# Protected $SL(2,\mathbb{R})$ Symmetry in Quantum Cosmology

## TL;DR

This paper introduces a new polymer quantization scheme for quantum cosmology based on an underlying SL(2,R) symmetry, offering a conformally invariant approach that addresses ambiguities and connects with conformal quantum mechanics.

## Contribution

It proposes a novel SL(2,R)-based regularization and quantization method for homogeneous isotropic quantum cosmology, preserving conformal symmetry and providing an alternative to traditional loop quantum cosmology.

## Key findings

- Developed a new regularization scheme respecting SL(2,R) symmetry.
- Constructed a polymer quantization with SL(2,R) invariance.
- Established connections with conformal quantum mechanics and matrix models.

## Abstract

The polymer quantization of cosmological backgrounds provides an alternative path to the original Wheeler-de Witt (WdW) quantum cosmology, based on a different representation the commutation relations of the canonical variables. This polymer representation allows to capture the lattice like structure of the quantum geometry and leads to a radically different quantum cosmology compared to the WdW construction. This new quantization scheme has attracted considerable attention due to the singularity resolution it allows in a wide class of symmetry reduced gravitational systems, in particular where the WdW scheme fails. However, as any canonical quantization scheme, ambiguities in the construction of the quantum theory, being regularization or factor-ordering ones, can drastically modify the resulting quantum dynamics. In this work, we propose a new criteria to restrict the quantization ambiguities in the simplest model of polymer quantum cosmology, for homogeneous and isotropic General Relativity minimally coupled to a massless scalar field. This new criteria is based on an underlying $\mathfrak{sl}(2,\mathbb{R})$ structure present in the phase space of this simple cosmological model. By preserving the symmetry of this cosmological system under this 1d conformal group, we derive a new regularization of the phase space. We perform both its polymer quantization and a quantization scheme directly providing a representation of the SL$(2,\mathbb{R})$ group action. The resulting quantum cosmology can be viewed as a lattice-like quantum mechanics with an SL$(2,\mathbb{R})$ invariance. This provides a new version of Loop Quantum Cosmology consistent with the conformal symmetry. This alternative construction opens new directions, among which a possible mapping with the conformal quantum mechanics as well as with recent matrix or tensor models constructions for quantum cosmological space-time.