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The quantized free Dirac field is considered on Minkowski spacetime (ofgeneral dimension). The Dirac field is coupled to an external scalar potentialwhose support is finite in time and which acts by a Moyal-deformedmultiplication with respect to the spatial variables. The Moyal-deformedmultiplication corresponds to the product of the algebra of a Moyal planedescribed in the setting of spectral geometry. It will be explained how thisleads to an interpretation of the Dirac field as a quantum field theory onMoyal-deformed Minkowski spacetime (with commutative time) in a setting ofLorentzian spectral geometries of which some basic aspects will be sketched.The scattering transformation will be shown to be unitarily implementable inthe canonical vacuum representation of the Dirac field. Furthermore, it will beindicated how the functional derivatives of the ensuing unitary scatteringoperators with respect to the strength of the non-commutative potential induce,in the spirit of Bogoliubov's formula, quantum field operators (correspondingto observables) depending on the elements of the non-commutative algebra ofMoyal-Minkowski spacetime.

Dirac field on Moyal-Minkowski spacetime and non-commutative potential scattering