Second Order Calculations of the O(N) sigma-Model Laplacian

TL;DR

This paper develops a second-order expansion of the functional Laplacian in the Schrodinger equation for the O(N) sigma-model, enhancing the understanding of quantum field theory vacuum functionals.

Contribution

It extends previous work by constructing the next-to-leading order terms of the functional Laplacian, ensuring invariance and algebraic consistency.

Findings

Derived the next-to-leading order terms of the functional Laplacian.

Ensured rotational invariance and Poincare algebra closure in the extended Laplacian.

Provided a more accurate framework for analyzing vacuum functionals in quantum field theory.

Abstract

For slowly varying fields 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 Schrodinger equation, the principal ingredient of which is a regulated functional Laplacian. We extend a previous work to construct the next to leading order terms of the Laplacian for the Schrodinger equation that acts on such local functionals. Like the leading order the next order is completely determined by imposing rotational invariance in the internal space together with closure of the Poincare algebra.