  We argue that the exact single-particle Green's function ($G$) in quantum
many-body theory does not conserve particle number because the single-particle
basis is incomplete. We conclude that the exact $G$ is not a probability
amplitude and is not $\Phi$-derivable in the Kadanoff-Baym sense. This sets up
a number of inconsistencies involving normalization, the definition of $G$,
interpretation of the spectral function, and $\Phi$-derivability. Our result
suggests that, in the most general case and in the most literal sense, $G$ is
not suitable for computing particle addition/removal spectra.
