Hermiticity of the Dirac Hamiltonian in gravitational field

The hermiticity properties of the Dirac Hamiltonian are discussed, both on the quantum mechanical and on the quantum field level. In a first step, it is shown that the Hamiltonian generating the time evolution of the Dirac wave function in relativistic quantum mechanics is not hermitian with respect to the covariantly defined inner product. A hermitian Hamiltonian is then defined and is shown to be directly related to the canonical field energy. In a second step, we use a manifestly covariant form of canonical Hamiltonian field theory in curved spacetime and show that, for the Dirac field, the canonical field momentum does not coincide with the generators of spacetime translations. Moreover, 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