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 42 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 ()2 theories. We consider also the (β )2 model for β2 < 8 π (the so called full L1 regime) and prove uniqueness of the infinite volume limit and a variational characterization of the unique infinite volume measure. The corresponding characterization for P ()2 theories is lacking due to the difficulty of studying the stability of the wFBSDE against local perturbations.
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