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Hamiltonian formulation of the $1+1$-dimensional $φ^4$ theory in a momentum-space Daubechies wavelet basis

Mrinmoy Basak, Debsubhra Chakraborty, Nilmani Mathur, Raghunath Ratabole

TL;DR

This work develops a Hamiltonian formulation of the 1+1 dimensional φ^4 theory using a momentum-space Daubechies wavelet basis to implement a nonperturbative truncation of both infrared and ultraviolet degrees of freedom. It constructs a wavelet-based Fock space with basis labels given by resolution and translation, derives expressions for the free and interacting Hamiltonians in this basis, and diagonalizes finite matrices to obtain energy spectra for both the free field and the interacting theory. The results show convergence of low-lying energies with increasing momentum resolution and reproduce the strong-coupling symmetry-breaking transition, with the extracted critical coupling gc approaching established values as the resolution grows. This demonstrates the viability and potential of a wavelet-based Hamiltonian approach for nonperturbative quantum field theory and points toward extensions to higher dimensions and gauge theories.

Abstract

We apply the wavelet formalism of quantum field theory to investigate nonperturbative dynamics within the Hamiltonian framework. In particular, we employ Daubechies wavelets in momentum space, whose basis functions are labeled by resolution and translation indices, providing a natural nonperturbative truncation of both infrared and ultraviolet truncation of quantum field theories. As an application, we compute the energy spectra of a free scalar field theory and the interacting $1+1$-dimensional $φ^4$ theory. This approach successfully reproduces the well-known strong-coupling phase transition in the $m^2 > 0$ regime. We find that the extracted critical coupling systematically converges toward its established value as the momentum resolution is increased, demonstrating the effectiveness of the wavelet-based Hamiltonian formulation for nonperturbative field-theoretic calculations.

Hamiltonian formulation of the $1+1$-dimensional $φ^4$ theory in a momentum-space Daubechies wavelet basis

TL;DR

This work develops a Hamiltonian formulation of the 1+1 dimensional φ^4 theory using a momentum-space Daubechies wavelet basis to implement a nonperturbative truncation of both infrared and ultraviolet degrees of freedom. It constructs a wavelet-based Fock space with basis labels given by resolution and translation, derives expressions for the free and interacting Hamiltonians in this basis, and diagonalizes finite matrices to obtain energy spectra for both the free field and the interacting theory. The results show convergence of low-lying energies with increasing momentum resolution and reproduce the strong-coupling symmetry-breaking transition, with the extracted critical coupling gc approaching established values as the resolution grows. This demonstrates the viability and potential of a wavelet-based Hamiltonian approach for nonperturbative quantum field theory and points toward extensions to higher dimensions and gauge theories.

Abstract

We apply the wavelet formalism of quantum field theory to investigate nonperturbative dynamics within the Hamiltonian framework. In particular, we employ Daubechies wavelets in momentum space, whose basis functions are labeled by resolution and translation indices, providing a natural nonperturbative truncation of both infrared and ultraviolet truncation of quantum field theories. As an application, we compute the energy spectra of a free scalar field theory and the interacting -dimensional theory. This approach successfully reproduces the well-known strong-coupling phase transition in the regime. We find that the extracted critical coupling systematically converges toward its established value as the momentum resolution is increased, demonstrating the effectiveness of the wavelet-based Hamiltonian formulation for nonperturbative field-theoretic calculations.
Paper Structure (13 sections, 58 equations, 7 figures, 3 tables)

This paper contains 13 sections, 58 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Daubechies wavelet basis
  3. Wavelet-based Fock space
  4. Scalar Field Theory: Fourier Modes
  5. Scalar Field Theory: Wavelet Modes
  6. Hamiltonian Eigenvalues
  7. The $\phi^4$ Hamiltonian in Daubechies wavelet based framework
  8. Interactive $\phi^4$-term: Fourier Modes
  9. Interactive $\phi^4$-term: Wavelet Modes
  10. Hamiltonian ($\mathrm{H}_0$+$\mathrm{H_I}$) Eigenvalues: the critical point
  11. Conclusions and outlooks
  12. The $\phi^4$ overlap integral
  13. The interacting Hamiltonian matrix element within the Daubechies wavelet based framework

Figures (7)

  • Figure 1: The scaling functions for (a) different values of resolution and (b) different values of translation index are presented.
  • Figure 2: The wavelet functions for (a) different values of resolution and (b) different values of translation index are presented.
  • Figure 3: The hopping strength between two different scaling-modes located at two positions are presented.
  • Figure 4: (a) The divergence of the ground state energy with the increasing value of the coupling constant $\lambda$, (b) the variation of the first excited state with the increasing value of the coupling constant $\lambda$.
  • Figure 5: The negative divergence of the ground state energy with the increasing value of the coupling constant $\lambda$.
  • ...and 2 more figures