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Multiresolution analysis of quantum theories using Daubechies wavelet basis

Mrinmoy Basak

TL;DR

This work develops a wavelet-based framework for quantum theories by embedding flow equation (SRG) methods within the Daubechies multiresolution basis. It demonstrates how a wavelet representation naturally supports systematic volume and resolution truncations, enabling block-diagonalization of Hamiltonians with improved truncation and controlled decoupling of scales. The thesis applies this to 1D quantum-mechanical problems (ISWP, SHO, delta and triangular potentials) and to 1+1D QFTs, showing renormalization and asymptotic freedom phenomena in a nonperturbative setting. It further extends to a two-field 1+1D model and a 2D Delta-function potential, illustrating how flow equations yield effective Hamiltonians while preserving low-energy physics, and outlining pathways toward 3+1D extensions and potential quantum-computation applications.

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

Multiresolution analysis of quantum theories using Daubechies wavelet basis

TL;DR

This work develops a wavelet-based framework for quantum theories by embedding flow equation (SRG) methods within the Daubechies multiresolution basis. It demonstrates how a wavelet representation naturally supports systematic volume and resolution truncations, enabling block-diagonalization of Hamiltonians with improved truncation and controlled decoupling of scales. The thesis applies this to 1D quantum-mechanical problems (ISWP, SHO, delta and triangular potentials) and to 1+1D QFTs, showing renormalization and asymptotic freedom phenomena in a nonperturbative setting. It further extends to a two-field 1+1D model and a 2D Delta-function potential, illustrating how flow equations yield effective Hamiltonians while preserving low-energy physics, and outlining pathways toward 3+1D extensions and potential quantum-computation applications.

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

Paper Structure

This paper contains 46 sections, 349 equations, 25 figures, 20 tables.

Table of Contents

  1. Introduction
  2. Multiresolution analysis (MRA) of function space ($\mathscr{L}^2(\mathbb{R})$) using Daubechies wavelet basis
  3. MRA of $\mathscr{L}^2(\mathbb{R})$
  4. Multiresolution generator
  5. Wavelets
  6. Basis with compact support
  7. Properties of elements in Daubechies wavelet family
  8. Quantum mechanics in Daubechies wavelet basis
  9. Formalism
  10. The infinite square well potential
  11. The Simple Harmonic Oscillator
  12. The Dirac-delta function potential
  13. The triangular potential
  14. Quantum Field Theory (QFT) in wavelet basis
  15. An early development, a brief history of QFT
  16. ...and 31 more sections

Figures (25)

  • Figure 2.1: Consider any generic function $f(x)$ (depicted in blue). The impact of (a) the scaling operator $\hat{D}$ and (b) the translation operator $\hat{T}$ on that function is illustrated in red.
  • Figure 2.2: The red dotted line illustrates the mother scaling function $s(x)$ for $K=6$, generated by combining $12$ translated instances of $s(x)$, each scaled to half of its original support.
  • Figure 2.3: Euler diagram for spanning of Hilbert space with wavelet basis.
  • Figure 2.4: Scaling and wavelet functions for different values of $K$
  • Figure 3.5: Log error $\left(\log\frac{|\epsilon_n^{s,k}-\epsilon_n|}{\epsilon_n}\right)$ of different eigenstates with increasing resolution $(k)$ plot for the infinite square well potential.
  • ...and 20 more figures