The O(N) sigma-model Laplacian

TL;DR

This paper constructs a leading-order O(N) sigma-model Laplacian acting on local functionals, based on symmetry principles and Poincaré algebra closure, to facilitate analysis of slowly varying fields in quantum field theory.

Contribution

It introduces a novel, symmetry-based construction of a regulated Laplacian for the O(N) sigma-model applicable to slowly varying fields.

Findings

Derived a leading-order Laplacian respecting internal rotational invariance.

Ensured the Laplacian's consistency with Poincaré algebra closure.

Provides a tool for analyzing vacuum functionals in quantum field theories.

Abstract

For fields that vary slowly on the scale of the lightest mass the logarithm of the vacuum functional of a massive quantum field theory can be expanded in terms of local functionals satisfying a form of the Schr\"odinger equation, the principal ingredient of which is a regulated functional Laplacian. We construct to leading order a Laplacian for the O(N) sigma-model that acts on such local functionals. It is determined by imposing rotational invariance in the internal space together with closure of the Poincar\'e algebra.