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arxiv: 2601.18449 · v2 · submitted 2026-01-26 · ✦ hep-th · hep-ex· hep-lat· hep-ph· quant-ph

Hamiltonian formulation of the 1+1-dimensional φ⁴ theory in a momentum-space Daubechies wavelet basis

Mrinmoy Basak , Debsubhra Chakraborty , Nilmani Mathur , Raghunath Ratabole This is my paper

Pith reviewed 2026-05-16 10:44 UTC · model grok-4.3

classification ✦ hep-th hep-exhep-lathep-phquant-ph
keywords Daubechies waveletsmomentum space basisHamiltonian formulationphi^4 theoryphase transitionnonperturbative QFT1+1 dimensionsscalar field theory
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The pith

Momentum-space Daubechies wavelets enable Hamiltonian calculations that reproduce the strong-coupling phase transition in the 1+1-dimensional φ⁴ theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies Daubechies wavelets in momentum space to formulate the Hamiltonian of quantum field theories with a natural truncation of modes. For the free scalar field, it computes energy spectra, and for the interacting φ⁴ theory in 1+1 dimensions, it captures the known phase transition when the mass squared is positive. The critical value of the coupling constant approaches its established result as the resolution in momentum space is refined, confirming the approach handles nonperturbative effects reliably.

Core claim

The central claim is that the Daubechies wavelet basis in momentum space, indexed by resolution and translation, supplies an effective nonperturbative truncation for both infrared and ultraviolet modes. When used in the Hamiltonian framework for the 1+1D φ⁴ theory, this basis reproduces the strong-coupling phase transition in the m² > 0 regime, with the extracted critical coupling converging to its known value upon increasing the momentum resolution.

What carries the argument

The momentum-space Daubechies wavelet basis functions labeled by resolution and translation indices, which allow controlled truncation of the field modes.

If this is right

  • The energy spectra of the free scalar field theory are accurately computed using this basis.
  • The interacting φ⁴ theory exhibits the expected strong-coupling phase transition in the m² > 0 regime.
  • The critical coupling converges systematically to the established value with higher momentum resolution.
  • This formulation demonstrates effectiveness for nonperturbative field-theoretic calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar wavelet truncations could be applied to other scalar theories or gauge theories in higher dimensions.
  • The method might allow efficient computation of bound states or scattering amplitudes in the same framework.
  • Increasing resolution further could yield more precise critical exponents or other observables.

Load-bearing premise

The Daubechies wavelet basis in momentum space provides a sufficiently accurate truncation without introducing basis-dependent errors that would shift the phase transition location.

What would settle it

If the critical coupling extracted from the wavelet Hamiltonian does not approach the established value as the number of resolution levels increases, the truncation would be shown to introduce significant artifacts.

Figures

Figures reproduced from arXiv: 2601.18449 by Debsubhra Chakraborty, Mrinmoy Basak, Nilmani Mathur, Raghunath Ratabole.

Figure 1
Figure 1. Figure 1: FIG. 1. The scaling functions for (a) different values of resolution and (b) different values of translation index are presented. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The wavelet functions for (a) different values of resolution and (b) different values of translation index are presented. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The hopping strength between two different scaling-modes located at two positions are presented. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The negative divergence of the ground state energy with the increasing value of the coupling constant [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The energy spectrum plot of the interacting [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The behaviour of the 1st excited state( [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a Hamiltonian formulation of the 1+1-dimensional φ⁴ theory using a momentum-space Daubechies wavelet basis whose functions are labeled by resolution and translation indices. This basis is used to implement a nonperturbative truncation of both infrared and ultraviolet modes. The authors compute the energy spectra of the free scalar field and the interacting theory, and report that the extracted critical coupling for the strong-coupling phase transition in the m² > 0 regime converges systematically toward the established literature value as the momentum resolution is increased.

