  The quantum theory of the Maxwell free field in Coulomb gauge on the de
Sitter expanding universe is completed with the technical elements needed for
building a coherent quantum theory of redshift. Paying a special attention to
the conserved observables and defining the projection operator selecting the
detected momenta it is shown that the expectation values of the energies of the
emitted and detected photons comply with the Lema\^ itre rule of Hubble's law.
Moreover, the quantum corrections to the dispersions of the principal
observables and new uncertainty relations are derived.
