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A sketch of an approach towards Lorentzian spectral geometry (based on jointwork with Mario Paschke) is described, together with a general way to defineabstractly the quantized Dirac field on such Lorentzian spectral geometries.Moyal-Minkowski spacetime serves as an example. The scattering of the quantizedDirac field by a non-commutative (Moyal-deformed) action of an external scalarpotential is investigated. It is shown that differentiating the S-matrix withrespect to the strength of the scattering potential gives rise to quantum fieldoperators depending on elements of the non-commutative algebra entering thespectral geometry description of Moyal-Minkowski spacetime, in the spirit of"Bogoliubov's formula", in analogy to the situation found in external potentialscattering by a usual scalar potential.

Quantum Dirac Field on Moyal-Minkowski Spacetime - Illustrating Quantum Field Theory over Lorentzian Spectral Geometry