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We dwell upon certain points concerning the meaning of quantum field theory,among these the problems with the perturbative approach, and the questionraised by tHooft of the existence of the theory in a well defined mathematicalsense, as well as some of the few existent mathematically precise results onfully quantized field theories. Emphasis is brought on how the mathematicalcontributions help to elucidate or illuminate certain conceptual aspects of thetheory when applied to real physical phenomena, in particular, the singularnature of quantum fields. In a first part, we present a comprehensive review ofdivergent versus asymptotic series, with qed as background example, as well asa method due to Terence Tao which conveys mathematical sense to divergentseries. In a second part we apply the method of Tao to the Casimir effect inits simplest form, consisting of perfectly conducting parallel plates, arguingthat the usual theory, which makes use of the Euler-MacLaurin formula, stillcontains a residual infinity, which is eliminated in our approach. In the thirdpart, we revisit the general theory of nonperturbative quantum fields, in theform of newly proposed Wightman axioms for interacting field theories, togetherwith Christian Jaekel. Applications to dressed electrons in a theory withmassless particles, such as qed, as well as unstable particles, are included,and various problems (mostly open) are discussed in connection with concretemodels.

Perturbative versus non-perturbative quantum field theory: the method of Tao, the Casimir effect and interacting Wightman theories