Communications in Analysis and Geometry

Volume 28 (2020)

Number 4

The First of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

A Euclidean signature semi-classical program

Pages: 979 – 1056

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n4.a6

Authors

Antonella Marini (Department of Mathematics, Yeshiva University, New York, N.Y., U.S.A.)

Rachel Maitra (Department of Applied Mathematics, Wentworth Institute of Technology, Boston, Massachusetts, U.S.A.)

Vincent Moncrief (Departments of Physics and Mathematics, Yale University, New Haven, Connecticut, US.A.)

Abstract

In this article we discuss our ongoing program to extend the scope of certain, well-developed microlocal methods for the asymptotic solution of Schrödinger’s equation (for suitable ‘nonlinear oscillatory’ quantum mechanical systems) to the treatment of several physically significant, interacting quantum field theories. Our main focus is on applying these ‘Euclidean-signature semi-classical’ methods to self-interacting (real) scalar fields of renormalizable type in $2$, $3$ and $4$ spacetime dimensions and to Yang-Mills fields in $3$ and $4$ spacetime dimensions. A central argument in favor of our program is that the asymptotic methods for Schrödinger operators developed in the microlocal literature are far superior, for the quantum mechanical systems to which they naturally apply, to the conventional WKB methods of the physics literature and that these methods can be modified, by techniques drawn from the calculus of variations and the analysis of elliptic boundary value problems, to apply to certain (bosonic) quantum field theories. Unlike conventional (Rayleigh/ Schrödinger) perturbation theory these methods avoid the artificial decomposition of an interacting system into an approximating ‘unperturbed’ system and its perturbation and instead keep the nonlinearities (and, if present, gauge invariances) of an interacting system intact at every level of the analysis.

Full Text (PDF format)

Received 21 December 2018

Accepted 19 June 2019

Published 1 October 2020