Perturbative and non-perturbative physics from singularities

Abstract

A function that representing a physical quantity has singularities which con tain perturbative or non-perturbative information about the physical system under investigation. Moreover, the theory of resurgence tells us that these perturbative and non-perturbative parts are intimately connected and it is possible to use one of them to obtain the other one. In this thesis, we combine these two ideas with a focus on the functions formulated in integral representations. Specifically, first consider ing two different examples on the semi-classical expansion in quantum mechanics and the pair production problem in electromagnetic backgrounds, we will concentrate on the quantum action which we express in the Schwinger’s integral representation. We will show that the perturbative and non-perturbative information about the physical system is hidden in singularities of the propagator TrU(t). The way we obtain the non-perturbative one is very similar to the Borel method which is used to handle the divergent perturbation series. Contrary to the Borel method, by probing the singular ities of TrU(t) directly and using the iε prescription, we will be able to prevent the Borel ambiguity problem in the physical cases that it leads to the violation of the uni tarity. Later, we will turn our attention to the renormalon problem in non-relativistic quantum mechanics. After presenting the existence of the renormalon divergence in a scattering problem with a background potential consisting of 2D δ-potential perturbed with a tilted 1D δ-potential, we will argue that the Borel ambiguity in the summation of the divergent series can be prevented again by a careful application of the iε pre scription and the resulting non- perturbative contribution due to the renormalon obeys the causality condition.