Wave equations with non-commutative space and time

Abstract

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 $|\lambda|$ 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 which links the scattering transformations, which thereby arise from non-commutative potentials, 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.