  In this article, we revisit the question of the validity of Hartree-Fock and
random-phase approximations. We show that there is a connection between the two
and while the RPA as it is known in much of the physics literature is of
limited validity, there is a generalised sense in which the random phase
approximation is of much wider applicability including to systems that do not
possess Fermi surfaces. The main conclusion is that the Hartree-Fock
approximation is a mean-field idea applied to the density operator, and the
random-phase approximation is a mean-field idea applied to the number operator.
The generalised RPA is used to compute single-particle properties such as
momentum distribution and spectral functions. It is found that we have to go
beyond the generalised RPA and include fluctuations in the momentum
distribution in order to recover a nonzero imaginary part of the one-particle
self energy, which is also explicitly computed, all this in any number of
spatial dimensions and no bosonization is needed.
