Linear hyperbolic PDEs with noncommutative time

Lechner G, Verch R (2015)

Publication Type: Journal article

Publication year: 2015


Book Volume: 9

Pages Range: 999-1040

Journal Issue: 3

DOI: 10.4171/JNCG/214


Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form (D C λW)f = 0 are studied, where = is a normal or prenormal hyperbolic differential operator on Rn, λ ⊂ 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 λ, the hyperbolic character of = is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in λ, 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.

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Lechner, G., & Verch, R. (2015). Linear hyperbolic PDEs with noncommutative time. Journal of Noncommutative Geometry, 9(3), 999-1040.


Lechner, Gandalf, and Rainer Verch. "Linear hyperbolic PDEs with noncommutative time." Journal of Noncommutative Geometry 9.3 (2015): 999-1040.

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