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Mathematical Issues in Loop Quantum Cosmology

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New Trends in Mathematical Physics

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Abstract

Focusing on mathematical aspects, this article gives a review of loop quantum cosmology as an application of background independent quantization techniques to cosmological models. Mathematical issues arise at two different levels. First, the kinematical basis of loop quantum cosmology is derived as an induced representation of loop quantum gravity. The discrete spatial geometry exhibited by quantum gravity as a consequence of the loop quantization is then realized also in cosmological models. Dynamical equations formulated in such models are difference rather than differential equations, whose analysis provides the second class of mathematical applications. Suitable solutions display typical features in quantum regimes, where they can resolve classical space-time singularities, but should also approach semiclassical behavior in classical regimes. Such solutions can be found using generating function or continued fraction techniques. Semiclassical behavior and corrections to the classical one are derived using effective equations which approximate partial difference equations by ordinary differential equations.

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Authors and Affiliations

  1. Institute for Gravitational Physics and Geometry, The Pennsylvania State University, 104 Davey Lab, University Park, PA, 16802, USA

    Martin Bojowald

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  1. Martin Bojowald
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Correspondence to Martin Bojowald .

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  1. Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, RJ, Brazil

    Vladas Sidoravičius

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Bojowald, M. (2009). Mathematical Issues in Loop Quantum Cosmology. In: Sidoravičius, V. (eds) New Trends in Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2810-5_6

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