Quantum Spaces are Modular

Understand

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.