Scaling Limits of Lattice Quantum Fields by Wavelets

Vincenzo Morinelli,Gerardo Morsella,Alexander Stottmeister,Yoh Tanimoto

doi:10.1007/s00220-021-04152-5

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.

Highlights

The Wilson–Kadanoff renormalization group [48,84,85] is a cornerstone in the understanding of classical and quantum many-body system
We present a rigorous renormalization group scheme for lattice quantum field theories in terms of operator algebras
It provides a conceptual framework that unifies the theory of critical phenomena and universality in statistical mechanics with quantum field theory via the existence of infrared fixed points under scale-changing operations

Summary

Introduction

The Wilson–Kadanoff renormalization group [48,84,85] is a cornerstone in the understanding of classical and quantum many-body system. Scaling maps from one-particle spaces A renormalization group for lattice scalar fields Apart from an infinite field-strength renormalization accounting for the rescaling of the symplectic structure σN ,L with N , the inductive limit map, R∞N : hN,L → h∞,L = l−→ imN hN,L , is given by multiplication with the Fourier transform ε−N φ0(εN in momentum space. Block-spin renormalization in terms of orthogonal Haar wavelets The above interpretation (3.15) and (3.16) of lattice fields as continuum field smeared with characteristic functions, {ε−Nd χx+[0,εN )d }x∈ N , can be understood as a special instance of the general scheme of wavelet scaling discussed above. In order to obtain a continuum field as the scaling limit of lattice fields, we need to choose a scaling function φ which is both localized and sufficiently regular This goal is achieved by the so-called Daubechies wavelets [23, Chapter 6].

The Continuum Limit of the Free Vacua

Other Scaling-Map Schemes