Wave Equations with Non-commutative Space and Time

The behaviour of solutions to the partial differential equation $$(D +\lambda W)f_{\lambda } = 0$$ is discussed, where D is a normal hyperbolic partial differential operator, or pre-normal hyperbolic operator, on n-dimensional Minkowski spacetime. The potential term W is a $$C_{0}^{\infty }$$ kernel operator which, in general, will be non-local in time, and $$\lambda$$ is a complex parameter. A result is presented which states that there are unique advanced and retarded Green’s operators for this partial differential equation if $$\vert \lambda \vert$$ is small enough (and also for a larger set of $$\lambda$$ values). Moreover, a scattering operator can be defined if the $$\lambda$$ values admit advanced and retarded Green operators. In general, however, the Cauchy-problem will be ill-posed, and examples will be given to that effect. It will also be explained that potential terms arising from non-commutative products on function spaces can be approximated by $$C_{0}^{\infty }$$ kernel operators and that, thereby, scattering by a non-commutative potential can be investigated, also when the solution spaces are (2nd) quantized. Furthermore, a discussion will be given in which the scattering transformations arising from non-commutative potentials will be linked to observables of quantum fields on non-commutative spacetimes through “Bogoliubov’s formula”. In particular, this helps to shed light on the question how observables arise for quantum fields on Lorentzian spectral geometries.