Thiemann complexifier in classical and quantum FLRW cosmology

Abstract

In the context of Loop Quantum Gravity (LQG), we study the fate of Thiemann complexifier in homogeneous and isotropic FRW cosmology. The complexifier is the dilatation operator acting on the canonical phase space for gravity and generates the canonical transformations shifting the Barbero-Immirzi parameter. We focus on the closed algebra consisting in the complexifier, the 3d volume and the Hamiltonian constraint, which we call the CVH algebra. In standard cosmology, for gravity coupled to a scalar field, the CVH algebra is identified as a su(1, 1) Lie algebra, with the Hamiltonian as a null generator, the complexifier as a boost and the su(1, 1) Casimir given by the matter density. The loop gravity cosmology approach introduces a regularization length scale λ and regularizes the gravitational Hamiltonian in terms of SU (2) holonomies. We show that this regularization is compatible with the CVH algebra, if we suitably regularize the complexifier and inverse volume factor. The regularized complexifier generates a generalized version of the Barbero's canonical transformation which reduces to the classical one when λ → 0. This structure allows for the exact integration of the actions of the Hamiltonian constraints and the complexifier. This straightforwardly extends to the quantum level: the cosmological evolution is described in terms of SU(1, 1) coherent states and the regularized complexifier generates unitary transformations. The Barbero-Immirzi parameter is to be distinguished from the regularization scale λ, it can be rescaled unitarily and the Immirzi ambiguity ultimately disappears from the physical predictions of the theory. Finally, we show that the complexifier becomes the effective Hamiltonian when deparametrizing the dynamics using the scalar field as a clock, thus underlining the deep relation between cosmological evolution and scale transformations. Contents

Conformal structure of FLRW cosmology: spinorial representation and the $$ \mathfrak{so} $$ (2, 3) algebra of observables

“…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]. As explained in [4] and reviewed below, the inclusion of matter preserves the sl(2, R) structure and simply induces a shift in the Casimir.…”

“…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]. See also [2][3][4] for details.…”

“…Since the volume v and the Hamiltonian constraint H g are respectively of dimension +1 and −1 with respect to scale transformations, it turns out that, together with the dilatation generator C, they form a closed sl(2, R) Lie algebra: 13) which is referred to as the CVH algebra [2][3][4]. To set it in the standard sl(2, R) ∼ su(1, 1) basis,…”

“…Symmetries play a crucial role in theoretical physics, where they not only fix quantization ambiguities and constrain quantum correlators, but often entirely define the theory (up to dualities), especially in the context of quantum gravity. Previous works [1][2][3][4] highlighted a hidden conformal invariance of the simplest gravitational system, the Friedman-Lemaître-Robertson-Walker (FLRW) cosmology of homogeneous and isotropic gravity coupled to a massless free scalar field 1 . This conformal invariance of the cosmological action under Mobius transformation in proper time, is distinct from the time reparametrization invariance, which is the remnant of the diffeomorphism invariance of general relativity in this cosmological setting.…”

“…At the classical level, this new symmetry leads by Noether theorem to conserved charges, encoding the evolution in proper time of the Hamiltonian, the extrinsic curvature and the volume [1]. These charges form a sl(2, R) Lie algebra, called the CVH algebra, a structure which was initially discussed in [3,4]. This conformal invariance allows one to reformulate the phase space of the gravitational sector as a AdS 2 space carrying the natural action of the 1D conformal group SL(2, R) ∼ SU (1,1).…”