Conservation, crossing symmetry, and completeness in diagrammatic theories

The diagrammatic analysis of interacting particle assemblies harbors a fundamental mismatch between two of its main implementations: $\mathrm{\ensuremath{\Phi}}$-derivable (conserving) approximations and parquet (crossing-symmetric) models. No termwise expansion, short of the exact theory itself, can be both conserving and crossing symmetric. This work applies the Kraichnan embedded-Hamiltonian formalism for strongly coupled systems to investigate consistency of the interplay between purely pair-mediated correlations and pair-irreducible ones. The approach sheds a different light on the issue of crossing symmetry versus conservation. In the process, the parquet equations acquire a different formulation.