Abstract
We dwell upon certain points concerning the meaning of quantum field theory: the problems with the perturbative approach, and the question raised by ’t Hooft of the existence of the theory in a well-defined (rigorous) 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 Tao’s method 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 (with Christian Jaekel) Wightman axioms for interacting field theories, with applications to “dressed” electrons in a theory with massless particles (such as qed), as well as unstable particles. Various problems (mostly open) are finally discussed in connection with concrete models.
Highlights
One of the most impressive measurements in physics is of the electron anomalous magnetic moment of the electron, which is known with a precision of a few parts in 1014 [1]
We come back to the perturbation series for the renormalized gyromagnetic ratio g of the electron, which is a prototype of divergent series in physics:
One of the most important features of relativistic quantum field theory is the behavior of the theory at large momenta
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. One of the problems with quantum field theory (qft) is that it seems to have lost contact with its main object of study, viz., explaining the observed phenomena in the theory of elementary particles One instance of this fact is that, except for a few of the lightest particles, all the remaining ones are unstable, and there exists up to the present time no single mathematically rigorous model of an unstable particle (this is reviewed in Reference [6], and we come back to this point in Sections 4 and 5).
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