  We study the Fock quantization of a compound classical system consisting of
point masses and a scalar field. We consider the Hamiltonian formulation of the
model by using the geometric constraint algorithm of Gotay, Nester and Hinds.
By relying on this Hamiltonian description, we characterize in a precise way
the real Hilbert space of classical solutions to the equations of motion and
use it to rigorously construct the Fock space of the system. We finally discuss
the structure of this space, in particular the impossibility of writing it in a
natural way as a tensor product of Hilbert spaces associated with the point
masses and the field, respectively.
