Hybrid quantization: From Bianchi I to the Gowdy model

Abstract

We complete the quantization of the vacuum Bianchi I model within the framework of loop quantum cosmology adopting a new improved dynamics scheme put forward recently. In addition, we revisit the hybrid quantization of the Gowdy T 3 cosmologies with linear polarization using that scheme, proving with rigor some steps that remained unconcluded. The family of Gowdy T 3 cosmologies is an inhomogeneous model whose subset of homogeneous solutions is given precisely by the vacuum Bianchi I model. Our hybrid approach combines the new loop quantum cosmology description of this homogeneous sector with a Fock quantization of the inhomogeneities. Both in the Bianchi I model and in the Gowdy model the Hamiltonian constraint provides an evolution equation with respect to the volume of the Bianchi I universe, which is a discrete variable with a strictly positive minimum. We show that, in vacuo, this evolution is well defined inasmuch as the associated initial value problem is well posed: physical solutions are completely determined by the data on the initial section of constant Bianchi I volume. This fact allows us first to carry out to completion the quantization of the vacuum Bianchi I model which had not yet been achieved and then to confirm the feasibility of the hybrid procedure when the homogeneous sector is quantized with the new improved dynamics scheme.

Approximation methods in loop quantum cosmology: from Gowdy cosmologies to inhomogeneous models in Friedmann–Robertson–Walker geometries

“…The hybrid quantization of this model has already been carried out by the authors in Ref. [31], by extending to the case with matter the hybrid quantization of the model in vacuo [27][28][29][30]. In this section, we summarize the main steps of this quantization.…”

“…This is a midisuperspace with three-torus spatial topology that contains inhomogeneities (corresponding to gravitational waves) varying in a single direction [26]. The quantization of this model has been carried out adopting a hybrid approach, which combines techniques of LQC when representing the homogeneous sector of the model with a Fock quantization of the inhomogeneities [27][28][29][30].…”

Inhomogeneous loop quantum cosmology: Hybrid quantization of the Gowdy model

“…Actually, a similar structure appears also in the solutions to the Hamiltonian constraint for the Bianchi I model, inasmuch as the solution on the section v + 4 is determined in terms of the same kind of linear combinations. Furthermore, it has been recently shown [31] that, for all v > 0, the set of linear combinations…”

Loop Quantum Cosmology

“…Due to space restrictions, there are some topics that we shall not cover in this chapter, such as the path integral formulation and its relation with spin foams [6], spin foam cosmology [7], the Gowdy models [8][9][10][11][12], nor numerical issues [13]. We do not provide either a review of all the existing ideas to study LQC effects on cosmic perturbations.…”