Quantum mechanics

Theoretical physics

Philosophy

Mathematical physics

Biology

Perturbation theory (quantum mechanics)

Particle physics

Telecommunications

Computer science

Simple (philosophy)

Dentistry

Observable

Paleontology

Physics

Calculus (dental)

Morse code

Epistemology

Quantum

Series (stratigraphy)

Medicine

Quantum chromodynamics

Class (philosophy)

Resummation

Path integral formulation

Annals of Physics

In these notes we give an overview of different topics in resurgence theory from a physics point of view, but with particular mathematical flavour. After a short review of the standard Borel method for the resummation of asymptotic series, we introduce the class of simple resurgent functions, explaining their importance in physical problems. We define the Stokes automorphism and the alien derivative and discuss these objects in concrete examples using the notion of trans-series expansion. With all the tools introduced, we see how resurgence and alien calculus allow us to extract non-perturbative physics from perturbation theory. To conclude, we apply Morse theory to a toy model path integral to understand why physical observables should be resurgent functions.

An introduction to resurgence, trans-series and alien calculus