Polymer quantum cosmology: Lifting quantization ambiguities using a $\mathrm{SL}\mathbf{(}2,\mathbb{R}\mathbf{)}$ conformal symmetry

In this paper, we stress that the simplest cosmological model consisting in a massless scalar field minimally coupled to homogeneous and isotropic gravity has an built-in $\mathrm{SL}(2,\mathbb{R})$ symmetry. Protecting this symmetry naturally provides an efficient way to constrain the quantization of this cosmological system whatever the quantization scheme and allows one, in particular, to fix the quantization ambiguities arising in the canonical quantization program. Applying this method to the loop quantization of the Friedmann-Lemaitre-Robertson-Walker cosmology leads to a new loop quantum cosmology model that preserves the $\mathrm{SL}(2,\mathbb{R})$ symmetry of the classical system. This new polymer regularization consistent with the conformal symmetry can be derived as a nonlinear canonical transformation of the classical Friedmann-Lemaitre-Robertson-Walker phase space, which maps the classical singular dynamics into a regular effective bouncing dynamics. This improved regularization preserves the scaling properties of the volume and Hamiltonian constraint. Three-dimensional scale transformations, generated by the dilatation operator, are realized as unitary transformations despite the minimal length scale hard coded in the theory. Finally, we point out that the resulting cosmological dynamics exhibits an interesting duality between short and long distances, reminiscent of the T-duality in string theory, with the near-singularity regime dual to the semiclassical regime at large volume. The technical details of the construction of this model are presented in a longer companion paper [J. B. Achour and E. R. Livine, arXiv:1904.06149].