2015/04/05 by Rainer Verch, Verch, Rainer · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Mathematical Physics Problems #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.1504.01115
openalex publication_date 2015/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The behaviour of solutions to the partial differential equation (D + λW)fλ= 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 C0^∞ kernel operator which, in general, will be non-local in time, and λ 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 |λ| is small enough (and also for a larger set of λ values). Moreover, a scattering operator can be defined if the λ 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 C0^∞ 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.