Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models
The mathematical structure of homogeneous loop quantum cosmology is analyzed, starting with and taking into account the general classification of homogeneous connections not restricted to be Abelian. As a first consequence, it is seen that the usual approach of quantizing Abelian models using spaces...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2013 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2013
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149374 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models / M. Bojowald// Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 101 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862728841383903232 |
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| author | Bojowald, M. |
| author_facet | Bojowald, M. |
| citation_txt | Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models / M. Bojowald// Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 101 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The mathematical structure of homogeneous loop quantum cosmology is analyzed, starting with and taking into account the general classification of homogeneous 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 homogeneous 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 properties, including lattice refinement seen as a direct consequence of state-dependent regularizations of the Hamiltonian constraint of loop quantum gravity. The representation of basic operators - holonomies and fluxes - can be derived from the full theory specialized to lattices. With the new methods of this article, loop quantum 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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| first_indexed | 2025-12-07T19:11:13Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149374 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:11:13Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bojowald, M. 2019-02-21T07:28:04Z 2019-02-21T07:28:04Z 2013 Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models / M. Bojowald// Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 101 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R10; 39A14 DOI: http://dx.doi.org/10.3842/SIGMA.2013.082 https://nasplib.isofts.kiev.ua/handle/123456789/149374 The mathematical structure of homogeneous loop quantum cosmology is analyzed, starting with and taking into account the general classification of homogeneous 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 homogeneous 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 properties, including lattice refinement seen as a direct consequence of state-dependent regularizations of the Hamiltonian constraint of loop quantum gravity. The representation of basic operators - holonomies and fluxes - can be derived from the full theory specialized to lattices. With the new methods of this article, loop quantum 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. The author is grateful to the anonymous referees for several helpful comments and suggestions.
 This work was supported in part by NSF grants PHY-0748336 and PHY-1307408. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models Article published earlier |
| spellingShingle | Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models Bojowald, M. |
| title | Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models |
| title_full | Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models |
| title_fullStr | Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models |
| title_full_unstemmed | Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models |
| title_short | Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models |
| title_sort | mathematical structure of loop quantum cosmology: homogeneous models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149374 |
| work_keys_str_mv | AT bojowaldm mathematicalstructureofloopquantumcosmologyhomogeneousmodels |