Non-perturbative \lambda\Phi^4 in D=1+1: an example of the constructive quantum field theory approach in a schematic way

TL;DR

This paper schematically illustrates the constructive quantum field theory approach applied to the simple                 in a schematic way, highlighting conceptual insights and difficulties of the non-perturbative construction in a simplified setting.

Contribution

The paper provides an oversimplified schematic exposition of the constructive quantum field theory approach for   in D=1+1, emphasizing conceptual insights and non-perturbative difficulties.

Findings

Highlights the conceptual insights of the constructive approach.

Demonstrates the non-perturbative difficulties in a simplified  model.

Connects schematic exposition with perturbative language.

Abstract

During the '70, several relativistic quantum field theory models in $D=1+1$ and also in $D=2+1$ have been constructed in a non-perturbative way. That was done in the so-called {\it constructive quantum field theory} approach, whose main results have been obtained by a clever use of Euclidean functional methods. Although in the construction of a single model there are several technical steps, some of them involving long proofs, the constructive quantum field theory approach contains conceptual insights about relativistic quantum field theory that deserved to be known and which are accessible without entering in technical details. The purpose of this note is to illustrate such insights by providing an oversimplified schematic exposition of the simple case of $\lambda\Phi^4$ (with $m>0$) in $D=1+1$. Because of the absence of ultraviolet divergences in its perturbative version, this simple example -although does not capture all the difficulties in the constructive quantum field theory approach- allows to stress those difficulties inherent to the non-perturbative definition. We have made an effort in order to avoid several of the long technical intermediate steps without missing the main ideas and making contact with the usual language of the perturbative approach.