  Non-commutative quantum physics at the atom scale can arise from coarse
graining of a classical statistical ensemble at the Planck scale. Position and
momentum of an isolated particle are classical observables which remain
computable in terms of the coarse grained information. However, the commuting
classical product of position and momentum observables is no longer defined in
the coarse grained system, which is therefore described by incomplete
statistics. The microphysical classical statistical ensemble at the Planck
scale admits an alternative non-commuting product structure for position and
momentum observables which is compatible with the coarse graining. Measurement
correlations for isolated atoms are based on this non-commutative product
structure. We present an explicit example for these ideas. It also realizes the
discreteness of the spin observable within a microphysical classical
statistical ensemble.
