  We derive the exact form of the bosonized Hamiltonian for a many-body fermion
system in one spatial dimension with arbitrary dispersion relations, using the
droplet bosonization method. For a single-particle Hamiltonian polynomial in
the momentum, the bosonized Hamiltonian is a polynomial of one degree higher in
the bosonic "boundary" field and includes subleading lower-order and derivative
terms. This generalizes the known results for massless relativistic and
nonrelativistic fermions (quadratic and cubic bosonic Hamiltonians,
respectively). We also consider two-body interactions and demonstrate that they
lead to interesting collective behavior and phase transitions in the Fermi sea.
