Hamiltonian Flow Equations in Daubechies Wavelet Basis

TL;DR

This paper extends the wavelet-based flow equation approach to analyze the low energy dynamics of coupled scalar fields in 1+1 dimensions, demonstrating effective scale separation and accurate reproduction of normal mode frequencies.

Contribution

It introduces a higher-resolution extension of the wavelet-based flow equations for coupled scalar fields, enabling block diagonalization and scale separation in Hamiltonian analysis.

Findings

Flow equations diagonalize Hamiltonian into fixed resolution blocks.

Wavelet basis transforms fields into localized oscillators.

Effective Hamiltonian reproduces normal mode frequencies.

Abstract

We study the low energy dynamics of a system of two coupled real scalar fields in 1+1 dimensions using the flow equation approach of Similarity Renormalization Group (SRG) in a wavelet basis. This paper presents an extension of the work by Michlin and Polyzou \cite{PhysRevD.95.094501} at one resolution higher. We also present the analysis of a model of two scalar fields coupled through a generally quadratic interaction in $1+1$ dimensions using wavelet-based flow equations. We demonstrate that the specifically chosen generator flows the Hamiltonian into a block diagonal form with each diagonal block being associated with a fixed resolution. The wavelet basis is known to transform the scalar field theory into a model of coupled localized oscillators, each of which is labelled by location and resolution indices. The chosen interaction represents the coupling between two types of oscillators at the same location and resolution index. There is no coupling between oscillators across locations and resolutions. We show that wavelet-based flow equations carry out scale separation while maintaining the interactions between the two scalar fields at each resolution. The effective Hamiltonian associated with the coarsest resolution is shown to correctly reproduce the normal mode frequencies of the model.