# Journal of Noncommutative Geometry

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**Volume 9, Issue 3, 2015, pp. 999–1040**

**DOI: 10.4171/JNCG/214**

Published online: 2015-10-29

Linear hyperbolic PDEs with noncommutative time

Gandalf Lechner^{[1]}and Rainer Verch

^{[2]}(1) Cardiff University, UK

(2) Universität Leipzig, Germany

Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form $(D+\lambda W)f=0$ are studied, where $D$ is a normal or prenormal hyperbolic differential operator on $\mathbb R^n$, $\lambda \in \mathbb C$ is a coupling constant, and $W$ is a regular integral operator with compactly supported kernel. In particular, $W$ can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small $|\lambda|$, the hyperbolic character of $D$ is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in $\lambda$, and the solution spaces are analyzed. It is shown that the acausal behavior of the solutions is well-controlled, but the Cauchy problem is ill-posed in general. Nonetheless, a scattering operator can be calculated which describes the effect of $W$ on the space of solutions to $Df=0$.

It is also described how these structures occur in the context of noncommutative Minkowski space, and how the results obtained here can be used for the analysis of classical and quantum field theories on such spaces.

*Keywords: *Linear hyperbolic PDEs, noncommutative time, noncommutative quantum field theory, quantization

Lechner Gandalf, Verch Rainer: Linear hyperbolic PDEs with noncommutative time. *J. Noncommut. Geom.* 9 (2015), 999-1040. doi: 10.4171/JNCG/214