Quantum Geometry Effects Research Articles

Path integrals and the WKB approximation in loop quantum cosmology

We follow the Feynman procedure to obtain a path integral formulation of loop quantum cosmology starting from the Hilbert space framework. Quantum geometry effects modify the weight associated with each path so that the effective measure on the space of paths is different from that used in the Wheeler-DeWitt theory. These differences introduce some conceptual subtleties in arriving at the WKB approximation. But the approximation is well defined and provides intuition for the differences between loop quantum cosmology and the Wheeler-DeWitt theory from a path integral perspective.

Loop quantum cosmology of Bianchi type I models

The "improved dynamics" of loop quantum cosmology is extended to include anisotropies of the Bianchi I model. As in the isotropic case, a massless scalar field serves as a relational time parameter. However, the extension is non-trivial because one has to face several conceptual subtleties as well as technical difficulties. These include: a better understanding of the relation between loop quantum gravity (LQG) and loop quantum cosmology (LQC); handling novel features associated with the non-local field strength operator in presence of anisotropies; and finding dynamical variables that make the action of the Hamiltonian constraint manageable. Our analysis provides a conceptually complete description that overcomes limitations of earlier works. We again find that the big bang singularity is resolved by quantum geometry effects but, because of the presence of Weyl curvature, Planck scale physics is now much richer than in the isotropic case. Since the Bianchi I models play a key role in the Belinskii, Khalatnikov, Lifshitz (BKL) conjecture on the nature of generic space-like singularities in general relativity, the quantum dynamics of Bianchi I cosmologies is likely to provide considerable intuition about the fate of generic space-like singularities in quantum gravity. Finally, we show that the quantum dynamics of Bianchi I cosmologies projects down \emph{exactly} to that of the Friedmann model. This opens a new avenue to relate more complicated models to simpler ones, thereby providing a new tool to relate the quantum dynamics of LQG to that of LQC.

Consistent loop quantum cosmology

A consistent combination of quantum geometry effects rules out a large class of models of loop quantum cosmology and their critical densities as they have been used in the recent literature. In particular, the critical density at which an isotropic universe filled with a free, massless scalar field would bounce must be well below the Planck density. In the presence of anisotropy, no model of the Schwarzschild black hole interior analyzed so far is consistent.

Covariant entropy bound and loop quantum cosmology

We examine Bousso's covariant entropy bound conjecture in the context of radiation filled, spatially flat, Friedmann-Robertson-Walker models. The bound is violated near the big bang. However, the hope has been that quantum gravity effects would intervene and protect it. Loop quantum cosmology provides a near ideal setting for investigating this issue. For, on the one hand, quantum geometry effects resolve the singularity and, on the other hand, the wave function is sharply peaked at a quantum corrected but smooth geometry which can supply the structure needed to test the bound. We find that the bound is respected. We suggest that the bound need not be an essential ingredient for a quantum gravity theory but may emerge from it under suitable circumstances.

The dark side of a patchwork universe

While observational cosmology has recently progressed fast, it revealed a serious dilemma called dark energy: an unknown source of exotic energy with negative pressure driving a current accelerating phase of the universe. All attempts so far to find a convincing theoretical explanation have failed, so that one of the last hopes is the yet to be developed quantum theory of gravity. In this article, loop quantum gravity is considered as a candidate, with an emphasis on properties which might play a role for the dark energy problem. Its basic feature is the discrete structure of space, often associated with quantum theories of gravity on general grounds. This gives rise to well-defined matter Hamiltonian operators and thus sheds light on conceptual questions related to the cosmological constant problem. It also implies typical quantum geometry effects which, from a more phenomenological point of view, may result in dark energy. In particular the latter scenario allows several non-trivial tests which can be made more precise by detailed observations in combination with a quantitative study of numerical quantum gravity. If the speculative possibility of a loop quantum gravitational origin of dark energy turns out to be realized, a program as outlined here will help to hammer out our ideas for a quantum theory of gravity, and at the same time allow predictions for the distant future of our universe.

Quantum nature of the big bang: An analytical and numerical investigation

Analytical and numerical methods are developed to analyze the quantum nature of the big bang in the setting of loop quantum cosmology. They enable one to explore the effects of quantum geometry both on the gravitational and matter sectors and significantly extend the known results on the resolution of the big bang singularity. Specifically, the following results are established for the homogeneous isotropic model with a massless scalar field: (i) the scalar field is shown to serve as an internal clock, thereby providing a detailed realization of the 'emergent time' idea; (ii) the physical Hilbert space, Dirac observables, and semiclassical states are constructed rigorously; (iii) the Hamiltonian constraint is solved numerically to show that the big bang is replaced by a big bounce. Thanks to the nonperturbative, background independent methods, unlike in other approaches the quantum evolution is deterministic across the deep Planck regime. Our constructions also provide a conceptual framework and technical tools which can be used in more general models. In this sense, they provide foundations for analyzing physical issues associated with the Planck regime of loop quantum cosmology as a whole.

Quantum geometry and the Schwarzschild singularity

In homogeneous cosmologies, quantum geometry effects lead to a resolution of the classical singularity without having to invoke special boundary conditions at the singularity or introduce ad hoc elements such as unphysical matter. The same effects are shown to lead to a resolution of the Schwarzschild singularity. The resulting quantum extension of spacetime is likely to have significant implications for the black hole evaporation process. Similarities and differences with the situation in quantum geometrodynamics are pointed out.

Black hole evaporation: a paradigm

A paradigm describing black hole evaporation in non-perturbative quantum gravity is developed by combining two sets of detailed results: (i) resolution of the Schwarzschild singularity using quantum geometry methods and (ii) time evolution of black holes in the trapping and dynamical horizon frameworks. Quantum geometry effects introduce a major modification in the traditional spacetime diagram of black hole evaporation, providing a possible mechanism for recovery of information that is classically lost in the process of black hole formation. The paradigm is developed directly in the Lorentzian regime and necessary conditions for its viability are discussed. If these conditions are met, much of the tension between expectations based on spacetime geometry and structure of quantum theory would be resolved.

Observational limits on quantum geometry effects

Using a form of modified dispersion relations derived in thecontext of quantum geometry, we investigate limits set bycurrent observations on potential corrections to Lorentzinvariance. We use a phenomological model in which there areseparate parameters for photons, leptons and hadrons. Constraints on these parameters are derived using thresholds forthe processes of photon stability, photon absorption, vacuumCerenkov radiation, pion stability and the GZK cutoff. Although the allowed region in parameter space is tightlyconstrained, non-vanishing corrections to Lorentz symmetry dueto quantum geometry are consistent with current astrophysicalobservations.