An efficient Wavelet-Based Hamiltonian Formulation of Quantum Field Theories using Flow-Equations

TL;DR

This paper introduces a wavelet-based Hamiltonian formulation for quantum field theories, utilizing flow-equation methods to efficiently analyze low-energy spectra by decoupling degrees of freedom across scales.

Contribution

It combines Daubechies wavelet basis with flow-equation techniques to systematically reduce Hamiltonian complexity and decouple scales in quantum field theories.

Findings

Reformulated free scalar field theory in wavelet basis as coupled localized oscillators.

Applied flow equations to decouple resolution modes, achieving block-diagonal Hamiltonian.

Extracted low-energy spectrum from the lowest-resolution block, reducing computational cost.

Abstract

We propose an effective Hamiltonian formulation of quantum field theories using a Daubechies wavelet basis in position space. Combined with flow-equation methods of the similarity renormalization group (SRG), this approach provides an efficient framework for analyzing quantum field theories by reducing the dimensionality of the Hamiltonian and systematically decoupling degrees of freedom across scales. As an application, the free scalar field theory has been reformulated within this framework to calculate the low-lying energy spectrum of the theory. These basis elements are known to transform the free scalar field theory into a theory of coupled localized oscillators, each of which is labeled by a location and a resolution index. In this representation, the Hamiltonian is naturally organized into fixed-resolution blocks, alongside blocks associated with the interactions between different resolutions. To decouple the different resolution modes and obtain a block diagonalized Hamiltonian with each block associated with a fixed resolution, the flow equation approach of SRG is applied. Finally, we demonstrate that with increasing resolution, the low-energy spectrum can be extracted from the effective lowest-resolution block of the Hamiltonian, leading to a significant reduction in computational cost.