Quantum Finite Elements for Lattice Field Theory

TL;DR

This paper introduces a novel Quantum Finite Element approach for lattice quantum field theories on curved manifolds, enabling non-perturbative analysis and convergence testing against known solutions.

Contribution

It develops a new simplicial lattice QFE Lagrangian incorporating counter terms and constructs a lattice Dirac operator on curved spaces.

Findings

QFE successfully reproduces phi4 theory at the Wilson-Fisher fixed point

Convergence of the Dirac equation on a Riemann sphere is demonstrated

Method shows promise for applications in Conformal Field Theories

Abstract

Viable non-perturbative methods for lattice quantum field theories on curved manifolds are difficult. By adapting features from the traditional finite element methods (FEM) and Regge Calculus, a new simplicial lattice Quantum Finite Element (QFE) Lagrangian is constructed for fields on a smooth Riemann manifold. To reach the continuum limit additional counter terms must be constructed to cancel the ultraviolet distortions. This is tested by the comparison of phi 4-th theory at the Wilson-Fisher fixed point with the exact Ising (c =1/2) CFT on a 2D Riemann sphere. The Dirac equation is also constructed on a simplicial lattice approximation to a Riemann manifold by introducing a lattice vierbein and spin connection on each link. Convergence of the QFE Dirac equation is tested against the exact solution for the 2D Riemann sphere. Future directions and applications to Conformal Field Theories are suggested.