  We re-examine the way in which Bogoliubov's theory of a dilute Bose gas at
$T=0$ has been extended to describe the statistical mechanics of interacting
bosons at finite temperature. We show explicitly that the field-theoretic
calculation of the grand partition function in this formulation amounts to a
canonical trace over the eigenfunctions of the Bogoliubov Hamiltonian at fixed
total number of bosons $N$, and that the additional trace over $N$ that is
required in the grand-canonical formalism is never carried out. This implies
that what usually passes as the grand-canonical treatment of the Bogoliubov
Hamiltonian is not quite grand-canonical, and is in fact a canonical one. We
also show that the discontinuity in the condensate density predicted by
previous formulations of this theory as the temperature $T$ goes past the
critical transition temperature $T_c$ is a direct consequence of an
inappropriate generalization of the Bogoliubov prescription to finite
temperatures, and that this discontinuity disappears when this prescription is
either used as a zero temperature approximation or avoided altogether. Armed
with the above findings, we reformulate the statistical mechanics of
interacting bosons in the canonical ensemble and derive the thermodynamics of
the system. We then show how the canonical treatment can be used to setup a
truly grand-canonical description of the statistical mechanics of a weakly
interacting Bose gas where the average number of bosons in the system varies
with temperature, unlike existing formulations where the total number of bosons
$N$ is taken to be a constant that does not depend on $T$. Consequences on the
physics of interacting bosons are briefly discussed.
