Information geometry of Euclidean quantum fields

Résumé : Information geometry provides differential geometric concepts like a Riemannian metric, connections and covariant derivatives on spaces of probability distributions. We discuss here how these concepts apply to quantum field theories in the Euclidean domain which can also be seen as statistical field theories. The geometry has a dual affine structure corresponding to sources and field expectation values seen as coordinates. A key concept is a new generating functional, which is a functional generalization of the Kullback-Leibler divergence. From its functional derivatives one can obtain connected as well as one-particle irreducible correlation functions. It also encodes directly the geometric structure, i. e. the Fisher information metric and the two dual connections, and it determines asymptotic probabilities for field configurations through Sanov's theorem.