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

Mrinmoy Basak, Debsubhra Chakraborty+2 more|arXiv (Cornell University)|Jan 26, 2026

Quantum, superfluid, helium dynamics被引用 0

一句话总结

该论文建立了动量空间 Daubechies 小波基哈密顿量框架，以非摊开方式研究1+1维 ϕ^4理论，计算能谱并定位对称性破缺的临界耦合。

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.

研究动机与目标

• Motivate a Hamiltonian, nonperturbative approach to ϕ^4 theory in 1+1 dimensions.
• Introduce a momentum-space Daubechies wavelet basis to implement systematic IR/UV truncation.
• Demonstrate energy spectrum computation for free and interacting theories within the wavelet framework.
• Show convergence of the critical coupling with increasing momentum resolution.

提出的方法

• Represent scalar fields and their momenta in a Daubechies wavelet basis labeled by resolution k and translation m.
• Construct a wavelet-based Fock space and express the Hamiltonian (free and φ^4 interaction) in this basis.
• Compute Hamiltonian matrix elements via overlap integrals of wavelet modes (Γ^k and E^k terms) and diagonalize the finite matrix.
• Implement truncation by limiting momentum and energy expectations, using periodic boundary conditions.
• Analyze convergence of energy eigenvalues with increasing resolution k and identify the phase transition at m^2>0.

Figure 1: The scaling functions for (a) different values of resolution and (b) different values of translation index are presented.

实验结果

研究问题

• RQ1 Can a momentum-space Daubechies wavelet basis provide a controlled nonperturbative truncation for ϕ^4 theory?
• RQ2 Do energy spectra and the critical coupling converge toward known values as momentum resolution increases?
• RQ3 How does the wavelet-based Hamiltonian reproduce the strong-coupling symmetry-breaking phase in 1+1 dimensions?

主要发现

lambda	E0(k=0)	E1(k=0)	E2(k=0)	E3(k=0)	E0(k=1)	E1(k=1)	E2(k=1)	E3(k=1)
0	0.00000	1.04351	1.41908	1.45087	0.00000	1.01149	1.12243	1.13259
10	-0.00058	1.04111	1.41720	1.44918	-0.00115	1.00844	1.11967	1.13001
20	-0.00231	1.03413	1.41143	1.44426	-0.00464	0.99923	1.11115	1.12239
30	-0.00522	1.02256	1.40120	1.43623	-0.01056	0.98323	1.09579	1.10984
40	-0.00940	1.00588	1.38506	1.42525	-0.01919	0.95863	1.07106	1.09226
50	-0.01500	0.98277	1.35983	1.41150	-0.03105	0.92062	1.03085	1.06864
60	-0.02231	0.95024	1.31787	1.39504	-0.04740	0.85115	0.95189	1.03579
70	-0.03186	0.90016	1.23400	1.37542	-0.07340	0.58345	0.62606	0.70603
80	-0.04494	0.80139	1.01208	1.27131	-0.27996	-0.12752	-0.00315	0.03254
90	-0.06562	0.47094	0.63609	0.79445	-1.13289	-0.88706	-0.78133	-0.59739
100	-0.12726	-0.11981	0.17024	0.24067	-2.03649	-1.74348	-1.60327	-1.38075
110	-0.78552	-0.46776	-0.40604	-0.25310	-2.95791	-2.62094	-2.44741	-2.18812
120	-1.46849	-1.08937	-1.02572	-0.80018	-3.89041	-3.51085	-3.30540	-3.00996
130	-2.16576	-1.74832	-1.66658	-1.37816	-4.83051	-4.40916	-4.17293	-3.84155
140	-2.87248	-2.42234	-2.32014	-1.96964	-5.77612	-5.31350	-5.04738	-4.68012
150	-3.58589	-3.10588	-2.98226	-2.56988	-6.72585	-6.22233	-5.92707	-5.52393

• The wavelet-based Hamiltonian reproduces the strong-coupling symmetry-breaking phase in the m^2>0 regime.
• Extracted critical coupling λ_c/24 trends toward established values as resolution increases (λ_c≈100 at k=0 and λ_c≈79 at k=1; g_c around 4.17 and 3.29 respectively).
• Energy eigenvalues converge toward exact values with higher momentum resolution in the free case (Table 2).
• Ground-state energy shows negative divergence with increasing λ, interpreted as a normal-ordered vacuum reference effect (Table 3 notes).
• The framework demonstrates finite-range inter-mode hopping due to Daubechies support, preserving locality while compressing degrees of freedom.

Figure 2: The wavelet functions for (a) different values of resolution and (b) different values of translation index are presented.