Some progress on the use of the variational method in quantum field theory

TL;DR

This paper introduces relativistic continuous matrix product states (RCMPS), a variational approach tailored for (1+1)-dimensional quantum field theories, demonstrating its effectiveness in approximating ground states and spectral data in strongly coupled models.

Contribution

The paper develops and applies RCMPS with Riemannian optimization to non-perturbatively study strongly coupled quantum field theories in (1+1) dimensions, including extensions to multi-field models.

Findings

Achieved competitive approximations for ground state energies in φ^4, Sine-Gordon, and Sinh-Gordon models.

Extended the RCMPS framework to models with multiple interacting fields.

Demonstrated the ability to evaluate non-local observables and extract spectral data such as particle masses.

Abstract

Strongly coupled quantum field theories in $(1+1)$ 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 $(1+1)$-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 $\phi^4$, 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.