Hermitian Dirac Hamiltonian in time dependent gravitational field

It is shown by a straightforward argument 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 whenever the background metric is time dependent. An alternative, hermitian, Hamiltonian is found and is shown to be directly related to the canonical field Hamiltonian used in quantum field theory.