolet divergences.
The traditional formalism of quantum field theory (QFT) is known to
have problems with ultraviolet divergences. These occur because of misuse
of the mathematics. The problem lies in the fact that QFT deals with distributions
instead of functions and that the rules of calculus for these are
not the same. One cannot simply multiply a distribution by a discontinuous
step-function and other simple rules of calculus like multiplication of distributions
can be a very complicated matter. In fact, this is the main problem
of [4].
In practice the ultraviolet divergencies are dealt with by regularization
and renormalization FIXME: explain these. But instead of using these firstaid
features on an ill-defined theory one could introduce QFT using the
mathematics in the right way from the beginning. This is the basic aim of
this dissertation.
Structure
Chapter 1
In Chapter 1 I introduce the tools needed in our later work. Section 1.2
is about formal power series which is an abstraction of the usual power
series used in calculus. A generalization which lets us use most of the wellknown
machinery from calculus to settings which have no natural notion
of convergence. This will become useful later on when we introduce the
scattering matrix as a formal power series.
In section 1.3 we will introduce Affine Transformations of distributions.
2