Conformal structure of FLRW cosmology: spinorial representation and the $$ \mathfrak{so} $$ (2, 3) algebra of observables
Jibril Ben Achour, Etera R. Livine
Abstract
A bstract 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 $$ \mathfrak{sl}\left(2,\mathbb{R}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>sl</mml:mi> <mml:mfenced> <mml:mn>2</mml:mn> <mml:mi>ℝ</mml:mi> </mml:mfenced> </mml:math> 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 CFT 1 terms. Here, we show that this conformal structure is embedded in a larger $$ \mathfrak{so} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>so</mml:mi> </mml:math> (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 $$ \mathfrak{so} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>so</mml:mi> </mml:math> (3, 2), but this also gives access to a scalar field operator $$ \hat{\phi} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> 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.