TY - JOUR

T1 - Mathematical structure of loop quantum cosmology

T2 - Homogeneous models

AU - Bojowald, Martin

N1 - Publisher Copyright:
© 2013, Institute of Mathematics. All rights reserved.

PY - 2013/1/1

Y1 - 2013/1/1

N2 - The mathematical structure of homogeneous loop quantum cosmology is ana-lyzed, starting with and taking into account the general classification of homoge-neous connections not restricted to be Abelian. As a first consequence, it is seen that the usual approach of quantizing Abelian models using spaces of functions on the Bohr compactification of the real line does not capture all properties of homo-geneous connections. A new, more general quantization is introduced which applies to non-Abelian models and, in the Abelian case, can be mapped by an isometric, but not unitary, algebra morphism onto common representations making use of the Bohr compactification. Physically, the Bohr compactification of spaces of Abelian connections leads to a degeneracy of edge lengths and representations of holonomies. Lifting this degeneracy, the new quantization gives rise to several dynamical prop-erties, including lattice refinement seen as a direct consequence of state-dependent regularizations of the Hamiltonian constraint of loop quantum gravity. The repre-sentation of basic operators — holonomies and fluxes — can be derived from the full theory specialized to lattices. With the new methods of this article, loop quan-tum cosmology comes closer to the full theory and is in a better position to produce reliable predictions when all quantum effects of the theory are taken into account.

AB - The mathematical structure of homogeneous loop quantum cosmology is ana-lyzed, starting with and taking into account the general classification of homoge-neous connections not restricted to be Abelian. As a first consequence, it is seen that the usual approach of quantizing Abelian models using spaces of functions on the Bohr compactification of the real line does not capture all properties of homo-geneous connections. A new, more general quantization is introduced which applies to non-Abelian models and, in the Abelian case, can be mapped by an isometric, but not unitary, algebra morphism onto common representations making use of the Bohr compactification. Physically, the Bohr compactification of spaces of Abelian connections leads to a degeneracy of edge lengths and representations of holonomies. Lifting this degeneracy, the new quantization gives rise to several dynamical prop-erties, including lattice refinement seen as a direct consequence of state-dependent regularizations of the Hamiltonian constraint of loop quantum gravity. The repre-sentation of basic operators — holonomies and fluxes — can be derived from the full theory specialized to lattices. With the new methods of this article, loop quan-tum cosmology comes closer to the full theory and is in a better position to produce reliable predictions when all quantum effects of the theory are taken into account.

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U2 - 10.3842/SIGMA.2013.082

DO - 10.3842/SIGMA.2013.082

M3 - Article

AN - SCOPUS:84891539343

SN - 1815-0659

VL - 9

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

M1 - 082

ER -