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 mathfrak{sl}left(2,mathbb{R}right) 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 CFT1 terms. Here, we show that this conformal structure is embedded in a larger mathfrak{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 mathfrak{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.
Highlights
Contrary to the CVH structure discussed in previous works, it allows one to account for the whole phase space of the FLRW cosmology sourced by a scalar field and of the gravitational sector at fixed scalar momentum
This technical paper extends the sl(2, R) formalism for FRW cosmology introduced in [2,3,4] and proven to reflect the conformal symmetry for the homogeneous sector of gravity coupled to a massless free scalar field [1]
The sl(2, R) Lie algebra describes the observables for the gravitational sector of the theory at fixed scalar field momentum
Summary
We start with a quick review of the FLRW cosmology of general relativity coupled to a homogeneous and isotropic massless free scalar field, focusing on the sl(2, R) framework and related conformal invariance introduced in the previous work [1]. While the dilatation C is a pure boost generator, the volume v and the Hamiltonian constraint Hg are null vectors in the Lie algebra sl(2, R) This isomorphism between the CVH algebra and the sl(2, R) Lie algebra leads to a vanishing sl(2, R)-Casimir: csl(2,R) = jz2 − kx2 − ky2 = −2vHg − C2 = 0. This implies that vacuum homogeneous and isotropic general relativity can be described at the quantum level by a null representation of sl(2, R), as advocated already in [4] and shown in details in [2]. It was understood in [1] that this sl(2, R) structure descends from a conformal invariance of the action for gravity plus scalar matter allowing to map it the cosmological system onto the conformal quantum mechanics introduced by de Alfaro, Fubini and Furlan [10]
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