The relevant Hilbert space in which to diagonalize the quantum field Hamiltonian and get a partially discrete spectrum is the space in which the system under consideration is in its rest frame; hence the spatial momentum vanishes. Then the discrete eigenvalues of H are the bound state masses $m$ (times $c^2$).
Extending the Hilbert space to arbitrary Lorentz frames then amounts to allowing eigenstates with arbitrary spatial momentum, obtained by an appropriate Lorentz transform, and this changes the energy $E$ to satisfy the relation $E^2=m^2+p^2$ (if $c=1$ or $E^2=(mc^2)^2+(cp)^2$ in arbitrary units. Thus each mass becomes a mass shell. Arbitrary states are then given by a momentum integral over spatial momentum of a sum over bound state masses, including an integral over the mass for the continuum part of the mass spectrum.