Particle conservation in the single-particle Green's function

TL;DR

The paper demonstrates that the exact single-particle Green's function in quantum many-body theory does not conserve particle number due to basis incompleteness, challenging its interpretation and use in spectral calculations.

Contribution

It reveals fundamental limitations of the single-particle Green's function, showing it is not particle-number conserving nor $ ext{Φ}$-derivable, and questions its physical interpretation.

Findings

Exact $G$ does not conserve particle number.

$G$ is not $ ext{Φ}$-derivable in the Kadanoff-Baym sense.

$G$ is not suitable for particle spectra calculations.

Abstract

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.