On the variational method for Euclidean quantum fields in infinite volume
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 forwardbackwards 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 DOI:
 10.48550/arXiv.2112.05562
 arXiv:
 arXiv:2112.05562
 Bibcode:
 2021arXiv211205562B
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 60H30;
 81T08
 EPrint:
 39 pages, some corrections and remarks on the FBSDE formulation