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Particle conservation in the single-particle Green's function

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 $Φ$-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 $Φ$-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.