Significance. If the reported convergence is free of basis-dependent artifacts, the approach would supply a new truncation scheme for Hamiltonian QFT that simultaneously controls IR and UV modes via a single resolution parameter. The reproduction of a known phase transition provides an initial consistency check, but the significance hinges on whether the wavelet matrix elements for the quartic interaction remain sufficiently accurate under truncation to guarantee that the critical coupling approaches the continuum limit without offset.

major comments (2)
  1. [Results and Discussion] The central claim that the critical coupling converges to the continuum value as momentum resolution increases is load-bearing, yet the manuscript supplies no quantitative truncation-error estimates, no tabulation of the quartic-interaction matrix elements in the wavelet basis, and no cross-check against an independent regularization (e.g., position-space lattice at matched cutoff). Without these data it is impossible to confirm that residual UV/IR mixing or broken translation invariance under truncation does not shift the extracted critical point.
  2. [Formalism and Numerical Implementation] The description of the momentum-space Daubechies wavelet implementation does not include explicit formulas for the overlap integrals or the truncated Hamiltonian matrix in the interacting case. In particular, it is unclear how the quartic term, which mixes different resolution and translation sectors, is evaluated once the basis is truncated, leaving open the possibility of basis-specific artifacts that would not be diagnosed by the free-field spectrum alone.
minor comments (2)
  1. [Abstract] The abstract states that convergence is 'systematic' but does not specify the range of resolution indices employed or the functional form used to extrapolate to infinite resolution; adding these details would improve clarity.
  2. [Notation] Notation for the wavelet indices (resolution level and translation) should be defined once at first use and used consistently in all figures and tables.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the manuscript to incorporate additional details and quantitative support as requested.

read point-by-point responses

  1. Referee: [Results and Discussion] The central claim that the critical coupling converges to the continuum value as momentum resolution increases is load-bearing, yet the manuscript supplies no quantitative truncation-error estimates, no tabulation of the quartic-interaction matrix elements in the wavelet basis, and no cross-check against an independent regularization (e.g., position-space lattice at matched cutoff). Without these data it is impossible to confirm that residual UV/IR mixing or broken translation invariance under truncation does not shift the extracted critical point.

    Authors: We agree that quantitative truncation-error estimates and additional documentation of the interaction matrix elements would strengthen the presentation. In the revised manuscript we have added explicit estimates of the truncation error obtained from the variation of the extracted critical coupling between successive resolution levels. A table of representative quartic-interaction matrix elements (for the lowest few resolution and translation indices) has been included in a new appendix. A direct cross-check against a position-space lattice regularization at matched cutoff is not performed in the present work, as it would require an independent code base and lies outside the scope of establishing the wavelet method; such a comparison is planned for a follow-up study. The exact reproduction of the free-field spectrum together with the observed systematic convergence of the critical coupling already constrain possible UV/IR mixing or translation-invariance artifacts, and we have added a short discussion of these issues in the revised text. revision: yes

  2. Referee: [Formalism and Numerical Implementation] The description of the momentum-space Daubechies wavelet implementation does not include explicit formulas for the overlap integrals or the truncated Hamiltonian matrix in the interacting case. In particular, it is unclear how the quartic term, which mixes different resolution and translation sectors, is evaluated once the basis is truncated, leaving open the possibility of basis-specific artifacts that would not be diagnosed by the free-field spectrum alone.

    Authors: We appreciate the request for greater technical detail. The revised manuscript now contains explicit expressions for the overlap integrals between momentum-space Daubechies wavelet basis functions. We also provide the explicit form of the truncated Hamiltonian matrix elements for the quartic interaction, showing how the sums are restricted to the retained resolution and translation indices and how the required four-wavelet integrals are evaluated (analytically where possible and by numerical quadrature otherwise). These additions make the implementation fully reproducible and allow direct inspection of any potential basis-specific effects beyond the free-field check. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central result benchmarks against external literature value

full rationale

The paper constructs a momentum-space Daubechies wavelet basis to truncate the Hamiltonian for the free scalar and interacting φ⁴ theories, then numerically extracts the critical coupling from the computed spectra. This extracted value is reported to converge toward an independently established literature result as resolution increases, rather than being fitted internally or defined in terms of itself. No load-bearing step reduces by construction to a self-citation, ansatz smuggled via prior work, or renaming of a known pattern; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to claims stated there. The central result rests on the assumption that the wavelet basis truncation is faithful for the interacting theory.

free parameters (1)
  • momentum resolution level
    The paper varies resolution to demonstrate convergence but does not specify how the maximum resolution is chosen or whether it is fitted.
axioms (1)
  • domain assumption Daubechies wavelets in momentum space form a complete basis that permits simultaneous IR and UV truncation without significant artifacts for the phi^4 interaction.
    Invoked when the authors state that the basis provides a natural nonperturbative truncation.

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