Quantum Spaces are Modular

TL;DR

This paper proposes a quantum definition of Euclidean space as modular spaces derived from polarizations of the Heisenberg algebra, revealing a fundamental length scale and topologically distinct spectra, with implications for quantum geometry.

Contribution

It introduces modular spaces as a quantum notion of space, showing they naturally include a fundamental length scale and relate to string theory and generalized geometry.

Findings

Modular spaces contain a fundamental length scale.

They have topologically distinct spectra from Schrödinger space.

Classical space emerges as a thermodynamical limit of modular space.

Abstract

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.