Nonperturbative Ambiguities and the Reality of Resurgent Transseries
Abstract
In a wide range of quantum theoretical settings—from quantum mechanics to quantum field theory, from gauge theory to string theory—singularities in the complex Borel plane, usually associated to instantons or renormalons, render perturbation theory illdefined as they give rise to nonperturbative ambiguities. These ambiguities are associated to choices of an integration contour in the resummation of perturbation theory, along (singular) Stokes directions in the complex Borel plane (rendering perturbative expansions nonBorel summable along any Stokes line). More recently, it has been shown that the proper framework to address these issues is that of resurgent analysis and transseries. In this context, the cancelation of all nonperturbative ambiguities is shown to be a consequence of choosing the transseries median resummation as the appropriate family of unambiguous real solutions along the couplingconstant real axis. While the median resummation is easily implemented for oneparameter transseries, once one considers more general multiparameter transseries the procedure becomes highly dependent upon properly understanding Stokes transitions in the complex Borel plane. In particular, all Stokes coefficients must now be known in order to explicitly implement multiparameter median resummations. In the cases where quantumtheoretical physical observables are described by resurgent functions and transseries, the methods described herein show how one may cancel nonperturbative ambiguities, and define these observables nonperturbatively starting out from perturbation theory. Along the way, structural results concerning resurgent transseries are also obtained.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 April 2015
 DOI:
 10.1007/s002200142165z
 arXiv:
 arXiv:1308.1115
 Bibcode:
 2015CMaPh.335..183A
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Classical Analysis and ODEs;
 Quantum Physics
 EPrint:
 62 pages, 4 figures