  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.
