Thiemann complexifier in classical and quantum FLRW cosmology

In the context of loop quantum gravity (LQG), we study the fate of Thiemann complexifier in homogeneous and isotropic Friedmann-Lemaitre-Roberston-Walker (FLRW) 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 (for Complexier-Volume-Hamiltonian constraint algebra). In standard cosmology, for gravity coupled to a scalar field, the CVH algebra is identified as a $\mathfrak{su}(1,1)$ Lie algebra, with the Hamiltonian as a null generator, the complexifier as a boost and the $\mathfrak{su}(1,1)$ Casimir given by the matter density. The loop gravity cosmology approach introduces a regularization length scale $\ensuremath{\lambda}$ 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 $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. 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 $\ensuremath{\lambda}$, 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.

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