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      Alternative quantization of the Hamiltonian in isotropic loop quantum cosmology

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          Abstract

          Since there are quantization ambiguities in constructing the Hamiltonian constraint operator in isotropic loop quantum cosmology, it is crucial to check whether the key features of loop quantum cosmology, such as the quantum bounce and effective scenario, are robust against the ambiguities. In this paper, we consider a typical quantization ambiguity arising from the quantization of the field strength of the gravitational connection. An alternative Hamiltonian constraint operator is constructed, which is shown to have the correct classical limit by the semiclassical analysis. The effective Hamiltonian incorporating higher order quantum corrections is also obtained. In the spatially flat FRW model with a massless scalar field, the classical big bang is again replaced by a quantum bounce. Moreover, there are still great possibilities for the expanding universe to recollapse due to the quantum gravity effect. Thus, these key features are robust against this quantization ambiguity.

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          Quantum Nature of the Big Bang: Improved dynamics

          An improved Hamiltonian constraint operator is introduced in loop quantum cosmology. Quantum dynamics of the spatially flat, isotropic model with a massless scalar field is then studied in detail using analytical and numerical methods. The scalar field continues to serve as `emergent time', the big bang is again replaced by a quantum bounce, and quantum evolution remains deterministic across the deep Planck regime. However, while with the Hamiltonian constraint used so far in loop quantum cosmology the quantum bounce can occur even at low matter densities, with the new Hamiltonian constraint it occurs only at a Planck-scale density. Thus, the new quantum dynamics retains the attractive features of current evolutions in loop quantum cosmology but, at the same time, cures their main weakness.
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            Robustness of key features of loop quantum cosmology

            Loop quantum cosmology of the k=0 FRW model (with a massless scalar field) is shown to be exactly soluble if the scalar field is used as the internal time already in the classical Hamiltonian theory. Analytical methods are then used i) to show that the quantum bounce is generic; ii) to establish that the matter density has an absolute upper bound which, furthermore, equals the critical density that first emerged in numerical simulations and effective equations; iii) to bring out the precise sense in which the Wheeler DeWitt theory approximates loop quantum cosmology and the sense in which this approximation fails; and iv) to show that discreteness underlying LQC is fundamental. Finally, the model is compared to analogous discussions in the literature and it is pointed out that some of their expectations do not survive a more careful examination. An effort has been made to make the underlying structure transparent also to those who are not familiar with details of loop quantum gravity.
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              Corrections to the Friedmann Equations from LQG for a Universe with a Free Scalar Field

              In loop quantum cosmology the quantum dynamics is well understood. We approximate the full quantum dynamics in the infinite dimensional Hilbert space by projecting it on a finite dimensional submanifold thereof, spanned by suitably chosen semiclassical states. This submanifold is isomorphic with the classical phase space and the projected dynamical flow provides effective equations incorporating the leading quantum corrections to the classical equations of motion. Numerical work has been done using quantum states which are semiclassical at late times. These states follow the classical trajectory until the density is on the order of 1% of the Planck density then deviate strongly from the classical trajectory. The effective equations we obtain reproduce this behavior to surprising accuracy.
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                Author and article information

                Journal
                11 February 2009
                2009-04-28
                Article
                0902.1913
                4ad7049e-2e3f-4843-a075-676ee5c4fd27

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                7 pages, 2 figures; a few modifications, references added
                gr-qc

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