A Euclidean Signature Semi-Classical Program

Abstract

In this article we discuss our ongoing program to extend the scope of certain, welldeveloped 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 allow one to avoid the artificial decomposition of an interacting system into an approximating 'unperturbed' system and its perturbation and instead to keep the nonlinearities (and, if present gauge invariances) of an interacting system intact at every level of the analysis.

Closed Form Solutions For The Quantum LRS Bianchi IX and VIII Wheeler DeWitt Equations For An Arbitrary Hartle-Hawking Ordering Parameter

“…Via superposition we constructed interesting looking wave functions and using an admittingly naive interpretation extrapolated some interesting rough statements on the quantum evolution of LRS Bianchi IX and VIII universes from them. The author looks forward in the future to comparing these results to those obtained by applying the Euclidean-signature semi classical method [24] [25] to the LRS Bianchi IX and VIII models.…”