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arXiv:2501.13618 · Physical review. D/Physical review. D.

Hamiltonian flow equations in the Daubechies wavelet basis

Mrinmoy Basak, Raghunath Ratabole A

Published May 27, 2025• 2 citations
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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 the similarity renormalization group in a wavelet basis. This paper presents an extension of the work by Michlin [Multiresolution decomposition of quantum field theories using wavelet bases, .] 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 labeled 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.

DOI
10.1103/physrevd.111.096024

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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.
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We investigate both the theoretical and computational aspects of using wavelet bases to perform an exact decomposition of a local field theory by spatial resolution. The decomposition admits natural volume and resolution truncations. We demonstrate that flow equation methods can be used to eliminate short-distance degrees of freedom in truncated theories. The method is tested on a free scalar field in one dimension, where the spatial derivatives couple the degrees of freedom on different scales, although the method is applicable to more complex field theories. The flow equation method is shown to decouple both distance and energy scales in this example. The response to changing the volume and resolution cutoffs and the mass is discussed.
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We describe a multi-scale resolution approach to analyzing problems in Quantum Mechanics using Daubechies wavelet basis. The expansion of the wavefunction of the quantum system in this basis allows a natural interpretation of each basis function as a quantum fluctuation of a specific resolution at a particular location. The Hamiltonian matrix constructed in this basis describes couplings between different length scales and thus allows for intuitive volume and resolution truncation. In quantum mechanical problems with a natural length scale, one can get approximate solution of the problem through simple matrix diagonalization. We illustrate this approach using the example of the standard quantum mechanical simple harmonic oscillator.
A novel Hamiltonian formulation of 1+11+11+1 dimensional φ4φ^4φ4 theory in Daubechies wavelet basis: momentum space analysis
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