Functional information geometry of Euclidean quantum fields

Abstract

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 extend in a functional sense 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 source fields or expectation value fields seen as coordinates. In the latter version the coordinates label the macrostates of the classical field theory. 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 functional Fisher information metric and the two dual connections, and it determines asymptotic probabilities for field configurations through Sanov’s theorem. Based on the two dual connections one can construct covariant functional derivatives which allow one to calculate connected and one-particle irreducible correlation functions in general functional coordinate systems. Published by the American Physical Society 2024