Probing Probability Geometry with Schwinger--Dyson Identities: Score Mismatch, Fisher Information, and Configurational Temperature

Abstract

We develop a geometric interpretation of Schwinger--Dyson identities by showing that their violations are controlled by a single score-mismatch field $\hat{I}´s$. For an arbitrary sampled probability distribution $Q$ and equilibrium measure $P_{\mathrm{eq}}$, every Schwinger--Dyson violation is determined by $\hat{I}´s=\nabla \log (Q/P_{\mathrm{eq}})$, which characterizes the departure from equilibrium. Each Schwinger--Dyson identity measures a projection of this field onto a probe direction in configuration space. The relative Fisher information is its squared norm. This gives a universal bound relating Fisher information to the complete Schwinger--Dyson hierarchy, thus implying that convergence in Fisher information restores all Schwinger--Dyson identities. We further obtain a variational characterization of the relative Fisher information in terms of Schwinger--Dyson violations, leading to a natural tomographic interpretation in which increasingly rich families of probe fields encode progressively more information about the underlying probability distortion. The configurational temperature, within this framework, emerges as a distinguished Schwinger--Dyson probe. The Stein operators and score-function methods arise naturally from the same probability-geometric structure. The score-mismatch field, therefore, provides a unified geometric language for understanding Schwinger--Dyson identities, configurational temperature, Fisher information, and non-equilibrium sampling in stochastic processes.

34 pages