Conformal structure of FLRW Cosmology: Spinorial representation and the so(3,2) algebra of observables

TL;DR

This paper reveals a larger conformal algebra structure in FLRW cosmology, enabling a unified quantum description of gravity and matter, and potentially facilitating group quantization of inflationary models.

Contribution

It extends the known $sl(2,R)$ conformal symmetry to an $so(3,2)$ algebra, unifying all observables and allowing for inclusion of scalar potentials in quantum cosmology.

Findings

Identifies an $so(3,2)$ algebra of observables in FLRW cosmology.

Provides a framework for quantizing the entire gravity-matter system as a single representation.

Opens possibilities for group quantization of inflationary cosmological backgrounds.

Abstract

It was recently shown that the homogeneous and isotropic cosmology of a massless scalar field coupled to general relativity exhibits a new hidden conformal invariance under Mobius transformation of the proper time, additionally to the invariance under time-reparamterization. The resulting Noether charges form a $sl(2,\mathbb{R})$ Lie algebra, which encapsulates the whole kinematics and dynamics of the geometry. This allows to map FLRW cosmology onto conformal mechanics and formulate quantum cosmology in $\text{CFT}_1$ terms. Here, we show that this conformal structure is embedded in a larger $so(3,2)$ algebra of observables, which allows to present all the Dirac observables for the whole gravity plus matter sectors in a unified picture. Not only this allows one to quantize the system and its whole algebra of observables as a single irreducible representation of $so(3,2)$, but this also gives access to a scalar field operator $\hat{\phi}$ opening the door to the inclusion of non-trivial potentials for the scalar field. As such, this extended conformal structure might allow to perform a group quantization of inflationary cosmological backgrounds.