  We carry out a model-theoretic analysis of the Heisenberg algebra. To this
end, a geometric structure is associated to the Heisenberg algebra and is shown
to be a Zariski geometry. Furthermore, this Zariski geometry is shown to be
non-classical, in the sense that it is not interpretable in an algebraically
closed field. On assuming self-adjointness of the position and momentum
operators, one obtains a discrete substructure of which the original Zariski
geometry is seen as the complexification.
