Non-local perturbations of hyperbolic PDEs and QFT models on non-commutative spacetimes

Full text: In the context of formulating quantum field theories over non-commutative spacetimes, such as Moyal Minkowski space or locally non-commutative versions thereof, one is led to consider a class of partial differential equations of the form (D+W)f=0, where D is a normally hyperbolic differential operator or Dirac operator, and W is some non-local perturbation (non-commutative potential). Despite the non-locality, we construct advanced and retarded fundamental solutions for such PDEs, discuss the structure of their solutions, and compute the scattering behaviour. We also describe the quantization of these systems, and explain how observables of a non-commutative QFT model can be extracted from the formalism by means of a variant of Bogoliubov's formula. (author)