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A novel Hamiltonian formulation of $1+1$ dimensional $φ^4$ theory in Daubechies wavelet basis: momentum space analysis

Mrinmoy Basak

TL;DR

This work develops a nonperturbative Hamiltonian framework for the $1+1$-dimensional $\,phi^4$ theory using a momentum-space Daubechies wavelet basis, enabling controlled truncations of the Fock space. By expanding field operators in wavelet modes and constructing the free Hamiltonian $H_0$ and interaction $H_I$, the authors obtain a band-diagonal, locality-preserving representation that converges with increasing resolution $k$. The study demonstrates a nonperturbative strong-coupling transition through the behavior of the energy gap and provides estimates for the critical coupling $g_c$ that improve with higher $k$, illustrating the method’s effectiveness despite coarse truncations. The approach offers a scalable, ab initio tool for nonperturbative QFT, with potential extensions to higher dimensions and gauge theories, complementing traditional lattice methods.

Abstract

We employ the wavelet formalism of quantum field theory to study field theories in the nonperturbative Hamiltonian framework. Specifically, we make use of Daubechies wavelets in momentum space. These basis elements are characterised by a resolution and a translation index that provides for a natural nonperturbative infrared and ultraviolet truncation of the quantum field theory. As an application, we consider the $φ^4$ theory and demonstrate the emergence of the well-known nonperturbative strong-coupling phase transition in the $m^2>0$ sector.

A novel Hamiltonian formulation of $1+1$ dimensional $φ^4$ theory in Daubechies wavelet basis: momentum space analysis

TL;DR

This work develops a nonperturbative Hamiltonian framework for the -dimensional theory using a momentum-space Daubechies wavelet basis, enabling controlled truncations of the Fock space. By expanding field operators in wavelet modes and constructing the free Hamiltonian and interaction , the authors obtain a band-diagonal, locality-preserving representation that converges with increasing resolution . The study demonstrates a nonperturbative strong-coupling transition through the behavior of the energy gap and provides estimates for the critical coupling that improve with higher , illustrating the method’s effectiveness despite coarse truncations. The approach offers a scalable, ab initio tool for nonperturbative QFT, with potential extensions to higher dimensions and gauge theories, complementing traditional lattice methods.

Abstract

We employ the wavelet formalism of quantum field theory to study field theories in the nonperturbative Hamiltonian framework. Specifically, we make use of Daubechies wavelets in momentum space. These basis elements are characterised by a resolution and a translation index that provides for a natural nonperturbative infrared and ultraviolet truncation of the quantum field theory. As an application, we consider the theory and demonstrate the emergence of the well-known nonperturbative strong-coupling phase transition in the sector.
Paper Structure (10 sections, 17 equations, 2 figures, 3 tables)

This paper contains 10 sections, 17 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Daubechies wavelet basis
  3. Wavelet-based Fock Space
  4. Scalar Field Hamiltonian
  5. Fock Space Construction
  6. Hamiltonian Eigenvalues
  7. The $\phi^4$ Hamiltonian in Wavelet Basis
  8. Interaction Hamiltonian Construction
  9. Spectrum and Criticality
  10. Conclusion and Outlook

Figures (2)

  • Figure 1: The Daubechies-$3$ scaling functions (left) and wavelet functions (right) for resolution $0$ and translation $0$.
  • Figure 2: The energy gap ($E_1 - E_0$) as a function of coupling $\lambda$. The sharp drop indicates the onset of the $\mathbb{Z}_2$ symmetry breaking phase transition.