  The diagrammatic theory of strongly correlated systems includes two types of
selfconsistent perturbative analysis: Phi derivability, or conserving
approximations, and iterative parquet theory. Becker and Grosser [W. Becker and
D. Grosser, Nuov. Cim. A 10, 343 (1972)] first showed that crossing symmetry
and elastic unitarity (conservation) could not both be satisfied in any
approximation to the two-particle Bethe-Salpeter equation for the transition
matrix. Jackson and Smith [A. D. Jackson and R. A. Smith, Phys. Rev. A 36, 2517
(1987)] later proved in particular that, despite their close affinity, Phi
derivability and parquet are fundamentally irreconcilable. Parquet theory
computes the two-body scattering amplitude, respecting its crossing symmetry.
Phi derivability computes the nonequilibrium one-body dynamics, respecting
conservation in the two-body response. Parquet cannot safeguard conservation
and Phi derivability cannot guarantee crossing symmetry, yet both are physical
requirements. We investigate these ``failure modes'' within a generalized
Hamiltonian approach. The two methods' respective relation to the exact ground
state sheds light on their complementary shortcomings.
