Multiresolution analysis of quantum theories using Daubechies wavelet basis

TL;DR

This paper develops a wavelet-based flow equation framework to analyze multiscale quantum theories, demonstrating improved Hamiltonian truncation and effective low-resolution physics separation in scalar field models.

Contribution

It introduces a novel wavelet-based formulation of flow equations for quantum theories, enabling better multiscale analysis and Hamiltonian simplification.

Findings

Flow equations block-diagonalize Hamiltonians with respect to wavelet resolution.

Couplings between low- and high-resolution degrees of freedom are effectively suppressed.

The method provides insights into constructing effective Hamiltonians using wavelet techniques.

Abstract

Flow equation methods, more generally known as Similarity Renormalization Group (SRG) techniques, were developed to address multiscale problems where multiple length or energy scales contribute simultaneously. In this Thesis, we formulate the flow equation method within a wavelet-based framework and apply it to study scale (resolution) separation in a two-dimensional scalar field theory. We demonstrate that the flow systematically block-diagonalizes the Hamiltonian with respect to wavelet resolution, achieving improved truncation compared to earlier studies. Using a model of two real scalar fields coupled through a quadratic interaction, we show that the flow equations effectively suppress couplings between low- and high-resolution degrees of freedom. This provides a clear mechanism for isolating low-resolution physics and offers insight into the construction of effective Hamiltonians using a wavelet-based flow equation approac