Scaling limits of lattice quantum fields by wavelets

TL;DR

This paper develops a rigorous renormalization group framework for lattice quantum field theories using wavelets, establishing a connection between lattice fields and continuum free fields through operator algebra techniques.

Contribution

It introduces a novel wavelet-based approach to construct scaling maps for lattice fields and demonstrates the existence of continuum limits within an operator algebra framework.

Findings

Existence of the inductive limit of free lattice ground states.

Lattice fields correspond to continuum fields smeared with Daubechies' scaling functions.

Comparison of wavelet-based scaling maps with traditional renormalization schemes.

Abstract

We present a rigorous renormalization group scheme for lattice quantum field theories in terms of operator algebras. The renormalization group is considered as an inductive system of scaling maps between lattice field algebras. We construct scaling maps for scalar lattice fields using Daubechies' wavelets, and show that the inductive limit of free lattice ground states exists and the limit state extends to the familiar massive continuum free field, with the continuum action of spacetime translations. In particular, lattice fields are identified with the continuum field smeared with Daubechies' scaling functions. We compare our scaling maps with other renormalization schemes and their features, such as the momentum shell method or block-spin transformations.