Euclidean signature semi-classical methods for bosonic field theories: interacting scalar fields

TL;DR

This paper develops a new Euclidean signature semi-classical approach using microlocal methods to analyze self-interacting scalar quantum fields in various dimensions, extending techniques traditionally used in Schrödinger problems.

Contribution

It introduces a novel semi-classical method for quantum field theories by solving Hamilton-Jacobi and transport equations in Euclidean signature, applicable to scalar fields with polynomial interactions.

Findings

Proves existence and uniqueness of fundamental solutions in suitable function spaces.

Establishes global regularity of solutions for scalar field models.

Extends microlocal analysis techniques to quantum field theory contexts.

Abstract

Elegant 'microlocal' methods have long since been extensively developed for the analysis of conventional Schroedinger eigenvalue problems. For technical reasons though these methods have not heretofore been applicable to quantum field theories. In this article however we initiate a 'Euclidean signature semi-classical' program to extend the scope of these analytical techniques to encompass the study of self-interacting scalar fields in 1 + 1, 2 + 1 and 3 + 1 dimensions. The basic microlocal approach entails, first of all, the solution of a single, nonlinear equation of Hamilton-Jacobi type followed by the integration (for both ground and excited states) of a sequence of linear 'transport' equations along the 'flow' generated by the 'fundamental solution' to the aforementioned Hamilton-Jacobi equation. Using a combination of the direct method of the calculus of variations, elliptic regularity theory and the Banach space version of the implicit function theorem we establish, in a suitable function space setting, the existence, uniqueness and global regularity of this needed 'fundamental solution' to the relevant, Euclidean signature Hamilton-Jacobi equation for the systems under study. Our methods are applicable to (massive) scalar fields with polynomial self-interactions of renormalizable type. They can, as we shall show elsewhere, also be applied to Yang-Mills fields in 2 + 1 and 3 + 1 dimensions.