  We dwell upon certain points concerning the meaning of quantum field theory,
among these the problems with the perturbative approach, and the question
raised by tHooft of the existence of the theory in a well defined mathematical
sense, as well as some of the few existent mathematically precise results on
fully quantized field theories. Emphasis is brought on how the mathematical
contributions help to elucidate or illuminate certain conceptual aspects of the
theory when applied to real physical phenomena, in particular, the singular
nature of quantum fields. In a first part, we present a comprehensive review of
divergent versus asymptotic series, with qed as background example, as well as
a method due to Terence Tao which conveys mathematical sense to divergent
series. In a second part we apply the method of Tao to the Casimir effect in
its simplest form, consisting of perfectly conducting parallel plates, arguing
that the usual theory, which makes use of the Euler-MacLaurin formula, still
contains a residual infinity, which is eliminated in our approach. In the third
part, we revisit the general theory of nonperturbative quantum fields, in the
form of newly proposed Wightman axioms for interacting field theories, together
with Christian Jaekel. Applications to dressed electrons in a theory with
massless particles, such as qed, as well as unstable particles, are included,
and various problems (mostly open) are discussed in connection with concrete
models.
