The Vacuum Functional at Large Distances

TL;DR

This paper develops a new approach to approximate the vacuum functional in quantum field theory, especially for slowly varying fields, by deriving an algebraic eigenvalue problem and proposing two solution schemes.

Contribution

It constructs an equation satisfied by the vacuum functional expansion in $\

Findings

Reduces the eigenvalue problem to algebraic equations

Introduces a semi-classical and a novel quantum-generated mass scheme

Applicable to classically massless theories with quantum mass generation

Abstract

For fields that vary slowly on the scale of the lightest mass the logarithm of the vacuum functional can be expanded as a sum of local functionals, however this does not satisfy the obvious form of the Schr\"odinger equation. For $\varphi^4$ theory we construct the appropriate equation that this expansion does satisfy. This reduces the eigenvalue problem for the Hamiltonian to a set of algebraic equations. We suggest two approaches to their solution. The first is equivalent to the usual semi-classical expansion whilst the other is a new scheme that may also be applied to theories that are classically massless but in which mass is generated quantum mechanically.