  Different finite difference replacements for the derivative are analyzed in
the context of the Heisenberg commutation relation. The type of the finite
difference operator is shown to be tied to whether one can naturally consider
$P$ and $X$ to be self-adjoint and skew self-adjoint or whether they have to be
viewed as creation and annihilation operators. The first class, generalizing
the central difference scheme, is shown to give unitary equivalent
representations. For the second case we construct a large class of examples,
generalizing previously known difference operator realizations of $[P,X]=Id$.
