  For linear scalar field theories, I characterize those classical Hamiltonian
vector fields which have self-adjoint operators as their quantum counterparts.
As an application, it is shown that for a scalar field in curved space-time (in
a Hadamard representation), a self-adjoint Hamiltonian for evolution along the
unit timelike normal to a Cauchy surface exists only if the second fundamental
form of the surface vanishes identically.
