On the variational method for Euclidean quantum fields in infinite volume

TL;DR

This paper develops a variational approach to analyze infinite volume limits of Euclidean quantum fields in two dimensions, establishing tightness, large deviation principles, and uniqueness results for several models.

Contribution

It extends the variational method to infinite volume quantum field theories, providing new results on tightness, large deviations, and measure uniqueness in 2D models.

Findings

Established tightness of $phi^4_2$ without cutoffs.

Proved a large deviation principle for infinite volume limits.

Demonstrated uniqueness of the infinite volume measure for the $eta^2<8\pi$ regime.

Abstract

We investigate the infinite volume limit of the variational description of Euclidean quantum fields introduced in a previous work. Focussing on two dimensional theories for simplicity, we prove in details how to use the variational approach to obtain tightness of $\varphi^4_2$ without cutoffs and a corresponding large deviation principle for any infinite volume limit. Any infinite volume measure is described via a forward--backwards stochastic differential equation in weak form (wFBSDE). Similar considerations apply to more general $P (\varphi)_2$ theories. We consider also the $\exp (\beta \varphi)_2$ model for $\beta^2 < 8 \pi$ (the so called full $L^1$ regime) and prove uniqueness of the infinite volume limit and a variational characterization of the unique infinite volume measure. The corresponding characterization for $P (\varphi)_2$ theories is lacking due to the difficulty of studying the stability of the wFBSDE against local perturbations.