  The quantum structure of Spacetime at the Planck scale suggests the use, in
defining interactions between fields, of the Quantum Wick product. The
resulting theory is ultraviolet finite, but subject to an adiabatic cutoff in
time which seems difficult to remove. We solve this problem here by another
strategy: the fields at a point in the interaction Lagrangian are replaced by
the fields at a quantum point, described by an optimally localized state on
QST; the resulting Lagrangian density agrees with the previous one after
spacetime integration, but gives rise to a different interaction hamiltonian.
But now the methods of perturbative Algebraic Quantum Field Theory can be
applied, and produce an ultraviolet finite perturbation expansion of the
interacting observables. If the obtained theory is tested in an equilibrium
state at finite temperature the adiabatic cutoff in time becomes immaterial,
namely it has no effect on the correlation function at any order in
perturbation theory. Moreover, the interacting vacuum state can be obtained in
the vanishing temperature limit. It is nevertheless important to stress that
the use of states which are optimally localized for a given observer brakes
Lorentz invariance at the very beginning.
