  Canonical Hamiltonian field theory in curved spacetime is formulated in a
manifestly covariant way. Second quantization is achieved invoking a
correspondence principle between the Poisson bracket of classical fields and
the commutator of the corresponding quantum operators. The Dirac theory is
investigated and it is shown that, in contrast to the case of bosonic fields,
in curved spacetime, the field momentum does not coincide with the generators
of spacetime translations. The reason is traced back to the presence of second
class constraints occurring in Dirac theory. Further, it is shown that the
modification of the Dirac Lagrangian by a surface term leads to a momentum
transfer between the Dirac field and the gravitational background field,
resulting in a theory that is free of constraints, but not manifestly
hermitian.
