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

TL;DR

This paper explores the mathematical foundations of quantum field theory, contrasting perturbative and non-perturbative approaches, applying Tao's method to the Casimir effect, and discussing recent axiomatic developments for interacting fields.

Contribution

It introduces Tao's method to give mathematical meaning to divergent series, applies it to the Casimir effect, and reviews nonperturbative axioms for interacting quantum fields.

Findings

Tao's method removes residual infinities in Casimir effect calculations.

Mathematically precise results illuminate conceptual issues in quantum field theory.

Discussion of open problems in nonperturbative quantum field models.

Abstract

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.