Wave equations with non-commutative space and time

TL;DR

This paper studies wave equations on non-commutative spacetimes, establishing conditions for unique Green's operators, defining scattering operators, and linking these to quantum field observables in non-commutative geometry.

Contribution

It introduces a framework for analyzing wave equations with non-commutative potentials, including existence of Green's and scattering operators, and connects these to quantum field observables.

Findings

Unique Green's operators exist for small enough coupling constants.

Scattering operators can be defined for certain non-commutative potentials.

The approach links non-commutative potentials to quantum field observables via Bogoliubov's formula.

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.