Hyperspace fermions,M\"{o}bius transformations, Krein space, fermion doubling, dark matter

We develop an approach to classical and quantum mechanics where continuous time is extended by an infinitesimal parameter $T$ and equations of motion converted into difference equations. These equations are solved and the physical limit $T \rightarrow 0$ then taken. In principle this strategy should recover all standard solutions to the original continuous time differential equations. We find this is valid for bosonic variables whereas with fermions, additional solutions occur. For both bosons and fermions, the difference equations of motion can be related to M\"{o}bius transformations in projective geometry. Quantization via Schwinger's action principle recovers standard particle-antiparticle modes for bosons but in the case of fermions, Hilbert space has to be replaced by Krein space. We discuss possible links with the fermion doubling problem and with dark matter.