Linear hyperbolic PDEs with noncommutative time

  • Gandalf Lechner

    Cardiff University, UK
  • Rainer Verch

    Universität Leipzig, Germany


Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form are studied, where is a normal or prenormal hyperbolic differential operator on , is a coupling constant, and is a regular integral operator with compactly supported kernel. In particular, can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small , the hyperbolic character of is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in , 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 on the space of solutions to .

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.

Cite this article

Gandalf Lechner, Rainer Verch, Linear hyperbolic PDEs with noncommutative time. J. Noncommut. Geom. 9 (2015), no. 3, pp. 999–1040

DOI 10.4171/JNCG/214