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arXiv · 2112.05562

On the variational method for Euclidean quantum fields in infinite volume

Nikolay Barashkov·Massimiliano Gubinelli
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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 $φ^4_2$ 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 $P (φ)_2$ theories. We consider also the $\exp (βφ)_2$ model for $β^2 < 8 π$ (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 $P (φ)_2$ theories is lacking due to the difficulty of studying the stability of the wFBSDE against local perturbations.

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Nikolay Barashkov, Massimiliano Gubinelli. 2022-11-21. On the variational method for Euclidean quantum fields in infinite volume. https://doi.org/10.2140/pmp.2023.4.761

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