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arXiv:2409.17887 · International Conference on Quarks and Nuclear Physics (QNP)

A Variational Approach to Quantum Field Theory

Martí Rovira, Robert J. Perry, A. Parreño

Published Feb 7, 2025• 0 citations
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Abstract

In strongly coupled field theories, perturbation theory cannot be employed to study the low-energy spectrum. Thus, non-perturbative techniques are required. We employ the variational method, a rigorous, non-perturbative approach which provides variational upper bounds on the energy eigenstates of the theory. An essential step in the variational method is the choice of trial wave function. In this work, we study the viability of employing a neural network as our variational ansatz. As a first step towards phenomenologically interesting strongly coupled theories like quantum chromodynamics, we study scalar field theories with quartic couplings.

DOI
10.22323/1.465.0036

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In strongly coupled field theories, perturbation theory cannot be employed to study the low-energy spectrum. Thus, non-perturbative techniques are required. We employ the variational method, a rigorous, non-perturbative approach which provides variational upper bounds on the energy eigenstates of the theory. An essential step in the variational method is the choice of trial wave function. In this work, we study the viability of employing a neural network as our variational ansatz. As a first step towards phenomenologically interesting strongly coupled theories like quantum chromodynamics, we study scalar field theories with quartic couplings.
Understanding strongly coupled quantum systems is one of the most profound challenges in modern theoretical physics, encompassing phenomena from quark confinement in Quantum Chromodynamics to emergent properties in strongly correlated materials. Traditional perturbative methods, reliant on small coupling constants, inherently fail in these regimes. This paper proposes a novel framework for "Variational-Asymptotic Unification," aiming to combine the non-perturbative strengths of variational principles with the analytical power of asymptotic expansions. Variational methods offer a direct route to approximating ground states and low-lying excitations, providing robust bounds and insights into the non-perturbative structure of a theory. Asymptotic expansions, on the other hand, offer systematic approximations in certain limits, such as large coupling, low energy, or large-N limits. We outline a methodology where variational ansatzes are constructed to incorporate leading asymptotic behaviors, and the variational parameters are subsequently determined by matching to known asymptotic limits or by extremizing an effective action derived from such an ansatz. This unified approach is posited to bridge the gap between perturbative and non-perturbative regimes, offering a more comprehensive and accurate description of strong-coupling physics. Potential applications range from calculating spectral properties of hadrons and glueballs in lattice gauge theories to elucidating phase transitions in condensed matter systems described by models like the Hubbard model. The paper discusses the theoretical foundations, outlines concrete methodological steps, and explores the advantages and challenges of this interdisciplinary approach, suggesting a pathway towards robust analytical tools for previously intractable problems.
We present a variational approach to quantum field theory based on wavefunctions parameterized by neural networks. While variational methods have a celebrated history across many fields, their application to quantum field theory has been limited by well-known challenges. We show that neural-network wavefunctions, combined with modern machine-learning techniques, enable competitive variational calculations in nontrivial field theories. As a demonstration, we reproduce the essential features of the two-dimensional nonlinear σσσ-model: asymptotic freedom, dynamical mass generation and the model's step-scaling function.
We develop a variational framework for addressing two-dimensional non-integrable quantum field theories through the exact structure of their integrable counterparts. Concentrating on the φ4\varphi^4φ4 Landau-Ginzburg model, we use the analytical Vacuum Expectation Values and Form Factors of local operators in the sinh-Gordon theory as the foundation of a variational ansatz. In this way, we obtain controlled estimates of central physical quantities of the φ4\varphi^4φ4 theory - such as the finite-volume ground-state energy and the physical mass as a function of the coupling constant. The strengths of the variational methods are leveraged in combination with the Hamiltonian truncation techniques and the LeClair-Mussardo formula, which also allow to probe the accuracy of the variational approximation varying the system size. Within the weak-coupling regime, a detailed numerical analysis reveals the behaviour of the finite-volume spectrum, the ground-state energy, and the elastic part of the scattering matrix, showing how the rigorous machinery of integrable models can serve as a guiding light into the complex landscape of non-integrable quantum field dynamics.
Strongly coupled quantum field theories in (1+1)(1+1)(1+1) dimensions are notoriously hard to solve non-perturbatively. Variational methods, despite their success for quantum many-body physics on the lattice, have long lacked a natural ansatz adapted to the relativistic setting. This monograph explains the intuition behind relativistic continuous matrix product states (RCMPS), a variational ansatz tailored to (1+1)(1+1)(1+1)-dimensional QFT, and reports on several years of progress in developing and applying this approach. Using Riemannian optimization on the manifold of RCMPS, we obtain competitive non-perturbative approximations to ground state energies and local observables in the φ4φ^4φ4, Sine-Gordon, and Sinh-Gordon models, including in strongly coupled regimes where perturbation theory fails. We then describe extensions to models with several interacting fields. Beyond energy density and local observables, we show how the framework can be used to evaluate non-local observables (defects) and, through an original linear programming approach, to extract spectral data such as particle masses. We close by discussing the current limitations of the method and the most promising directions for future work.
Physicists dating back to Feynman have lamented the difficulties of applying the variational principle to quantum field theories. In nonrelativistic quantum field theories, the challenge is to parametrize and optimize over the infinitely many n-particle wave functions comprising the state's Fock-space representation. Here we approach this problem by introducing neural-network quantum field states, a deep learning ansatz that enables application of the variational principle to nonrelativistic quantum field theories in the continuum. Our ansatz uses the Deep Sets neural network architecture to simultaneously parametrize all of the n-particle wave functions comprising a quantum field state. We employ our ansatz to approximate ground states of various field theories, including an inhomogeneous system and a system with long-range interactions, thus demonstrating a powerful new tool for probing quantum field theories.
This work describes a variational technique to solve non-Hermitian quantum systems, in particular those with negative coupling quartic self-interaction. Differences of applying variational methods from standard quantum mechanics to non-Hermitian quantum mechanics -- including potential pitfalls -- are highlighted. A successful implementation of the variational approach for the non-Hermitian case is presented. The technique is used to analyze scalar quantum field theory in d=2 with negative quartic self-interaction in the Hamiltonian formulation in lattice discretization.