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We consider a general class of approximations which guarantees theconservation of particle number in many-body perturbation theory. To do this weextend the concept of $\Phi$-derivability for the self-energy $\Sigma$ to alarger class of diagrammatic terms in which only some of the Green's functionlines contain the fully dressed Green's function $G$. We call the correspondingapproximations for $\Sigma$ partially $\Phi$-derivable. A special subclass ofsuch approximations, which are gauge-invariant, is obtained by dressing loopsin the diagrammatic expansion of $\Phi$ consistently with $G$. Theseapproximations are number conserving but do not have to fulfill otherconservation laws, such as the conservation of energy and momentum. From ourformalism we can easily deduce if commonly used approximations will fulfill thecontinuity equation, which implies particle number conservation. We furthershow how the concept of partial $\Phi$-derivability plays an important role inthe derivation of a generalized sum rule for the particle number, which reducesto the Luttinger-Ward theorem in the case of a homogeneous electron gas, andthe Friedel sum rule in the case of the Anderson model. To do this we need toensure that the Green's function has certain complex analytic properties, whichcan be guaranteed if the spectral function is positive semi-definite.The latterproperty can be ensured for a subset of partially $\Phi$-derivableapproximations for the self-energy, namely those that can be constructed fromsquares of so-called half-diagrams. In case the analytic requirements are notfulfilled we highlight a number of subtle issues related to branch cuts, polestructure and multi-valuedness. We also show that various schemes of computingthe particle number are consistent for particle number conservingapproximations.

Partial self-consistency and analyticity in many-body perturbation theory: particle number conservation and a generalized sum rule