A Brief Introduction to Loop Quantum Cosmology

In recent years, Loop Quantum Gravity has emerged as a solid candidate for a nonperturbative quantum theory of General Relativity. It is a background independent theory based on a description of the gravitational field in terms of holonomies and fluxes. In order to discuss its physical implications, a lot of attention has been paid to the application of the quantization techniques of Loop Quantum Gravity to symmetry reduced models with cosmological solutions, a line of research that has been called Loop Quantum Cosmology. We summarize its fundamentals and the main differences with respect to the more conventional quantization approaches employed in cosmology until now. In addition, we comment on the most important results that have been obtained in Loop Quantum Cosmology by analyzing simple homogeneous and isotropic models. These results include the resolution of the classical big-bang singularity, which is replaced by a quantum bounce.Comment: 15 pages, published in AIP Conference Proceedings, Volume 1130, Geometry and Physics: XVII International Fall Workshop on Geometry and Physic

Involutions on the Algebra of Physical Observables From Reality Conditions

Some aspects of the algebraic quantization programme proposed by Ashtekar are revisited in this article. It is proved that, for systems with first-class constraints, the involution introduced on the algebra of quantum operators via reality conditions can never be projected unambiguously to the algebra of physical observables, ie, of quantum observables modulo constraints. It is nevertheless shown that, under sufficiently general assumptions, one can still induce an involution on the algebra of physical observables from reality conditions, though the involution obtained depends on the choice of particular representatives for the equivalence classes of quantum observables and this implies an additional ambiguity in the quantization procedure suggested by Ashtekar.Comment: 19 pages, latex, no figure

CANONICAL QUANTIZATION OF THE BELINSKII-ZAKHAROV ONE-SOLITON SOLUTIONS

We apply the algebraic quantization programme proposed by Ashtekar to the analysis of the Belinski\v{\i}-Zakharov classical spacetimes, obtained from the Kasner metrics by means of a generalized soliton transformation. When the solitonic parameters associated with this transformation are frozen, the resulting Belinski\v{\i}-Zakharov metrics provide the set of classical solutions to a gravitational minisuperspace model whose Einstein equations reduce to the dynamical equations generated by a homogeneous Hamiltonian constraint and to a couple of second-class constraints. The reduced phase space of such a model has the symplectic structure of the cotangent bundle over $I\!\!\!\,R^+\times I\!\!\!\,R^+$. In this reduced phase space, we find a complete set of real observables which form a Lie algebra under Poisson brackets. The quantization of the gravitational model is then carried out by constructing an irreducible unitary representation of that algebra of observables. Finally, we show that the quantum theory obtained in this way is unitarily equivalent to that which describes the quantum dynamics of the Kasner model.Comment: 27 pages, latex, no figures

The Martin-Benito-Mena Marugan-Olmedo prescription for the Dapor-Liegener model of Loop Quantum Cosmology

Recently, an alternative Hamiltonian constraint for Loop Quantum Cosmology has been put forward by Dapor and Liegener, inspired by previous work on regularization due to Thiemann. Here, we quantize this Hamiltonian following a prescription for cosmology proposed by Mart\'{\i}n-Benito, Mena Marug\'an, and Olmedo. To this effect, we first regularize the Euclidean and Lorentzian parts of the Hamiltonian constraint separately in the case of a Bianchi I cosmology. This allows us to identify a natural symmetrization of the Hamiltonian which is apparent in anisotropic scenarios. Preserving this symmetrization in isotropic regimes, we then determine the Hamiltonian constraint corresponding to a Friedmann-Lema\^itre-Robertson-Walker cosmology, which we proceed to quantize. We compute the action of this Hamiltonian operator in the volume eigenbasis and show that it takes the form of a fourth-order difference equation, unlike in standard Loop Quantum Cosmology, where it is known to be of second order. We investigate the superselection sectors of our constraint operator, proving that they are semilattices supported only on either the positive or the negative semiaxis, depending on the triad orientation. Remarkably, the decoupling between semiaxes allows us to write a closed expression for the generalized eigenfunctions of the geometric part of the constraint. This expression is totally determined by the values at the two points of the semilattice that are closest to the origin, namely the two contributions with smallest eigenvolume. This is in clear contrast with the situation found for the standard Hamiltonian of Loop Quantum Cosmology, where only the smallest value is free. This result indicates that the degeneracy of the new geometric Hamiltonian operator is equal to two, doubling the possible number of solutions with respect to the conventional quantization considered until now.Comment: 15 pages, published in Physical Review