# Smooth interpolation of lattice gauge fields by signal processing methods

## TL;DR

This paper introduces a smooth, gauge-equivariant interpolation method for lattice gauge fields using Fourier filtering, with extensions to non-abelian gauge groups and considerations for gauge invariance.

## Contribution

It presents a novel Fourier-based interpolation technique for lattice gauge fields that is smooth, gauge-equivariant, and extends to non-abelian groups like SU(2).

## Key findings

- The interpolation is $C^$ and free from transition functions.
- A non-abelian generalization using Cayley parametrization is developed.
- Maximum entropy methods are proposed to handle gauge invariance.

## Abstract

We digitally filter the Fourier modes of the link angles of an abelian lattice gauge field which produces the Fourier modes of a continuum $A_\mu(x)$ that exactly reproduces the lattice links through their definition as phases of finite parallel transport. The constructed interpolation is smooth ($C^\infty$), free from transition functions, and gauge equivariant. After discussing some properties of this interpolation, we discuss the non-abelian generalization of the method, arriving for SU(2), at a Cayley parametrization of the links in terms of the Fourier modes of $A^c_\mu(x)$. We then discuss the use of a maximum entropy type method to address gauge invariance in the non-abelian case.