Annals of Mathematical Sciences and Applications
Volume 1 (2016)
Euclidean signature semi-classical methods for bosonic field theories: interacting scalar fields
Pages: 3 – 55
Elegant microlocal methods have long since been extensively developed for the analysis of conventional Schrödinger 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.
semi-classical methods for bosonic fields, Schrödinger quantum field theory, quantized scalar fields, elliptic theory on unbounded domains
2010 Mathematics Subject Classification
35J20, 35J25, 81Q20