Short Distance Properties from Large Distance Behaviour

TL;DR

This paper develops a method to derive short-distance properties of quantum field theories from their large-distance behavior, successfully reproducing known counter-terms and simplifying for specific models.

Contribution

It introduces an expansion of the vacuum functional for slowly varying fields that satisfies a Schrödinger-like equation, providing a new way to analyze short-distance properties.

Findings

Correctly reproduces short-distance counter-terms in 1+1D scalar theories

Provides an approximate simplification for Sine-Gordon and Sinh-Gordon models

Establishes a link between large-distance behavior and short-distance properties

Abstract

For slowly varying fields the vacuum functional of a quantum field theory may be expanded in terms of local functionals. This expansion satisfies its own form of the Schr\"odinger equation from which the expansion coefficents can be found. For scalar field theory in 1+1 dimensions we show that this approach correctly reproduces the short-distance properties as contained in the counter-terms. We also describe an approximate simplification that occurs for the Sine-Gordon and Sinh-Gordon vacuum functionals.