Resurgence and non-perturbative physics

Course description

The course will take place on Tuesdays from 10-12 in room B1 during the first half of the fall semester (21.09-16.11, excluding 12.10).

Asymptotic series expansions appear in many branches of physics and mathematics. These are found when solutions to a given problem are expanded in power series in a small parameter, such as coupling constants in quantum field theory. Asymptotic series have a distinctive feature: they do not converge for any value of the small parameter. This fact makes it difficult to assign a meaningful sum to the series. On top of that, observables in quantum physics admit non-perturbative contributions, sometimes called “instantons”, which cannot be captured by a power series ansatz.

In the 1970s, J.Écalle developed a theory known as “resurgence” which explains how to properly sum asymptotic series and consistently include instantons corrections. Moreover, it establishes rigorously a connection between the large-order behaviour of perturbation theory and non-perturbative effects. In this course I will give an overview of resurgence theory, considering applications to problems in quantum mechanics and quantum field theory.

I will start discussing the appearance of asymptotic series in perturbative solutions of quantum mechanics problems, using as a working example the quantum particle in a 1D constant force field. Solutions to the Schrodinger equation are found in terms of Airy functions, which can be defined in terms of asymptotic expansions. I will then describe in detail the summation procedure for asymptotic series, known as Borel resummation. For this summation to be consistent, one has to extend the power series ansatz to “transseries” which incorporate non-perturbative effect. I will also discuss the emergence of Stokes-like phenomena in this context and, if times allows it, its connection to Thimble decompositions.

The second part of the course will be devoted to applications of resurgence theory. These can range from the degeneracy-lifting of ground states in double-well potentials to the phenomena of non-perturbative particle production in electric field background (Schwinger effect). The selection can vary according to the tastes of the audience.

Basics of complex analysis (contour integrals, analytic continuations, branch cuts and Riemann sheets, etc.)

Nicola Dondi