A Hierarchical Finite Element Method for Quantum Field Theory

TL;DR

This paper introduces a hierarchical finite element approach to scalar quantum field theory on a subdivided triangular lattice, deriving a renormalization map and connecting it to finite element approximations of the Laplacian.

Contribution

It presents a novel hierarchical finite element framework for quantum field theory and derives an exact renormalization map linked to finite element methods.

Findings

Derived an exact renormalization map for subdivided triangles

Identified a fixed point corresponding to the cotangent finite element formula

Established a connection between quantum field theory discretization and finite element approximation

Abstract

We study a model of scalar quantum field theory in which space-time is a discrete set of points obtained by repeatedly subdividing a triangle into three triangles at the centroid. By integrating out the field variable at the centroid we get a renormalized action on the original triangle. The exact renormalization map between the angles of the triangles is obtained as well. A fixed point of this map happens to be the cotangent formula of Finite Element Method which approximates the Laplacian in two dimensions.