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.)