Javier Olmedo

Hybrid quantization of an inflationary universe

We quantize to completion an inflationary universe with small inhomogeneities in the framework of loop quantum cosmology. The homogeneous setting consists of a massive scalar field propagating in a closed, homogeneous scenario. We provide a complete quantum description of the system employing loop quantization techniques. After introducing small inhomogeneities as scalar perturbations, we identify the true physical degrees of freedom by means of a partial gauge fixing, removing all the local degrees of freedom except the matter perturbations. We finally combine a Fock description for the inhomogeneities with the polymeric quantization of the homogeneous background, providing the quantum Hamiltonian constraint of the composed system. Its solutions are then completely characterized, owing to the suitable choice of quantum constraint, and the physical Hilbert space is constructed. Finally, we consider the analog description for an alternate gauge and, moreover, in terms of gauge-invariant quantities. In the deparametrized model, all these descriptions are unitarily equivalent at the quantum level.

Further improvements in the understanding of isotropic loop quantum cosmology

Supported by the Spanish MICINN Project FIS2008-06078-C03-03 and the ConsoliderIngenio 2010 Program CPAN (CSD2007-00042). Financial aid by the I3P Program of CSIC and the European Social Fund under the grant I3PBPD2006. financial support under CSIC the grant JAEPre_08_00791.

Prescriptions in loop quantum cosmology: A comparative analysis

This work was supported in part by the MICINN under Project No. FIS2008-06078-C03-03 and the Consolider-Ingenio program CPAN, under Contract No. CSD2007-00042, from Spain, and by the Natural Sciences and Engineering Research Council of Canada. T. P. acknowledges also the hospitality of the Institute of Theoretical Physics of Warsaw University and the financial support of Minister Nauki i Szkolnictwa Wyzszego under Grant No. N N202 104838. J. O. acknowledges the support of CSIC under Grant No. JAE-Pre 0800791.

Quantum black holes in loop quantum gravity

We study the quantization of spherically symmetric vacuum spacetimes within loop quantum gravity. In particular, we give additional details about our previous work in which we showed that one could complete the quantization of the model and that the singularity inside black holes is resolved. Moreover, we consider an alternative quantization based on a slightly different kinematical Hilbert space. The ambiguity in kinematical spaces stems from how one treats the periodicity of one of the classical variables in these models. The corresponding physical Hilbert spaces solve the diffeomorphism and Hamiltonian constraint but their intrinsic structure is radically different depending on the kinematical Hilbert space one started from. In both cases there are quantum observables that do not have a classical counterpart. However, one can show that at the end of the day, by examining Dirac observables, both quantizations lead to the same physical predictions.

Cosmological perturbations in hybrid loop quantum cosmology: Mukhanov-Sasaki variables

We study cosmological perturbations in the framework of loop quantum cosmology, using a hybrid quantization approach and Mukhanov–Sasaki variables. The formulation in terms of these gauge invariants allows one to clarify the independence of the results on choices of gauge and facilitates the comparison with other approaches proposed to deal with cosmological perturbations in the context of loop quantum theory. A kind of Born–Oppenheimer ansatz is employed to extract the dynamics of the inhomogeneous perturbations, separating them from the degrees of freedom of the Friedmann–Robertson–Walker geometry. With this ansatz, we derive an approximate Schrodinger equation for the cosmological perturbations and study its range of validity. We also prove that, with an alternate factor ordering, the dynamics deduced for the perturbations is similar to the one found in the so-called dressed metric approach, apart from a possible scaling of the matter field in order to preserve its unitary evolution in the regime of quantum field theory in a curved background and some quantization prescription issues. Finally, we obtain the effective equations that are naturally associated with the Mukhanov–Sasakivariables, both with and without introducing the Born–Oppenheimer ansatz, and with the different factor orderings that we have studied.

Hybrid quantization of an inflationary model: The flat case

We present a complete quantization of an approximately homogeneous and isotropic universe with small scalar perturbations. We consider the case in which the matter content is a minimally coupled scalar field and the spatial sections are flat and compact, with the topology of a three-torus. The quantization is carried out along the lines that were put forward by the authors in a previous work for spherical topology. The action of the system is truncated at second order in perturbations. The local gauge freedom is fixed at the classical level, although different gauges are discussed and shown to lead to equivalent conclusions. Moreover, descriptions in terms of gauge-invariant quantities are considered. The reduced system is proven to admit a symplectic structure, and its dynamical evolution is dictated by a Hamiltonian constraint. Then, the background geometry is polymerically quantized, while a Fock representation is adopted for the inhomogeneities. The latter is selected by uniqueness criteria adapted from quantum field theory in curved spacetimes, which determine a specific scaling of the perturbations. In our hybrid quantization, we promote the Hamiltonian constraint to an operator on the kinematical Hilbert space. If the zero mode of the scalar field is interpreted as a relational time, a suitable ansatz for the dependence of the physical states on the polymeric degrees of freedom leads to a quantum wave equation for the evolution of the perturbations. Alternatively, the solutions to the quantum constraint can be characterized by their initial data on the minimum-volume section of each superselection sector. The physical implications of this model will be addressed in a future work, in order to check whether they are compatible with observations.

Unique Fock quantization of scalar cosmological perturbations

This work was supported by the research grants Nos. MICINN/MINECO FIS2011-30145-C03-02, MICINN FIS2008- 06078-C03-03, and CPAN CSD2007-00042 from Spain, and CERN/FP/116373/2010 from Portugal. J.O. acknowledges CSIC for financial support under the grant No. JAE-Pre 08 00791, and M.F.-M. acknowledges CSIC and the European Social Fund for support under the grant No. JAEPre 2010 01544.

A uniqueness criterion for the Fock quantization of scalar fields with time-dependent mass

A major problem in the quantization of fields in curved spacetimes is the ambiguity in the choice of a Fock representation for the canonical commutation relations. There exists infinite number of choices leading to different physical predictions. In stationary scenarios, a common strategy is to select a vacuum (or a family of unitarily equivalent vacua) by requiring invariance under the spacetime symmetries. When stationarity is lost, a natural generalization consists in replacing time invariance by unitarity in the evolution. We prove that when the spatial sections are compact, the criterion of a unitary dynamics, together with the invariance under the spatial isometries, suffices to select a unique family of Fock quantizations for a scalar field with time-dependent mass.

Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields with a time dependent mass in ultrastatic spacetimes

This work was supported by the research grants MICINN/MINECO FIS2011-30145-C03-02, MICINN FIS2008-06078-C03-03 and CPAN CSD2007-00042 from Spain, DGAPA-UNAM IN117012-3 from Mexico and CERN/FP/116373/2010 from Portugal. J.O. acknowledges CSIC by financial support under the grant JAE-Pre 08 00791.

Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes

is introduced as part of a linear, time dependent canonical transformation in phase space. In this context, we prove in full detail a uniqueness result about the Fock quantization requiring that the dynamics be unitary and the spatial symmetries of the field equations have a natural unitary implementation. The main conclusion is that, with those requirements, only one particular canonical transformation is allowed, and thus only one choice of the field-momentum pair (up to irrelevant constant scalings). This complements another previous uniqueness result for scalar fields with a time varying mass on S 3 , which selects a specific equivalence class of Fock representations of the canonical commutation relations under the conditions of a unitary evolution and the invariance of the vacuum under the background symmetries. In total, the combination of these two different statements of uniqueness picks up a unique Fock quantization for the system. We also extend our proof of uniqueness to other compact topologies and spacetime dimensions.