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

TL;DR

This paper develops a wavelet-based Hamiltonian approach using Daubechies wavelets in momentum space to study nonperturbative dynamics of 1+1 dimensional scalar field theories, successfully capturing phase transitions.

Contribution

It introduces a novel wavelet formalism for Hamiltonian quantum field theory that effectively handles infrared and ultraviolet truncations nonperturbatively.

Findings

Accurately reproduces the phase transition in 1+1D $\,\phi^4$ theory.

Critical coupling converges to known values with increased resolution.

Demonstrates wavelet basis as a powerful tool for nonperturbative QFT calculations.

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 $\phi^4$ 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.