The noncommutative space of light-like worldlines

Abstract

The noncommutative space of light-like worldlines that is covariant under the light-like (or null-plane) κ-deformation of the (3+1) Poincaré group is fully constructed as the quantization of the corresponding Poisson homogeneous space of null geodesics. This new noncommutative space of geodesics is five-dimensional, and turns out to be defined as a quadratic algebra that can be mapped to a non-central extension of the direct sum of two Heisenberg-Weyl algebras whose noncommutative parameter is just the Planck scale parameter κ −1 . Moreover, it is shown that the usual time-like κ-deformation of the Poincaré group does not allow the construction of the Poisson homogeneous space of light-like worldlines. Therefore, the most natural choice in order to model the propagation of massless particles on a quantum Minkowski spacetime seems to be provided by the light-like κ-deformation.

Yang–Mills solutions on Minkowski space via non-compact coset spaces

“…Here again, the algebra g can be decomposed into h ⊕ m, with m generating the coset G/H , but this coset is not reductive as in the previous cases. Indeed, P 3 and P 4 are orthogonal to themselves with respect to the Cartan-Killing metric (2.4), so there is no orthogonal complement to h. Moreover, m also forms a subalgebra of g (see also [13]), spanned by {I 1 , I 2 , I 3 } = {P 1 , P 2 , K 3 }, with…”