Continuity of Scalar Fields With Logarithmic Correlations

TL;DR

This paper investigates the continuity properties of scalar quantum fields with logarithmic correlations, using stochastic process theory and a novel averaging method to establish continuity and quantify short-distance variations.

Contribution

It introduces a new averaging approach to study scalar fields, proving their continuity and providing explicit bounds on their roughness, especially for fields with logarithmic correlations.

Findings

The averaged process is continuous.

Explicit modulus of continuity bounds are derived.

The approach applies to fields with mild singularities.

Abstract

We apply select ideas from the modern theory of stochastic processes in order to study the continuity/roughness of scalar quantum fields. A scalar field with logarithmic correlations (such as a massless field in 1+1 spacetime dimensions) has the mildest of singularities, making it a logical starting point. Instead of the usual inner product of the field with a smooth function, we introduce a moving average on an interval which allows us to obtain explicit results and has a simple physical interpretation. Using the mathematical work of Dudley, we prove that the averaged random process is in fact continuous, and give a precise modulus of continuity bounding the short-distance variation.