On the Tameness of Perturbative Expansions

Abstract

The notion of tameness is believed to bear a significant meaning for fundamental physics: according to the Tameness Conjecture, partition functions and correlation functions for UV-consistent theories are tame functions. In this thesis we first introduce the notion of tame structures and functions and illustrate their properties. Secondly, we present the theory of Borel summability and how it is necessary to understand the non-analytic behaviour of path integrals in the weak-coupling limit. These methods will be then applied to the study of partition and correlation functions of certain quantum field theories on a point-like spacetime: by viewing them as the Borel sums of their asymptotic expansions, we will show explicitly that they are tame functions of the tame structure known as R_G. Thereafter, we expound the basics of constructive field theory and, relying on results obtained thereby, we extend our arguments for the tameness of partition functions to more general theories. Finally, we explain how asymptotic expansions appear in quantum mechanics when treated with the exact WKB method; we discuss how the occurrence of Stokes phenomenon prevents us from establishing the tameness of the energy eigenvalues.