  At present, our notion of space is a classical concept. Taking the point of
view that quantum theory is more fundamental than classical physics, and that
space should be given a purely quantum definition, we revisit the notion of
Euclidean space from the point of view of quantum mechanics. Since space
appears in physics in the form of labels on relativistic fields or Schrodinger
wave functionals, we propose to define Euclidean quantum space as a choice of
polarization for the Heisenberg algebra of quantum theory. We show, following
Mackey, that generically, such polarizations contain a fundamental length scale
and that contrary to what is implied by the Schrodinger polarization, they
possess topologically distinct spectra. These are the modular spaces. We show
that they naturally come equipped with additional geometrical structures
usually encountered in the context of string theory or generalized geometry.
Moreover, we show how modular space reconciles the presence of a fundamental
scale with translation and rotation invariance. We also discuss how the usual
classical notion of space comes out as a form of thermodynamical limit of
modular space while the Schrodinger space is a singular limit.
