  Novel controlled non-perturbative techniques are a must in the study of
strongly correlated systems, especially near quantum criticality. One of these
techniques, bosonization, has been extensively used to understand
one-dimensional, as well as higher dimensional electronic systems at finite
density. In this paper, we generalize the theory of two-dimensional
bosonization of Fermi liquids, in the presence of a homogeneous weak magnetic
field perpendicular to the plane. Here, we extend the formalism of bosonization
to treat free spinless fermions at finite density in a uniform magnetic field.
We show that particle-hole fluctuations of a Fermi surface satisfy a {\em
covariant Schwinger algebra}, allowing to express a fermionic theory with
forward scattering interactions as a quadratic bosonic theory representing the
quantum fluctuations of the Fermi surface. By means of a coherent-state path
integral formalism we compute the fermion propagator as well as particle-hole
bosonic correlations functions. We analyze the presence of de Haas-van Alphen
oscillations and show how the quantum oscillations of the orbital
magnetization, the Lifshitz-Kosevich theory, are obtained by means of the
bosonized theory. We also study the effects of forward scattering interactions.
In particular, we obtain oscillatory corrections to the Landau zero sound
collective mode.
