The Observer's Ghost: a field-space connection-form and its application to gauge theories and general relativity

TL;DR

This work develops a covariant, field-space perspective on gauge theories and gravity by introducing a functional connection and a horizontal (covariant) derivative δ_H. It shows that in bounded regions, gauge invariance and diffeomorphism covariance demand nontrivial corner terms and a nonzero functional curvature 𝔉, generalizing the Vilkovisky–DeWitt framework to the classical presymplectic setting. The formalism yields gauge-invariant symplectic data for Yang–Mills and General Relativity, clarifies the BRST interpretation in terms of a field-space connection, and situates the Gribov problem as a fundamental obstruction to global flatness. A physically meaningful picture emerges in which the field-space connection defines an abstract observer, guiding how to separate gauge from physical changes and how boundaries encode relational information. These ideas have potential implications for boundary charges, BRST, and nonperturbative treatments in contexts with finite regions and background independence, including holographic and FRG analyses.

Abstract

We introduce a functional covariant differential as a tool for studying field space geometry in a manifestly covariant way. We then touch upon its role in gauge theories and general relativity over bounded regions, and in BRST symmetry. Due to the Gribov problem, we argue that our formalism ---allowing for a non-vanishing functional curvature---is necessary for a global treatment of gauge-invariance in field space. We conclude by suggesting that the structures we introduce satisfactorily implement the notion of a (non-asymptotic) observer in gauge theories and general relativity.