Convergent series for lattice models and QFTs

One of the key issues of modern theoretical physics is the description of interacting systems with a large number of degrees of freedom. The full information about such systems is naturally encoded in the infinite set of correlation functions, which can be formally expressed in terms of the path integral. The rigorous mathematical definition of the path integral is closely related to the foundation of quantum field theories and in most of the cases remains an open question. One of the standard approaches to the path integral is a perturbative expansion with an initial approximation given by the non-interacting part of the theory, describing the propagation of the free degrees of freedom. However, this form of perturbation theory leads to asymptotic series, with limited applicability. In the current project, we propose alternative perturbative expansions based on a non-standard initial approximation, given by a certain theory with interaction. These expansions allow one to construct convergent series, applicable in a wide range of the physical parameters. The proposal is aimed at developing new effective numerical schemes for lattice field theories and path integrals and also can be considered as an important step towards a `non-perturbative` definition and foundation of quantum field theories. In particular, the proposed project suggests a new `non-perturbative` approach for studying super-renormalizable quantum field theories and novel strategies for lattice computations in the infinite volume limit and for systems with a `sign problem`, which are not accessible by direct Monte Carlo simulations. The methods we plan to use in the current project range from lattice simulation techniques to the recent developments in mathematical physics and constructive field theory.

The central objects of studies in quantum field theory (QFT) and in all branches of science where the QFT methods can be applied, providing the description of the experimental data, are the correlation functions. They can be naturally expressed in terms of the path integral. However, the path integral itself gives only a formal solution and for the practical computations one has to find some efficient method of its evaluation. The standard approaches to the evaluation of the path integral include applications of the Monte Carlo methods and different perturbative expansions. All of these methods have limited applicability: Monte Carlo methods require the positive definition of the distribution density and perturbative expansions are in general divergent asymptotic series. In this project the alternative approaches to the path integral leading to the convergent expansions were studied. The convergent expansions were developed for a wide range of models including matrix models relevant to the description of the two dimensional quantum gravity, lattice models describing critical phenomena and tensor models applicable in the data analysis. These convergent expansions simultaneously give us new robust computational schemes and provide understanding of the analytic structures of QFT models.