Aspects of non-perturbative approximation schemes for local quantum field theory

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 $φ^4$ , 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.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 $φ^4$, 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.

hep-thCross-listedarXiv:2604.10284Apr 14, 20260 citations

Nonperturbative Methods in Low Dimensional Quantum Field Theories

G. Domokos, Z. Horvȧth, Susan Kovesi-Domokos

Mar 1, 199110 citations

Equal-time approach to real-time dynamics of quantum fields

R.J. Ott, Torsten V. Zache, Jürgen Berges

We employ the equal-time formulation of quantum field theory to derive effective kinetic theories, first for a weakly coupled non-relativistic Bose gas, and then for a strongly correlated system of self-interacting N N -component fields. Our results provide the link between state-of-the-art measurements of equal-time effective actions using quantum simulator platforms, as employed in [Phys. Rev. X 10, 011020 (2020); Nat. Phys. 16, 1012 (2020)], and observables underlying effective kinetic or hydrodynamic descriptions. New non-perturbative approximation schemes can be developed and certified this way, where the a priori time-local formulation of the equal-time effective action has crucial advantages over the conventional closed-time-path approach which is non-local in time.We employ the equal-time formulation of quantum field theory to derive effective kinetic theories, first for a weakly coupled non-relativistic Bose gas, and then for a strongly correlated system of self-interacting N N -component fields. Our results provide the link between state-of-the-art measurements of equal-time effective actions using quantum simulator platforms, as employed in [Phys. Rev. X 10, 011020 (2020); Nat. Phys. 16, 1012 (2020)], and observables underlying effective kinetic or hydrodynamic descriptions. New non-perturbative approximation schemes can be developed and certified this way, where the a priori time-local formulation of the equal-time effective action has crucial advantages over the conventional closed-time-path approach which is non-local in time.

arXiv:2204.06463SciPost PhysicsFeb 1, 20232 citations

Problems in the non-local quantum field theory

Гарий Владимирович Ефимов

Jan 1, 19743 citations

The status and prospects of quantum non-local field theory

Neil J. Cornish

Unknown eBooksJan 1, 19910 citations

A systematic localized approximation scheme for the coupled-cluster treatment of quantum spin systems

R. F. Bishop, John B. Parkinson, Yang Xian

Within the framework of the coupled-luster method a systematic localized approximation scheme, the so-called LSUBn scheme, is developed and applied to anisotropic quantum spin chains. The authors use computer generation of the terms in the coupled non-linear equations to obtain successive approximations up to order n=8. The results for the ground-state energy show systematic and rapid convergence in the Ising-like region. The method also gives good results in the planar region where perturbation theory is not valid, and only breaks down for values of the anisotropy parameter Delta <or approximately=-0.5. They show that for a given value of n the LSUBn scheme reproduces exactly the corresponding nth order of large- Delta perturbation theory around the Ising limit.Within the framework of the coupled-luster method a systematic localized approximation scheme, the so-called LSUBn scheme, is developed and applied to anisotropic quantum spin chains. The authors use computer generation of the terms in the coupled non-linear equations to obtain successive approximations up to order n=8. The results for the ground-state energy show systematic and rapid convergence in the Ising-like region. The method also gives good results in the planar region where perturbation theory is not valid, and only breaks down for values of the anisotropy parameter Delta <or approximately=-0.5. They show that for a given value of n the LSUBn scheme reproduces exactly the corresponding nth order of large- Delta perturbation theory around the Ising limit.

Journal of Physics Condensed MatterJun 29, 199213 citations

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

φ^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.